Collision Phenomena in Liquids and Solids 9781107147904

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Collision Phenomena in Liquids and Solids
 9781107147904

Table of contents :
Contents......Page 5
Preface......Page 12
1 Introduction......Page 16
1.1 History and Outlook......Page 17
1.2 Dimensionless Groups......Page 20
1.3 Mass and Momentum Balance Equations......Page 22
1.4 Inviscid and Viscous Newtonian Fluids: The Incompressible Euler and Navier–Stokes Equations......Page 24
1.5 Impact at Liquid Surface and Equations of Impulsive Motion......Page 27
1.6 Boundary Layer Equations......Page 28
1.7 Quasi-one-dimensional and Lubrication Approximations in Problems on Drop Impact and Spreading......Page 30
1.8 Wettability......Page 34
1.9 Rheological Constitutive Equations of Non-Newtonian Fluids and Solids......Page 37
1.10 Instabilities and Small Perturbations: Rayleigh Capillary Instability, Bending Instability, Kelvin–Helmholtz Instability, Rayleigh–Taylor Instability......Page 44
1.11 Total Mechanical Energy of Deforming Bodies: Where Is It Lost?......Page 52
1.12 References......Page 54
2.1 Inviscid Flow in a Thin Film on a Wall......Page 59
2.2 Propagation of Kinematic Discontinuity......Page 67
2.3 External Irrotational Flows About Blunt Bodies......Page 73
2.4 Flows Past Arbitrary Axisymmetric Bodies of Revolution......Page 76
2.5 Transient Motion in Inviscid Fluids and Forces Associated with the Added Masses......Page 78
2.6 Friction and Shape Drag......Page 85
2.7 Dynamics of a Rim Bounding a Free Liquid Sheet......Page 90
2.8 References......Page 97
Part I Collision of Liquid Jets and Drops with a Dry Solid Wall......Page 7
3.1 Normal and Inclined Impact of Inviscid Planar Jets onto a Plane Wall......Page 102
3.2 Normal Impact of Axisymmetric Impinging Jet......Page 106
3.3 Hydraulic Jump......Page 111
3.4 References......Page 113
4 Drop Impact onto a Dry Solid Wall......Page 115
4.1 Inviscid Flow on a Wall Generated by Inertia-Dominated Drop Impact......Page 117
4.2 Flow in a Spreading Viscous Drop, Including Description of Inclined Impact and Thermal Effects......Page 121
4.3 Initial Phase of Drop Impact......Page 135
4.4 Maximum Spreading Diameter......Page 138
4.5 Time Evolution of the Drop Diameter: Rim Dynamics on a Wall......Page 141
4.6 Drop Impact onto Spherical Targets and Encapsulation......Page 143
4.7 Outcomes of Drop Impact onto a Dry Wall......Page 145
4.8 The Effect of Reduced Pressure of the Surrounding Gas......Page 148
4.9 Drop Impact onto Hot Rigid Surfaces......Page 149
4.10 Drop Impact with Solidification and Icing......Page 155
4.11 References......Page 164
5 Drop Impact onto Dry Surfaces with Complex Morphology......Page 170
5.1 Drop Splashing on Rough and Textured Surfaces......Page 171
5.2 Drop Impact Close to a Pore......Page 174
5.3 Drop Impact onto Porous Surfaces......Page 180
5.4 Nano-textured Surfaces: Drop Impact onto Suspended Nanofiber Membranes......Page 192
5.5 Drop Impact onto Nanofiber Mats on Impermeable Substrates and Suppression of Splashing......Page 201
5.6 Hydrodynamic Focusing in Drop Impact onto Nanofiber Mats and Membranes......Page 204
5.7 Impact of Aqueous Suspension Drops onto Non-Wettable Porous Membranes: Hydrodynamic Focusing and Penetration of Nanoparticles......Page 215
5.8 Drop Impact onto Hot Surfaces Coated by Nanofiber Mats......Page 229
5.9 Nano-textured Surfaces: Suppression of the Leidenfrost Effect......Page 238
5.10 Bouncing Prevention: Dynamic Electrowetting......Page 246
5.11 References......Page 262
Part III Spray Formation and Impact onto Surfaces......Page 8
6.1 Drop Impact onto Thin Liquid Layer on a Wall: Weak Impacts and Self-similar Capillary Waves......Page 270
6.2 Strong Impacts of Drops onto Thin Liquid Layer: Crown Formation......Page 272
6.3 Drop Impact onto Thick Liquid Layers on a Wall: Cavity Expansion......Page 288
6.4 Residual Film Thickness......Page 298
6.5 Drop Impact onto a Deep Liquid Pool: Crater and Crown Formation, the Worthington Jets and Bubble Entrapment......Page 302
6.6 Bending Instability of a Free Viscous Rim on Top of the Crown: Mechanism of Splash......Page 308
6.7 Impact of Drop Train......Page 325
6.8 References......Page 330
7.1 Fundamentals......Page 338
7.2 Non-Optical Measurement Techniques......Page 344
7.3 Direct Imaging......Page 345
7.4 Non-Imaging Optical Measurement Techniques......Page 355
7.5 Measurement Techniques for Liquid Films......Page 362
7.6 References......Page 365
8 Atomization and Spray Formation......Page 369
8.1 Primary Atomization......Page 370
8.2 Secondary Aerodynamic Breakup......Page 381
8.3 Drop–Drop Binary Collisions in Sprays......Page 392
8.4 Secondary Drop Detachment from a Filament......Page 406
8.5 Secondary Electrically Driven Drop Breakup: The Rayleigh Limit......Page 416
8.6 References......Page 421
9 Spray Impact......Page 427
9.1 Spray Impact onto Liquid Films......Page 432
9.2 Description of the Secondary Spray......Page 455
9.3 Correlations for Spray Impact Phenomena......Page 477
9.4 References......Page 482
Part V Solid–Solid Collisions......Page 9
10.1 Impact of Rigid Body at Liquid Surface......Page 488
10.2 Rigid Body Entry and Penetration into Liquid: The Wagner Problem......Page 493
10.3 Rigid Sphere Entry and Penetration into Liquid......Page 497
10.4 References......Page 500
11.1 Motion of a Rigid Immersed Particle near a Wall......Page 502
11.2 Deformation of an Immersed Elastic Particle......Page 504
11.3 Restitution Coefficient......Page 506
11.4 Effect of Particle Material and Surface Properties......Page 508
11.5 References......Page 510
12.1 Relatively Weak and Strong Impacts, the Split Hopkinson Pressure Bar: Propagation of Elastic Waves in Long Rods – Inertial Effects and Anelastic Material Properties. Strong Impacts and Irreversible Plastic Effects......Page 514
12.2 Impingement of a Rigid/Semi-Brittle Ice Particle......Page 521
12.3 References......Page 528
13.1 Shaped-charge Jet Penetration Depth......Page 530
13.2 Crater Configuration due to Shaped-charge Jet Penetration......Page 532
13.3 Normal Penetration of an Eroding Projectile into an Elastic–Plastic Target......Page 536
13.4 High-Speed Penetration......Page 557
13.5 Quasi-Steady Penetration of an Eroding Projectile......Page 559
13.6 Normal and Oblique Penetration of a Rigid Projectile into an Elastic–Plastic Target......Page 560
13.7 Explosion Welding......Page 568
13.8 References......Page 579
14.1 Ice Particle Collision with a Dry Solid Wall......Page 581
14.2 Ice Particle: Fragmentation Threshold for an Impact Velocity......Page 585
14.3 Dynamic Fracture of a Deforming Elastic–Plastic Material......Page 588
14.4 Fragmentation of Thick Elastic–Plastic Targets......Page 592
14.5 Fragmentation of an Impacting Projectile......Page 602
14.6 Debris Cloud Produced by Projectile Impact, Vulnerability......Page 605
14.7 Effect of the Energy of the Plastic Dissipation on the Size of the Smallest Fragment......Page 613
14.8 References......Page 614
Index......Page 619

Citation preview

Collision Phenomena in Liquids and Solids A comprehensive account of the physical foundations of collision and impact phenomena and their applications in a multitude of engineering disciplines. In-depth explanations are included to reveal the unifying features of collision phenomena in both liquids and solids, and to apply them to disciplines including theoretical and applied mechanics, physics and applied mathematics, materials science, aerospace, mechanical and chemical engineering and terminal ballistics. Covering a range of examples from drops, jets and sprays, to seaplanes and ballistic projectiles, and detailing a variety of theoretical, numerical and experimental tools that can be used in developing new models and approaches, this is an ideal resource for students, researchers and practicing engineers alike. Alexander L. Yarin is a Distinguished Professor at the University of Illinois at Chicago, a Fellow of the American Physical Society and a recipient of the Rashi Foundation Fellowship of the Israel Academy of Sciences and Humanities in 1992–1995. His awards include the Gutwirth Award from the Technion-Israel Institute of Technology, the Hershel Rich Prize and the Prize for Technological Development for Defense against Terror from the American Technion Society. Ilia V. Roisman is based at the Institute of Fluid Mechanics at Technische Universität Darmstadt, where he leads the research group Dynamics of Drops and Sprays. He was the recipient of the STAB Research Prize for Fluid Mechanics in 2010. Cameron Tropea is Head of the Institute of Fluid Mechanics and Aerodynamics at Technische Universität Darmstadt. He is a member of the Scientific Commission of the Council of Science and Humanities in Germany and is Editor-in-Chief of Experiments in Fluids since 2002.

Collision Phenomena in Liquids and Solids ALEXANDER L. YARIN University of Illinois, Chicago, U.S.A.

ILIA V. ROISMAN Technische Universität, Darmstadt, Germany

CAMERON TROPEA Technische Universität, Darmstadt, Germany

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi - 110002, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107147904 DOI: 10.1017/9781316556580

© Alexander L. Yarin, Ilia V. Roisman, and Cameron Tropea 2017 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2017 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall. A catalog record for this publication is available from the British Library. ISBN 978-1-107-14790-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface 1

Introduction

1

1.1 1.2 1.3 1.4

2 5 7

History and Outlook Dimensionless Groups Mass and Momentum Balance Equations Inviscid and Viscous Newtonian Fluids: The Incompressible Euler and Navier–Stokes Equations 1.5 Impact at Liquid Surface and Equations of Impulsive Motion 1.6 Boundary Layer Equations 1.7 Quasi-one-dimensional and Lubrication Approximations in Problems on Drop Impact and Spreading 1.8 Wettability 1.9 Rheological Constitutive Equations of Non-Newtonian Fluids and Solids 1.10 Instabilities and Small Perturbations: Rayleigh Capillary Instability, Bending Instability, Kelvin–Helmholtz Instability, Rayleigh–Taylor Instability 1.11 Total Mechanical Energy of Deforming Bodies: Where Is It Lost? 1.12 References 2

page xi

9 12 13 15 19 22

29 37 39

Selected Basic Flows and Forces

44

2.1 2.2 2.3 2.4 2.5

44 52 58 61

2.6 2.7 2.8

Inviscid Flow in a Thin Film on a Wall Propagation of Kinematic Discontinuity External Irrotational Flows About Blunt Bodies Flows Past Arbitrary Axisymmetric Bodies of Revolution Transient Motion in Inviscid Fluids and Forces Associated with the Added Masses Friction and Shape Drag Dynamics of a Rim Bounding a Free Liquid Sheet References

63 70 75 82

vi

Contents

Part I Collision of Liquid Jets and Drops with a Dry Solid Wall 3

4

5

Jet Impact onto a Solid Wall

87

3.1 3.2 3.3 3.4

87 91 96 98

Normal and Inclined Impact of Inviscid Planar Jets onto a Plane Wall Normal Impact of Axisymmetric Impinging Jet Hydraulic Jump References

Drop Impact onto a Dry Solid Wall

100

4.1 4.2

102

Inviscid Flow on a Wall Generated by Inertia-Dominated Drop Impact Flow in a Spreading Viscous Drop, Including Description of Inclined Impact and Thermal Effects 4.3 Initial Phase of Drop Impact 4.4 Maximum Spreading Diameter 4.5 Time Evolution of the Drop Diameter: Rim Dynamics on a Wall 4.6 Drop Impact onto Spherical Targets and Encapsulation 4.7 Outcomes of Drop Impact onto a Dry Wall 4.8 The Effect of Reduced Pressure of the Surrounding Gas 4.9 Drop Impact onto Hot Rigid Surfaces 4.10 Drop Impact with Solidification and Icing 4.11 References

106 120 123 126 128 130 133 134 140 149

Drop Impact onto Dry Surfaces with Complex Morphology

155

5.1 5.2 5.3 5.4

156 159 165

Drop Splashing on Rough and Textured Surfaces Drop Impact Close to a Pore Drop Impact onto Porous Surfaces Nano-textured Surfaces: Drop Impact onto Suspended Nanofiber Membranes 5.5 Drop Impact onto Nanofiber Mats on Impermeable Substrates and Suppression of Splashing 5.6 Hydrodynamic Focusing in Drop Impact onto Nanofiber Mats and Membranes 5.7 Impact of Aqueous Suspension Drops onto Non-Wettable Porous Membranes: Hydrodynamic Focusing and Penetration of Nanoparticles 5.8 Drop Impact onto Hot Surfaces Coated by Nanofiber Mats 5.9 Nano-textured Surfaces: Suppression of the Leidenfrost Effect 5.10 Bouncing Prevention: Dynamic Electrowetting 5.11 References

177 186 189

200 214 223 231 247

Contents

vii

Part II Drop Impacts onto Liquid Surfaces 6

Drop Impacts with Liquid Pools and Layers 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Drop Impact onto Thin Liquid Layer on a Wall: Weak Impacts and Self-similar Capillary Waves Strong Impacts of Drops onto Thin Liquid Layer: Crown Formation Drop Impact onto Thick Liquid Layers on a Wall: Cavity Expansion Residual Film Thickness Drop Impact onto a Deep Liquid Pool: Crater and Crown Formation, the Worthington Jets and Bubble Entrapment Bending Instability of a Free Viscous Rim on Top of the Crown: Mechanism of Splash Impact of Drop Train References

255 255 257 273 283 287 293 310 315

Part III Spray Formation and Impact onto Surfaces 7

8

9

Drop and Spray Diagnostics

323

7.1 7.2 7.3 7.4 7.5 7.6

323 329 330 340 347 350

Fundamentals Non-Optical Measurement Techniques Direct Imaging Non-Imaging Optical Measurement Techniques Measurement Techniques for Liquid Films References

Atomization and Spray Formation

354

8.1 8.2 8.3 8.4 8.5 8.6

355 366 377 391 401 406

Primary Atomization Secondary Aerodynamic Breakup Drop–Drop Binary Collisions in Sprays Secondary Drop Detachment from a Filament Secondary Electrically Driven Drop Breakup: The Rayleigh Limit References

Spray Impact

412

9.1 9.2 9.3 9.4

417 440 462 467

Spray Impact onto Liquid Films Description of the Secondary Spray Correlations for Spray Impact Phenomena References

viii

Contents

Part IV Collisions of Solid Bodies with Liquid 10

11

Rigid Body Collision with Liquid Surface

473

10.1 10.2 10.3 10.4

473 478 482 485

Impact of Rigid Body at Liquid Surface Rigid Body Entry and Penetration into Liquid: The Wagner Problem Rigid Sphere Entry and Penetration into Liquid References

Particle Impact onto Wetted Wall

487

11.1 11.2 11.3 11.4 11.5

487 489 491 493 495

Motion of a Rigid Immersed Particle near a Wall Deformation of an Immersed Elastic Particle Restitution Coefficient Effect of Particle Material and Surface Properties References

Part V Solid–Solid Collisions 12

13

14

Particle and Long Bar Impact onto a Rigid Wall

499

12.1 Relatively Weak and Strong Impacts, the Split Hopkinson Pressure Bar: Propagation of Elastic Waves in Long Rods – Inertial Effects and Anelastic Material Properties. Strong Impacts and Irreversible Plastic Effects 12.2 Impingement of a Rigid/Semi-Brittle Ice Particle 12.3 References

499 506 513

Shaped-charge (Munroe) Jets and Projectile Penetration

515

13.1 Shaped-charge Jet Penetration Depth 13.2 Crater Configuration due to Shaped-charge Jet Penetration 13.3 Normal Penetration of an Eroding Projectile into an Elastic–Plastic Target 13.4 High-Speed Penetration 13.5 Quasi-Steady Penetration of an Eroding Projectile 13.6 Normal and Oblique Penetration of a Rigid Projectile into an Elastic–Plastic Target 13.7 Explosion Welding 13.8 References

515 517

545 553 564

Fragmentation

566

14.1 Ice Particle Collision with a Dry Solid Wall 14.2 Ice Particle: Fragmentation Threshold for an Impact Velocity 14.3 Dynamic Fracture of a Deforming Elastic–Plastic Material

566 570 573

521 542 544

Contents

14.4 14.5 14.6 14.7

ix

Fragmentation of Thick Elastic–Plastic Targets Fragmentation of an Impacting Projectile Debris Cloud Produced by Projectile Impact, Vulnerability Effect of the Energy of the Plastic Dissipation on the Size of the Smallest Fragment 14.8 References

577 587 590 598 599

Index

604

Preface

Collision phenomena can be ordinary like a rain drop impacting onto a window, a leaf or a puddle, or extraordinary such as a meteorite or a bolide collision with Earth. Some are frequently encountered in science and everyday life, others are extremely rare. Being very different at first sight, collision phenomena in liquids and solids share many underlying common features. The subject of the present book is highly cross-disciplinary with a very wide scope of applications in mind, and such a collection of topics in one book does not yet exist, as to our knowledge. One of the main motivations for providing such a collection of topics is to underline the commonality among the various occurrences of collision phenomena, which lead to similar physical and technological ideas and modeling approaches. An improved in-depth understanding of the phenomena can be expected after recognizing the common underlying physics involved. A second motivation is that the knowledge presently available on the subject is extremely widely scattered, mainly according to applications, and in a large number of different journals. For example, collisions in the solid mechanics context are considered as a totally different subject than impacts in the fluid mechanical context, whereas in reality inevitable geometric similarities dictate inevitable kinematic similarities, and in some cases similar rheological behavior, which greatly unifies these two fields to the extent still unrecognized by the majority of practitioners. This obscures the true state of the art, with the associated danger that research may be unintentionally and unnecessarily duplicated or some novel developments delayed. A further motivation can be found in the rapid progress made over the last decade in this field, partly attributed to the much improved means for visualization of collision phenomena with high-speed cameras. In this respect the proposed book is timely. There is sufficient material to justify a concise and coherent collection of recent advances, which may also help initiate further research in a complementary manner. Our personal research experience covers and spans practically all the topics covered in the book, which is a monograph significantly based on our own results published in peer-reviewed journals over the last 20 years. As the title suggests, we are dealing with collision phenomena in different combinations: individual jet or drop impacts onto solids, individual drop impacts onto liquid layers, spray-related impacts, solid–liquid collisions and solid–solid collisions. Accordingly, the book has been organized into five respective Parts I to V with the first additional introductory Chapter 1 devoted to the fundamental topics of hydrodynamics and

xii

Preface

solid mechanics, like the basic physical phenomena, the conservation equations, the governing dimensionless groups, rheological constitutive equations, asymptotical simplified approaches and a discussion of typical instabilities encountered in the field. While the introductory section on fundamentals does not present in-depth derivations, as these can be found in standard textbooks, the remaining sections do include derivations to a degree that a qualified person can reproduce them and learn the underlying mathematical approaches and physical principles. In this sense the book goes far beyond just being a review of the field, but should put the reader in a position to solve new problems on his/her own. In the second additional introductory Chapter 2 several basic flows are discussed in detail to ascertain flow patterns and forces relevant in the context of collision phenomena. The liquid–solid impacts (covered in Chapters 3, 4 and 5 in Part I) and liquid–liquid impacts (covered in Chapter 6 in Part II) discuss the problems associated with jet and drop collisions with dry or wetted solid substrates or the free surfaces of deep pools. The subsequent Chapters 7, 8 and 9 in Part III address collision phenomena involving liquid sprays. The description of sprays demands rather specific statistical methods. In this part we have therefore also included an introduction to drop and spray diagnostics, again, because our description of drop and spray collision phenomena is governed in many ways by our means of measurement and validation. As a quotation from Lord Kelvin goes: “When you cannot measure, your knowledge is meagre and unsatisfactory.” In Part IV (Chapters 10 and 11) the collisions of solid bodies with liquid are considered, where the body deformations are negligibly small and the hydrodynamics of the flow generated by body collision dominates the problem. Finally, in Part V (Chapters 12, 13 and 14) the collisions of solid bodies leading to significant deformation or erosion are in focus. In high-speed solid–solid collisions deformation of solid bodies resembles that in liquid–liquid impacts due to the geometric similarities, and hydrodynamic-like models for the former are close to the hydrodynamic models for the latter, which links these seemingly unrelated phenomena and is fruitful for their understanding and description. To a large part the numerous application fields of collision phenomena in liquids and solids have been reduced to generic problems in this book, whose approaches and solutions offer a higher degree of universality. However, also some very specific applications have been discussed to illustrate the use of such generic approaches and techniques to real-world problems. An introduction to each chapter is used to acquaint the reader with the scope of problems addressed in that chapter and their applications. Each chapter in the book can be read stand-alone after a reader has familiarized himself/herself with the introductory Chapters 1 and 2 devoted to the underlying fundamental ideas of fluid and solid mechanics. The book incorporates a wide range of relevant references to existing literature at the end of each chapter. Still the treatment of all the topics considered in the book is sufficiently self-contained, detailed and covered in-depth. This book is directed toward senior year undergraduate students, graduate students, researchers, engineers and practitioners in industry. It is expected that the reader would have been exposed to the fundamentals of fluid mechanics and some basic facts of solid mechanics, which are briefly reviewed in the introductory Chapters 1 and 2 and are important to the central topics of the book: drops, jets, sprays and solid collision

Preface

xiii

phenomena. These include some basic approximations involved, basic concepts of wettability and hydrodynamic instabilities, as well as rheological behavior. This first two chapters do not strive to replace textbooks; hence derivations are kept to a minimum, with appropriate reference to more comprehensive sources, while a non-typical comprehensive and unifying view of liquid and solid mechanics is adopted. Researchers from many different fields are addressed: theoretical and applied mechanics, materials science, aerospace, terminal ballistics, engineers from the fields of energy, chemistry, mechanical, automotive, environmental and chemical engineering, production, food science and fields involving domestic products, as well as physics and applied mathematics. The book can be of special importance to the researchers and engineers interested in developing exact or approximate models of the phenomena similar to those covered in this book, while benefiting from its in-depth and comprehensive exposition of physical foundations of such phenomena. Special thanks are directed to our students and colleagues to whom we are grateful for their collaboration and fruitful discussions, and whose names are frequently mentioned in this book. Thanks also go to Hannah Kittel for the image used on the cover of the book. Above all, we wish to thank our parents and families. Their encouragement and help made this book possible.

1

Introduction

This introductory chapter overviews the fundamentals of collision phenomena in liquids and solids. It begins with the physical estimates in Section 1.1, which ascertain the conditions of the commonality of phenomena characteristic of liquid and solid collisions and the historical and modern reasons for deep interest in them. Before embarking on a discussion of the governing equations some basic dimensionless groups are introduced in Section 1.2. Then, the reader encounters the basic laws of mechanics of liquids and solids formulated as the mass and momentum balance equations in Section 1.3. The distinction between liquids and solids can stem from rheological constitutive equations, which are to be added to the basic laws. Two rheological models, of an inviscid and Newtonian viscous liquid, are introduced in Section 1.4, which transforms the basic laws to the Laplace equation for the kinematics of potential flows of inviscid fluids accompanied by the Bernoulli integral of the momentum balance, as well as to the Navier–Stokes equations describing general flows of viscous fluids, or in the limiting case, to the Stokes equations for the creeping flows dominated by viscosity. A special case of a strong short impact of solid onto any type of liquid reveals the potential impulsive motions introduced in Section 1.5. On the other hand, high-speed flows of low-viscosity liquids near a solid surface reveal traditional boundary layers, while near free liquid surfaces the other, less frequently discussed, boundary layers arise. Both types of the boundary layers and the corresponding equations are considered in Section 1.6. Geometric peculiarities of flows in thin liquid layers on solid surfaces allow for such simplifications as the quasi-one-dimensional and lubrication approximations discussed in Section 1.7. Special physical conditions exist at the moving contact line where liquid surface is in contact with both the underlying solid surface and the surrounding gas, which involves such issues as the Navier slip also covered in Section 1.7. The static configurations of sessile and pendant liquid drops, in particular their contact angles with solid surfaces, can be significantly affected by the surface texture and chemical composition – the group of questions elucidated in Section 1.8 and associated with wettability. Rheological transition from traditional liquids to solids is gradual and spans Newtonian viscous liquids, various non-Newtonian liquids including viscoelastic liquids, the elastic Hookean solids, elastic-viscoplastic materials and then, paradoxically (at very high impact velocities) the inviscid materials characterized by inertia only (Section 1.9). A short exposition of some basic instabilities encountered in collision phenomena is given in Section 1.10. Finally, in Section 1.11 the correct use of the energy balance approach in the modeling of some hydrodynamic problems is discussed.

2

Introduction

1.1

History and Outlook Collision phenomena are common, spectacular and frequent in real life. People were always fascinated with water drops impacting soil, stone, puddles or plants during rain. Watching rain generates thoughts, some physical, some philosophical, or both: The rain to the wind said, ‘You push and I’ll pelt.’ They so smote the garden bed That the flowers actually knelt, And lay lodged–though not dead. I know how the flowers felt. “Lodged” by Robert Frost (1874–1963)1

Drops impacting onto a liquid layer are so attractive to the general public that they are regularly used in commercials aired on television, frequent on advertising billboards and shown on postcards. They motivated the famous poetic words of Edgerton and Killian (1954) in their book on ultra-high-speed photography: “In the land of splashes, what the scientist knows as Inertia and Surface Tension are the sculptors in liquids, and fashion from them delicate shapes none-the-less beautiful, because they are too ephemeral for any eye but that of the high-speed camera.” Drops impacting on liquid or solid surfaces can spread, or splash or even bounce back, as the detailed observations initiated in a series of brilliant works of Worthington in the late nineteenth century and summarized in his book Worthington (1908). To recognize the minute (actually, millisecond!) details of drop impact beyond those visible to poets, Worthington used high-speed photography, while the illumination was provided by a synchronized electric spark in air. The modern reincarnation of Worthington’s approach is the use of charge-coupled devices (CCD cameras) and light-emitting diodes (LEDs) as light sources (Yarin 2006, Thoroddsen et al. 2008, Josserand and Thoroddsen 2016). Drop spreading, splashing and bouncing imply an easy deformability characteristic of liquids, which are normally experienced as soft materials. However, folk wisdom expressed in the proverb “drop by drop wears away the stone” implies drop capabilities comparable to that of stones, for example, limestones located under leaky and dripping gutters. The characteristic time of water drop deformation during an impact, τde f , is mostly determined by the competition of the inertia (the driving mechanism) and the surface tension (the restraining mechanism), and thus is of the order of τde f ∼ (ρD3 /σ )1/2 , where ρ and σ are the density and surface tension, and D is the volume-equivalent drop diameter. For water drops, with ρ = 103 kg/m3 , σ = 0.0072 kg/s2 and D ∼ 0.001 m, τde f ∼ 4 ms, which indeed, requires a CCD camera for detailed observations. On the other hand, the impact time, τimp , is of the order of τimp ∼ D/V0 , where V0 is the impact velocity. Therefore, in the cases where the impact velocity is high enough for the inequality τimp < τde f to hold, an impacting drop does not have enough 1

Courtesy of Henry Holt and Company, LLC; The Random House Group, Penguin Random House, UK

1.1 History and Outlook

3

time to deform and can initially behave as an extremely rigid (very stiff) solid. This determines the critical (lowest) limit of such solid-like behavior as V0,crit ≈ (σ /ρD)1/2 , which is about 0.27 m/s. It should be emphasized that drops falling from a leaking gutter of a suburban home have velocities of the order of several meters per second, i.e. V0 > V0,crit , and thus exhibit an initially solid-like behavior. To evaluate the pressure they exert on an underlying surface, note that the information that the leading edge of a drop has impacted on an obstacle spreads with the speed of sound in water c, which is about 1497 m/s. During time t the mass of liquid, which is affected by the deceleration due to drop impact is thus m = ρctS, where S is the impact area. Accordingly, the momentum balance reads mV0 = F t, with F being the force exerted on the underlying surface. Thus, the pressure p experienced by the underlying surface during the time τel ∼ D/c ∼ 10−6 s before rarefaction proceeds from the trailing side of the drop (τel  τde f < τimp ), is p = F/S = ρcV0 , with V0 > V0,crit . For V0 = 4 m/s, this pressure is about p ≈ 60 atm. For limestone, marble and granite the ultimate strength in compression can be as low as 20, 50 and 70 MPa (about 200, 500 and 700 atm), respectively, which means that a prolonged dripping can definitely wear them away and drop impacts, indeed, reveal some solid-like phenomena on the liquid side. Cannon balls, bullets, projectiles and shaped-charge jets and their action on a target (a fortification or armor) attracted human attention not less intense than that devoted to rain, and especially their penetration capabilities were the focus of attention. In such cases one deals with sub-ordnance, ordnance and ultra-ordnance velocity ranges encompassing velocities from 25 to 3000 m/s (Backman and Goldsmith 1978). The field of terminal ballistics dealing with such questions was established by the classical works of Euler, Robins and Poncelet (Rosenberg and Dekel 2012), which were followed much later by the seminal works of Munroe (1900), Birkhoff et al. (1948) and Lavrentiev (1957). The early pioneers in the eighteenth and nineteenth centuries processed a wide variety of experimental data to establish the resistance experienced by cannon balls and bullets penetrating into solid targets, as well as the corresponding penetration depth. Only much later it was realized that in many cases solid–solid penetration reveals liquid-like properties of solids. For metals the yield stress Y and the ultimate strength σ∗ , which is typically of the order of Y , are much less than the pressure exerted initially by a projectile, p = ρcV0 (exactly due to the same reason as for liquids), or at a later stage when it reduces to the level of p = ρV02 due to the rarefaction emanating from the rear edge. Indeed, taking for steel ρ = 7.8 × 103 kg/m3 , Y = 690 MPa, c ≈ 5900 m/s and V0 =1000 m/s, one finds the following ratios Y/(ρcV0 ) = 0.015 and Y/(ρV02 ) = 0.088. Similarly, for tungsten when ρ = 19.25 × 103 kg/m3 , Y = 550 MPa and c ≈ 5220 m/s, one finds for the collision velocity of V0 =1000 m/s, the following ratios Y/(ρcV0 ) = 0.00547 and Y/(ρV02 ) = 0.0286. Therefore, in cases of collision of steel and tungsten with armor, the pressure in both projectile and target far exceeds their plasticity limits, which means that metals will flow. Moreover, the above-mentioned low values of the Y/p ratios reveal that plastic resistance to flow will be relatively small, and the dominant forces will be inertial (the situation quite similar to that in flows of such “inviscid” liquids as water, especially after sufficiently fast drop impacts).

4

Introduction

Neither the general public, nor the majority of the scientific community, realize that such spectacular phenomena as comet and asteroid collisions with planets, or projectile penetration into armor, can be “close relatives” of tiny drop impacts on the other end of the scale bar; however in fact, they are! The elucidation of this fact is the main motivation of the authors to write this book, since their personal research experiences spanned liquid–solid, liquid–liquid, solid–liquid and solid–solid impacts. Collision phenomena one encounters in real life, technology and nature span the entire spectrum from tiny drops to asteroids; to name a few: r r r r r r r r r r r r r r r r r r r r r r r r r r r

Ink-jet printing Spray cooling of hot surfaces Spray coating, spray painting Annealing, quenching of metal alloys Fire suppression Fuel injection Touchless cleaning with sprays Spray inhalation (impacts and deposition in the lungs) Encapsulation Domestic applications (e.g. hair spray) Near-net shape manufacturing Erosion of (steam) turbine blades Ice accretion on turbine components, power lines, aircraft Dilution of lubricating films due to fuel droplet impingement Spreading of plant diseases by rain Spore spreading by rain Criminal forensics Crop spraying Aeration of surface layers of lakes, seas and oceans Soil erosion Transport of granular materials Seaplane landing Shaped-charge jet penetration Ballistic penetration Military applications Explosion welding Solid material testing

Such a wide variety of fascinating and practically important situations typically involve a hidden common denominator dictated by “inviscid”-like flow and geometrical similarity of collision and impact phenomena. As the above-mentioned historical introduction shows, to a large part, the topics covered in this book have developed quite independently from one another in the sense that different communities were involved in the different collision phenomena: liquid–solid, liquid–liquid, solid–liquid and solid– solid. This book is an attempt to provide a unique vision of the underlying similarities

1.2 Dimensionless Groups

5

existing in collision phenomena, which greatly facilitate their understanding and modeling and which are still not fully recognized due to the scatter of these phenomena among different disciplines. The fact that each of these disciplines dealing with collision phenomena has undergone rapid development over the past years is indisputable. This can be attributed to a multitude of factors; however, there is no doubt that in the interest of improving numerous industrial processes (including those of modern high-tech industries), understanding the underlying physics, as opposed to relying on simple engineering correlations, is becoming a necessity and is increasingly being sought by industry. An in-depth understanding of various natural collision phenomena is also required to facilitate solid foundations of ecology, geology and other branches of science. It should be emphasized that joint consideration of fluid and solid mechanics including fracture mechanics is not uncommon for textbooks, as in the recent one of Barenblatt (2014), which shows that such an approach can be fruitful. Collision phenomena are becoming recognized as one of the fundamental events on which an entire production or natural process may depend. This is most easily illustrated by the above-mentioned examples – both in engineering and in nature – in which impact and collision phenomena play a vital role. Another factor contributing to the current interest in collision phenomena is undoubtedly the remarkable development in high-speed imaging over the past decade. This allows collision phenomena to be studied at unprecedented precision and resolution, revealing physics which were heretofore often only the subject of speculation or empirical modeling. Accordingly, new mathematical models of the phenomena can now be developed and validated to a much higher degree of certainty. Therefore, understanding and modeling of collision phenomena also form a challenging new domain in the fields of applied physics and mathematics, stimulating novel and classical experimental, theoretical and numerical approaches. Whereas the book underlines similarities among different collision phenomena, there are some restrictions in scales. At very large length scales, for instance the collision of galaxies, or at very high velocity scales (hyper-ordnance or cosmic) phase transition, nuclear physics, gravity and relativity affect the collision phenomena, going beyond the scope of this book. Therefore, we can say that the book is restricted to mesoscales and ordnance and ultra-ordnance velocities, although attention is definitely paid to the effect of nano-texture on solid surfaces on drop impact, i.e. phenomena at nano-scales.

1.2

Dimensionless Groups Dimensional analysis is a powerful tool for generalization of experimental data and uncovering hidden scalings and self-similarities in seemingly complicated hydrodynamic situations. The general ideas and multiple examples of the applications of dimensional analysis are discussed in several superb monographs, which an interested reader can easily find: Bridgman (1931), Barenblatt (1987, 2000), Sedov (1993) and Yarin (2012). Therefore, in the present section we briefly list the main dimensionless groups

6

Introduction

relevant in this book. The dimensionless groups governing drop impact onto a solid surface or a liquid layer are We =

ρDV02 ρDV0 μ We1/2 , Re = , Oh = = σ μ Re (ρσ D)1/2

K = We · Oh−2/5 , St =

ρg μg MV0 h0 , H = , R = , Vi = 6π μa2 D ρ μ

(1.1) (1.2)

where ρ, μ and σ denote liquid density, viscosity and surface tension, D and V0 the drop diameter and impact velocity, h0 thickness of the liquid film, ρg and μg are the surrounding gas density and viscosity. We, Re and Oh denote the Weber, Reynolds and Ohnesorge numbers, and H dimensionless film thickness; K is an important composite group. St is the Stokes number, where M is the mass of a spherical particle of radius a = D/2 impacting onto a thin viscous layer of viscosity μ at the wall. In addition, R and Vi denote the density and viscosity ratios. Also, gravity-related effects are characterized by the Bond number Bo = ρgD2 /σ , i.e. the ratio of D2 to the square of the capillary length  1/2 σ (1.3) λc = ρg (g being gravity acceleration), or by the Froude number Fr =

V02 We = . gD Bo

(1.4)

Further dimensionless parameters characterizing roughness and wettability effects will be relevant to drop impact on solid dry surfaces, as well as the equilibrium contact angle. Among them the capillary number Ca =

We μV0 = . σ Re

(1.5)

The capillary number is important if the dynamic contact angle influences significantly the considered flow. In this case the velocity of propagation of the contact line U is used in the expression (1.5) instead of V0 . Non-spherical drop aspect ratio and the Strouhal number characterizing transient phenomena can also appear. In relation to the discussion of the electrohydrodynamic aspects of drop impacts, the dimensionless charge relaxation time α, and the electric Bond number BoE naturally arise, with 2 DE∞ τCV0 , BoE = (1.6) D σ where τC is the charge relaxation time, and E∞ is the applied electric field strength, or alternatively, UE /D, where voltage UE is given. Moreover, and this is very important for understanding of the organization of Parts I to V of the book, the dimensionless groups related to the rheological behavior of colliding materials can be introduced. Namely, the Deborah number De and the dimensionless

α=

1.3 Mass and Momentum Balance Equations

7

stiffness S and plasticity P groups (or, alternatively, the modified Bingham number, Bn) can be introduced as θV0 D

(1.7)

E Y , P or Bn = , 2 ρV0 ρV02

(1.8)

De = S=

where θ is the viscoelastic relaxation time, E is Young’s modulus and Y is the yield stress. In addition, at the early stages of impact or collision when the compressibility effects are important, instead of Eqs. (1.8), the dimensionless groups S, P or Bn should involve the speed of sound in solid material c, i.e. S=

Y E , P or Bn = . ρcV0 ρcV0

(1.9)

Also, the expressions for the groups P and Bn can involve the ultimate strength σ∗ rather than the yield stress Y , as in Eqs. (1.8) and (1.9). The dimensionless groups (1.1)–(1.7) will determine the discussion in Parts I–IV of the book. In particular, the effects of Re, We, R, Vi and especially K groups on drop deposition, splashing and bouncing will be introduced in the ordered manner. It should be emphasized that the density and viscosity ratios would only be important in very specific situations, not all that common. The dimensionless groups (1.8) and (1.9) are relevant for the discussion in Part V. Note also, that the significant number of dimensionless groups listed above (even not including such dimensionless groups as the Stefan number (Ste) related to thermal effects in Section 4.2 in Chapter 4, or the Mach number (Ma) related to the gas compressibility in Section 4.8 of Chapter 4), does not allow one to strictly order the material according to only Re and We, but makes much more reasonable the present organization of material in the book, which ascends from the liquid-only to the solid-only phenomena, with the secondary details (and the corresponding dimensionless groups) discussed in the framework of this structure.

1.3

Mass and Momentum Balance Equations Here we restrict ourselves to the incompressible case mostly relevant to problems related to flows of liquids associated with drop impact. The mass balance equation for an incompressible liquid reduces in hydrodynamics to the so-called continuity equation in the following invariant form ∇ · v = 0.

(1.10)

This scalar equation is insufficient alone to describe fluid flows, since it incorporates two or three velocity components. Furthermore, not every arbitrary velocity vector will satisfy this equation and only those velocity fields which do not contradict Eq. (1.10)

8

Introduction

are kinematically admissible and thus can be realized, in principle. Those velocity fields which do not satisfy Eq. (1.10) are forbidden, since they lead to hole or fold formation in the flow field, i.e. to discontinuities, which are not permitted by the continuity equation (1.10). The momentum balance equation in hydrodynamics is nothing but the second law of Newton for an infinitesimally small material element (Lamb 1959, Loitsyanskii 1966, Landau and Lifshitz 1987, Batchelor 2002) ρ

Dv = ∇ · σ + ρa, Dt

(1.11)

where the incompressibility is assumed. The material time derivative is denoted as D(•)/Dt, ρ is the density, σ is the stress tensor (related to the surface forces) and a is the acceleration associated with a body force. If the body force is restricted to be the gravity force, then a = g, with g being acceleration due to gravity. The fluid particle acceleration expressed by the material time derivative Dv/Dt can be split into the temporal and convective parts, namely ∂v Dv = + (v · ∇)v. Dt ∂t

(1.12)

This expression shows that even if the velocity field is stationary and ∂v/∂t = 0, a material particle will still experience an acceleration when it is entrained by flow to a location with a different local velocity, which is expressed by the second term on the right-hand side of Eq. (1.12). In mechanics of incompressible fluids the stress tensor σ is traditionally split into two parts: an isotropic one associated with pressure p, and an additional, deviatoric tensor τ σ = −pI + τ ,

(1.13)

where I denotes the unit tensor. Substituting Eq. (1.13) into Eq. (1.11), and using Eq. (1.12), one arrives at the following form of the momentum balance equation   ∂v + (v · ∇)v = −∇p + ∇ · τ + ρa, (1.14) ρ ∂t which is known as the Cauchy momentum equation. The mass and momentum balance Eqs. (1.10) and (1.11) form a system of fundamental equations required to describe fluid flow. However, these equations are insufficient, since a statement about material behavior is required to relate the stress tensor σ with flow kinematics. It should be emphasized that Eqs. (1.10) and (1.11) apply equally to such different continua as incompressible elastic solids and incompressible fluids. There is therefore a need to distinguish different types of material behavior, which requires an additional rheological constitutive equation, which relates to flow kinematics. Sometimes (but very infrequently, e.g. for polymeric liquids; see Doi and Edwards 1986, Bird et al. 1987) such an equation can be derived from a micromechanical model of material of a certain type using methods of statistical physics. Alternatively (and much more frequently; cf. Loitsyanskii 1966, Landau and Lifshitz 1987, Larson 1988, Batchelor

9

1.4 Inviscid and Viscous Newtonian Fluids

2002) such an equation is postulated phenomenologically to mimic and generalize certain experimental observations of material behavior in some simplifying limiting cases. In Section 1.9 a detailed account of the phenomenological approach to the formulation of rheological constitutive equations of rheologically complex liquids and solids is given, whereas in the following section the two most important and simplest cases of the inviscid and viscous Newtonian fluids are covered.

1.4

Inviscid and Viscous Newtonian Fluids: The Incompressible Euler and Navier–Stokes Equations Historically the first phenomenological tensorial rheological constitutive equation was introduced by Euler. He assumed that the deviatoric stresses (already understood at that time as viscous stresses after Newton’s experiments) are negligibly small and thus the stress tensor is always isotropic, as in hydrostatics, even though the fluid is in motion τ = 0,

σ = −pI.

Bearing in mind Eq. (1.15), the momentum balance (1.14) reduces to   ∂v + (v · ∇)v = −∇p + ρa, ρ ∂t

(1.15)

(1.16)

which is known as the (incompressible) Euler equation. It should be emphasized that this nonlinear equation for an inviscid fluid can be analytically integrated if the body forces are conservative, i.e. possess a potential (which is true, for example, for the gravity force), irrespective of the fluid being incompressible or compressible [albeit barotropic, i.e. ρ = ρ(p)]. The integral is called the Bernoulli equation. For incompressible potential flows v = ∇φ (which is equivalent to irrotational flows with ∇ × v = 0), where φ is the hydrodynamic potential, the continuity equation (1.10) reduces to the Laplace equation for φ ∇ 2 φ = 0,

(1.17)

∂φ p (∇φ)2 + + + gz = f (t ), ∂t ρ 2

(1.18)

whereas the Bernoulli integral reads

with g being the magnitude of the gravity acceleration (i.e. a = g), z being the vertical coordinate and f (t ) being a function of time which can be established from the boundary conditions. In such cases the kinematics of any fluid mechanical problem is generated by the corresponding solutions of the Laplace equation (1.17), whereas the dynamics, i.e. the corresponding pressure, are immediately recovered from the algebraic Bernoulli equation (1.18). This simplifying approach, known as potential flow theory or ideal fluid flows, still may be rather involved when complicated free surface configurations are present and their evolution must be established. Such situations may require numerical solutions or

10

Introduction

further simplifications discussed below (see Section 1.5). For the numerical simulations of drop impact onto liquid surfaces (Weiss and Yarin 1999) it is convenient to use the integral equivalent of the Laplace equation (Lamb 1959, Tikhonov and Samarskii 1990), which allows one to find the normal velocity component at the free surface vn = ∂φ/∂n using the knowledge of the distribution of the potential φ at the free surface. Since the tangential velocity components can be found by differentiation of the known distribution of φ over the free surface, the entire velocity vector at the surface can be found using the information on φ only at the free surface. This forms the foundation of the Boundary Integral Method (or discretized, numerical equivalent, the Boundary Element Method, BEM). It is emphasized that the effect of the boundary associated with a solid wall underneath the liquid layer can be accounted for using the method of images (Weiss and Yarin 1999). Then, only the free liquid surface is left to be tackled. The time marching required to update the positions of the individual fluid elements at the free surface involves the kinematic condition there Dr = ∇φ, Dt

(1.19)

with r being the position vector, and the equation required to update the potential distribution at the free surface, which follows from the Bernoulli equation (1.18) (∇φ)2 σκ Dφ = − − gz. Dt 2 ρ

(1.20)

In this equation the pressure at the free surface is obtained invoking the Young– Laplace equation p = σ κ, with κ being the mean curvature of the free surface, σ the surface tension and f (t ) = 0, when a droplet is already connected to a liquid layer, which extends to infinity. Note that for potential flows, the material time derivative ∂φ ∂φ ∂φ ∂φ ∂φ Dφ = +u +v +w = + (∇φ)2 Dt ∂t ∂x ∂y ∂z ∂t

(1.21)

where x, y and z are the Cartesian coordinates, and u, v and w are the corresponding velocity components. Potential flow, which is identically irrotational, is rooted in the simplified rheological constitutive equation (1.15). Indeed, an initially potential/irrotational flow stays a potential/irrotational flow at any time under the conditions of Kelvin’s circulation theorem [zero viscosity, as in Eq. (1.15), conservative body forces, and fluid is barotropic] (Kochin et al. 1964, Batchelor 2002). Intuitively it is approximately valid for lowviscosity liquids (e.g. those like water), albeit, as was established much later in 1904 by Prandtl, only at some distance away from the solid boundaries (see Section 1.6 in the present chapter). For sufficiently viscous fluids and/or in cases where the flow development sufficiently close to a wall is studied, Eq. (1.15) is insufficient and hydrodynamics according to potential flow theory collapses. An alternative rheological constitutive equation is needed. This is the Newton–Stokes constitutive equation, which assumes a linear dependence between the deviatoric stress tensor τ and the rate-of-strain tensor D = (∇v + ∇vT )/2, with ∇v being the tensor gradient of velocity. Namely, for

1.4 Inviscid and Viscous Newtonian Fluids

11

liquids and the incompressible gases it is assumed that τ = 2μD,

(1.22)

with μ being the viscosity, and thus the stress tensor σ becomes σ = −pI + 2μD.

(1.23)

The phenomenological constitutive equations (1.22)–(1.23) are based on a few experiments in the simplest limiting cases, e.g. in the simple shear flow and, in addition, (but very infrequently) in the uniaxial elongational flow. If a fluid follows predictions of Eqs. (1.22)–(1.23) in such rheometric flows, the viscosity coefficient is found as a fitting parameter without detailing its real physical origin and the equations are assumed to be true in any other flow. Fortunately, this phenomenological approach works extremely well for many important fluids, in particular, for air, water, glycerol and multiple oils. Such fluids are called Newtonian when μ is constant for all rates of strain. Substituting the rheological constitutive equation (1.22) into the momentum balance equation (1.14) and accounting for the continuity equation (1.10), one arrives at the incompressible Navier–Stokes equation   ∂v + (v · ∇)v = −∇p + μ∇ 2 v + ρa. (1.24) ρ ∂t The Navier–Stokes equation (1.24) expresses the balance of the nonlinear inertial forces (on the left) with the surface forces (the viscous forces and pressure) and the body force on the right. Together with the scalar equation (1.10), this vector equation forms the system of four scalar equations, which, together with respective boundary conditions, allows all three components of velocity vector and pressure in the general case to be found. Solutions of the Navier–Stokes equations are subject to the initial conditions for the velocity field v and pressure in the transient cases, and to certain boundary conditions in all cases. The latter include several possible kinematic boundary conditions listed below as (i)–(iii). (i) The no-slip condition, which implies that liquid elements in direct contact with a solid boundary possess exactly the same velocity as the velocity of the solid boundary at the same point. Mathematically speaking, this is the Dirichlet-type boundary condition. It should be emphasized that any potential velocity field would satisfy the Navier–Stokes equation (1.24), since ∇ 2 (∇φ) = ∇(∇ 2 φ) = 0 due to Eq. (1.17). However, such solutions cannot satisfy the no-slip condition, and thus are excluded. (ii) At the free surface or other fluid/fluid interfaces the no-slip condition implies that the velocities of two adjacent immiscible fluids are equal. (iii) Also, there is the condition that the interface is material, i.e. the fluid boundary moves normal to itself together with the adjacent fluid elements, which arises in the time-dependent situations involving free surfaces and/or fluid/fluid interfaces. This kinematic boundary condition is supplemented by the following dynamic boundary conditions. (iv) At the free surface of a liquid, boundary conditions for stresses should be imposed, which implies that shear stresses are imposed by the overlying gas, which

12

Introduction

is also a Newtonian fluid, whereas the normal stress is not only imposed by gas but also experiences a jump due to the surface tension. Typically, the shear forces imposed by gas on liquid are negligibly small due to the much lower viscosity of gases compared to those of liquids. Hence, the shear stresses on the liquid side of the free surface should be zero, which results in the zero shear rates, i.e. velocity derivatives at the surface. Mathematically speaking, this results in the Neumann-type boundary conditions at the free surface, where, in addition, the normal stress is equal to the capillary pressure multiplied by −1 added to the outside atmospheric pressure. More generally, at the fluid/fluid interfaces the dynamic boundary conditions require that shear stresses are continuous and normal stresses experience a jump associated with the capillary pressure. (v) There might be additional boundary conditions for pressure, for example at the entrance and exit of a pipe. There are a few important non-trivial analytical solutions of the Navier–Stokes equations (Loitsyanskii 1966, Landau and Lifshitz 1987, Batchelor 2002). Even for direct numerical simulations the nonlinear partial differential Navier–Stokes equations are very complex even in steady-state cases without moving free surfaces. The situation becomes even more complicated in the case of collision phenomena, when rapid transient effects are important and small scale free surface features near and at the free surface appear. Still, there are a number of examples of direct numerical solutions of the Navier–Stokes equations relevant for collision phenomena using, for example, the Volume of Fluid (VOF) method (e.g. in Rieber and Frohn 1999, Bussmann et al. 1999, Renardy et al. 2003). In the case of highly viscous liquids, or very slow motion, or for extremely small scales, the inertial forces are negligibly small and Eq. (1.24) reduces to the Stokes equation − ∇p + μ∇ 2 v + ρa = 0.

(1.25)

This linear equation, together with the continuity equation (1.10), has a number of elegant and non-trivial analytical solutions (Loitsyanskii 1966, Happel and Brenner 1991, Batchelor 2002, Kim and Karilla 2005). The Stokes equation also admits an equivalent boundary integral formulation and thus relatively easy numerical solutions based on the Boundary Element Method (BEM) (Pozrikidis 1992). In the context of drop impact on a solid wall, such solutions can be found in Reznik and Yarin (2002a, 2002b).

1.5

Impact at Liquid Surface and Equations of Impulsive Motion Strong and short impacts of solid bodies onto a liquid surface or an impact onto a solid body already floating at the surface of an incompressible liquid result in an instantaneously established pressure field with high pressure gradients (Lamb 1959, Kochin et al. 1964, Batchelor 2002). The body might be a ball of several millimeters in diameter (Zenit and Hunt 1998), or a seaplane skid upon landing. For effectively inviscid liquids,

1.6 Boundary Layer Equations

13

in the case of a strong and short impact, in the Euler equation (1.16), the pressure gradient dominates the body force and the latter can be neglected. Then, the equation takes the form   p Dv . (1.26) = −∇ Dt ρ The integration in time, following a material particle, yields    1 τ v(r) = ∇ − p(r, t ) dt ρ 0

(1.27)

where it is assumed that at the moment of impact at t = 0 fluid was at rest; r is the position vector and τ is the impact duration. The impact duration τ is very short, i.e. τ → 0 but the impact is strong, i.e. p → ∞. Therefore, the pressure impulse  τ p(t ) dt (1.28)

= 0

is finite. Equations (1.27) and (1.28) show that the flow arising as a result of impact is potential (and thus, irrotational), i.e. v = ∇φ,

φ=−

ρ

(1.29)

with φ being the flow potential. The impact flows of incompressible liquids arise only under the action of pressure, and thus a potential flow stays potential according to Kelvin’s circulation theorem (Kochin et al. 1964, Batchelor 2002), as explained in Section 1.4 in the present chapter. Due to the continuity equation (1.10), the flow potential is found from the Laplace equation (1.17).

1.6

Boundary Layer Equations The case of high Reynolds number corresponds to inertial effects being dominant over viscous ones. Still, completely neglecting all viscous effects, as was done by Euler (see Section 1.4), or with impulsive motions (see Section 1.5), does not allow one to satisfy the no-slip conditions at the solid wall. A significant simplification of the problem formulation was achieved by Prandtl in 1904 (Prandtl 1952, Loitsyanskii 1966, Schlichting 1968, van Dyke 1964). He demonstrated that in the case of the high Reynolds number flows, thin boundary layers near the wall appear, within which the flow is significantly slowed down by the wall whereby the viscous forces then compete with the inertial ones due to the high velocity gradients generated. The thickness of laminar boundary layers is of the order of δ = LRe−1/2 , where L is the characteristic longitudinal length scale of the problem, say the initial drop diameter, and obviously δ  L. The original physical estimates of Prandtl (Prandtl 1952, Loitsyanskii 1966, Schlichting 1968), or a more formal asymptotic analysis (van Dyke 1964), show that the flow

14

Introduction

inside the boundary layer is governed by a reduced asymptotic version of the Navier– Stokes equations, which is called the Prandtl or the boundary layer equations. Since boundary layers should be thin compared to the wall radius of curvature (otherwise, the boundary layer approximation would be invalid), the Cartesian coordinates x and y, tangential and normal to the wall, respectively, will suffice, and in planar cases the incompressible boundary layer equations read ∂u ∂v + =0 ∂x ∂y   ∂u ∂u dU ∂ 2u ∂u +u +v = ρU +μ 2. ρ ∂t ∂x ∂y dx ∂y

(1.30) (1.31)

The first equation here is the continuity equation, while the second one is the momentum balance Prandtl equation. Equation (1.31) is the asymptotic form of the tangential to the solid surface projection of the momentum balance. In addition, the normal projection of the momentum balance reduces to the statement that pressure is constant across the boundary layer, i.e. dp/ dy = 0 there. The system of two equations (1.30) and (1.31) is used to find the two unknowns – the x- and y- velocity components u and v, respectively. It should be emphasized that the boundary layer approximation implies that the flow far from the wall at y → ∞ (i.e. at the distance of the order of δ from the wall) is identical to the potential flow field about the same body. Then, the velocity distribution U (x) at the outer boundary of the boundary layer can be found in advance. Since the pressure is constant across the boundary layer, the pressure-related term in the longitudinal momentum balance equation (1.31), −dp/ dx, can be found from the Bernoulli equation (1.18). For the steady-state outer flow, the latter equation yields −dp/dx = ρU dU/ dx, as in Eq. (1.31), which is the term known in advance. Solutions of the boundary layer equations satisfy the no-slip condition at the wall and the asymptotic matching condition at the outer boundary of the boundary layer, namely, y = 0, u = 0;

y → ∞, u → U.

(1.32)

The second condition (1.32) implies that the inner flow in the boundary layer is asymptotically matched with the outer potential flow. It should be emphasized that the x-momentum balance Prandtl equation (1.31) is a nonlinear partial differential equation, and the nonlinearity is exactly the same as in the original Navier–Stokes equation. However, neglecting the small linear viscous term μ∂ 2 u/∂x2 , which reduces the Navier–Stokes to the Prandtl equation, changes the type of the former. Namely, the Navier–Stokes equation is an elliptic equation, whereas the Prandtl boundary-layer equation is parabolic. Solution of nonlinear parabolic equations is immensely simpler (albeit still non-trivial) compared to elliptic ones. Flow systems governed by parabolic equations can be numerically solved by moving downstream from an initial flow state at the inlet. Boundary layer theory, i.e. parabolic equations, practically neglects the backward effect of the viscous boundary layer on the outer inviscid potential flow. If such an effect becomes important (e.g. recirculating flows), the outer flow can be dramatically displaced and, due to that change, boundary layer separation can occur as a result of an adverse pressure gradient arising in such cases. Then, the

1.7 Quasi-one-dimensional and Lubrication Approximations

15

entire asymptotical boundary layer approach fails, and there is no practical alternative to solving the full Navier–Stokes equations, typically numerically. The effect of wall curvature on the boundary layer equation (1.31) is negligible; however, the continuity equation is affected by the wall curvature. For example, in the case of an axisymmetric boundary layer developing from the stagnation point x = 0, y = 0 of an axisymmetric body of revolution with the generatrix at r = r0 (x), the continuity equation takes the following form ∂ (r0 u) ∂ (r0 v) + = 0. ∂x ∂y

(1.33)

Solving the boundary layer equations (1.33) and (1.31) is of interest, for example, in problems of a head-on collision of a spherical drop with a body of revolution of comparable radius of curvature. Equations (1.33) and (1.31) admit the so-called Mangler transformation of coordinates, which reduces them to the planar case Eqs. (1.30) and (1.31) in a fictitious plane (Loitsyanskii 1966, Schlichting 1968). Note, that free surface flows of low-viscosity liquids, e.g. impacts of water-like drops, also involve the formation of viscous boundary layers near the free surface. Indeed, any inviscid potential flow inside a drop cannot be reconciled with the boundary condition of zero shear stresses at the free surface (see Section 1.4), and a viscous boundary layer should arise there to match the shear imposed at the free surface. Since such boundary layers are associated with rapidly changing free surface shapes and should be matched to complicated transient flows inside impacting drops, their theory (Lundgren 1989) is much less developed compared to the near-wall boundary layers discussed above.

1.7

Quasi-one-dimensional and Lubrication Approximations in Problems on Drop Impact and Spreading Drop impact and spreading on solid walls and thin liquid layers at the wall are shorttime phenomena. The characteristic time is of the order of τ ∼1 ms. For low-viscosity liquids, e.g. for water, the kinematic viscosity ν ∼10−6 m2 /s, and viscous forces acting from the wall can affect a liquid layer of thickness δ of the order of δ ∼(ντ )1/2 ∼10 µm. This thickness is typically negligibly small compared to the thickness of the liquid lamella spreading over the wall from the impact location. Then, for such low-viscosity liquids viscous effects are negligible in the main body of liquid. As a result, the velocity profile across the spreading lamella is almost uniform, and the description of the flow admits a quasi-one-dimensional approach, a familiar tool in the theory of free liquid jets and films (Yarin 1993). In the framework of the quasi-one-dimensional approach, the two main flow parameters of interest, the lamella velocity along the wall V and the lamella thickness h are sought as functions of time and the longitudinal coordinate along the wall. Such equations for spreading liquid lamellae resulting from drop impacts were derived in Yarin and Weiss (1995). In the planar case (formally corresponding to normal impact of a liquid cylinder) the quasi-one-dimensional continuity (mass balance) and

16

Introduction

momentum balance equations read ∂h ∂hV + =0 ∂t ∂x    2  ∂V ∂V ∂ ∂ h ρh +V =σ h 2 . ∂t ∂x ∂x ∂x

(1.34)

(1.35)

In Eqs. (1.34) and (1.35) t denotes time, x is the longitudinal coordinate along the wall, ρ and σ are the density and surface tension of the liquid. In the axisymmetric case corresponding to normal impact of a spherical drop, the quasi-one-dimensional continuity and momentum balance equations take the following form ∂rh ∂rhV + =0 ∂t ∂r    2  ∂V ∂V ∂ ∂ h ρh +V =σ h 2 ∂t ∂r ∂r ∂r

(1.36)

(1.37)

where r is the radial coordinate with its origin at the center of impact. Since the momentum balance equations (1.35) and (1.37) completely disregard viscous forces and are applicable only for inertia-dominated flows, they can be considered, in a sense, to be similar to the Euler equation (1.16). In fact, they can also be obtained by integrating the Euler equation across the liquid lamella. When drop spreading on the wall is dominated by viscosity, which corresponds to highly viscous liquids, very thin lamellae, or very slow motion, one can neglect the inertial forces. Then the situation is kindred to the Stokes equation (1.25). In such cases, a further simplification is provided by the fact that the lamellae are sufficiently thin and do not carry any short-wavelength perturbations. However, the lamella thinness does not imply that the velocity profile is uniform across the lamella cross-section. In such creeping flow films, the no-slip condition at the wall affects the entire velocity profile, which becomes approximately semi-parabolic. Such velocity profiles not only satisfy the no-slip condition at the wall but also the condition of the zero-shear stress at the free surface. In the inertialess approximation the continuity and momentum balance equations can be combined into a single equation for the film thickness h. In addition to the viscous forces, this equation also accounts for the surface tension and in the general case takes the following form (e.g. Oron et al. 1997, Ristenpart et al. 2006) σ ρg ∂h + ∇ · (h3 ∇∇ 2 h) = ∇ · (h3 ∇h) ∂t 3μ 3μ

(1.38)

where nabla-operators are two-dimensional with both coordinates tangential to the wall. On the right-hand side in this equation the effect of gravity acceleration acting normally to the wall has been included. Equation (1.38) is frequently called the lubrication equation, since the physical assumptions involved in its derivation are kindred to those implemented by Reynolds

1.7 Quasi-one-dimensional and Lubrication Approximations

17

in his theory of lubrication layers in journal bearings (Loitsyanskii 1966, Tadmor and Gogos 2013). It should be emphasized that the lubrication films in journal bearings are confined by two walls, with one of them moving, and in many cases, curved. Therefore, the original lubrication equation of Reynolds differs from the lubrication equation (1.38). The planar and axisymmetric equivalents to Eq. (1.38) are given by the following equations    3  ρg ∂ σ ∂ ∂h 3∂ h 3 ∂h = + h h (1.39) ∂t 3μ ∂x ∂x3 3μ ∂x ∂x     ∂h ρg 1 ∂ σ 1 ∂ ∂ 3h ∂h + rh3 3 = rh3 (1.40) ∂t 3μ r ∂r ∂r 3μ r ∂r ∂r where x and r are the longitudinal Cartesian and radial coordinates, respectively (the latter having its origin at the center of drop impact). For a lamella surrounded by a contact line, boundary conditions at the contact line (CL) should be specified. In steadystate, the contact lines are pinned, and the equilibrium contact angles can be imposed as boundary conditions at the contact lines for Eqs. (1.39) and (1.40) (see Section 1.8 in the present chapter). For transient cases, strict no-slip boundary conditions at the wall in contact with a moving CL result in a non-integrable force singularity (Dussan V 1979, Dussan V and Davis 1974). To eliminate this singularity, different slip conditions at moving contact lines have been proposed, for example, in the form of the Navier slip boundary condition y = 0,

u−λ

∂u =0 ∂y

(1.41)

with λ being the molecular slip length. The boundary condition (1.41) can be used to determine the flow near a moving CL, as well as the dependence of the apparent dynamic contact angle on the velocity of a moving CL. A number of existing experimental and theoretical studies imply that in a creeping flow regime such dependence is determined by the liquid viscosity, surface tension and molecular factors such as the London–van der Waals forces, significant in the nearest vicinity of the CL (de Gennes 1985). The experiments of Hoffman (1975) revealed a nearly universal relation between the advancing CL velocity and the apparent dynamic contact angle (θ ) over a wide range of velocities of the CL, given by Kistler (1993) as:   (1.42) θ = fHoff Ca + f −1 (θstatic ) Hoff with







x fHoff (x) = arccos 1 − 2 tanh 5.16 1 + 1.31x0.99

0.706 ,

(1.43)

and Ca being the capillary number [cf. Eq. (1.5) in this chapter]. Cox (1986) showed that the CL velocity depends on the flow at some distance from it, namely, the law of CL motion has a universal form but contains a term depending

18

Introduction

Figure 1.1 Contact line hysteresis.

on the flow. Note that, except the purely hydrodynamic local effects considered by Cox (1986), also local molecular-kinetic effects discussed by Blake (1993) could affect the dependence of the apparent contact angle on the CL velocity, and most probably both mechanisms contribute. For sufficiently low velocities both theories describe the CL motion equally well and in good agreement with Hoffman’s empirical law, whereby deviations increase as the flow velocity increases. It has been shown that CL motion behaves close to Hoffman’s law for small and moderate Bond numbers for gravitydriven drop spreading (the Bond number is Bo=ρgD2 /σ , with D being the volumeequivalent drop diameter, see Section 1.2 in this chapter). Another flow regime was found for large Bond numbers, for which strong gravity forcing is present (Reznik and Yarin 2002b). The CL motion in this case is affected by the bulk flow resulting from gravity; the apparent contact angle rapidly reaches the value of 180◦ and rolling motion sets in. There is a significant difference between the advancing and receding contact lines and the corresponding contact angles, which is demonstrated by the so-called contact angle hysteresis. The dependence of the contact angle θ on the CL velocity U presented as the dimensionless capillary number, Ca=μU/σ (with μ and σ being fluid viscosity and surface tension, respectively) is sketched in Fig. 1.1. Positive Ca corresponds to the advancing CL, while negative Ca corresponds to the receding CL. The dependencies of θ on Ca are different for positive and negative values of Ca. Moreover, they do not necessarily tend to the same limit as Ca→ ±0, thus determining two different equilibrium contact angle values, one which is achieved by the incipient advancing CL, θae , and another one, which is achieved by the incipient receding CL, θre . Any static contact angle in the range θre ≤ θ ≤ θae is possible, thus determining the contact-angle hysteresis, which is characteristic for many systems (Dussan V 1979). Note, that on very smooth and chemically homogeneous surfaces the contact-angle hysteresis disappears and θae = θre .

19

1.8 Wettability

Figure 1.2 (a) Partially wettable substrate. (b) Non-wettable substrate. Small drops in the images

are in equilibrium (Dror et al. 2007). Reproduced with permission. Copyright (2007) by John Wiley and Sons.

1.8

Wettability Liquid drops in equilibrium exhibit a certain apparent contact angle with the underlying surface (Fig. 1.2). If the contact angle is less than 90◦ , the surface is partially wettable, if more than 90◦ , it is non-wettable. In the extreme case of zero contact angle (water on clean glass), the surface is fully wettable. In another extreme case of contact angles greater than 150◦ , the surfaces are called superhydrophobic. At the angle tips in Figs. 1.2a and 1.2b three phases, liquid, gas and solid, are in contact (see the sketch in Fig. 1.3a), and accordingly, the force balance projected onto the horizontal direction yields Young’s equation (Butt et al. 2013, de Gennes et al. 2004) cos θ =

σSG − σSL σ

(1.44)

which determines the equilibrium contact angle θ through the surface tension coefficients (the surface energies) σSG , σSL and σ =σLG corresponding to the solid–gas, solid– liquid and liquid–gas interfaces (see Fig. 1.3a). It should be emphasized that the contact line equilibrium implies not only the horizontal balance of Eq. (1.44) but also the force balance in the vertical direction. The latter implies an inevitable deformation of solid substrates, which is hardly visible when they are stiff, but is visible on soft substrates where a wetting rim appears (Chen et al. 2013). Young’s equation implies an absolutely smooth substrate, which consists of a single material. The experimental evidence shows that roughness and different impurities present at the surface or inhomogeneities embedded in it can significantly change the equilibrium contact angle. The correction in the case of the rough surfaces is introduced in the form of the Wenzel equation (Butt et al. 2013) cos θrough = R cos θ

(1.45)

where θrough is the apparent contact angle at a rough surface, and R ≥ 1 is the ratio of the real surface area of the rough substrate to the projected one (see Fig. 1.3b). This equation shows that roughness always diminishes the apparent contact angle.

20

Introduction

Figure 1.3 (a) Contact line of a drop in equilibrium on a perfectly smooth and homogeneous

surface. (b) Drop in the Wenzel state on a rough surface. (c) Drop in the Cassie–Baxter state on an inhomogeneous surface.

On a surface with impurities, the contact angle is predicted using the Cassie–Baxter equation (Butt et al. 2013) cos θinhomogeneous = f1 cos θ1 + f2 cos θ2

(1.46)

where f1 and f2 are the relative parts of the surface occupied by materials 1 and 2, respectively ( f1 + f2 = 1), and θ1 and θ2 their corresponding equilibrium contact angles when the surface would be entirely occupied by each one of them, determined by Eq. (1.44); see Fig. 1.3c. If a surface has grooves filled with a fluid, flow of another fluid over such a surface happens in the Cassie–Baxter state. It reveals an effective slip length of the primary (the outer) fluid affected by viscosities of both fluids, including the one in the grooves (Schönecker et al. 2014).

1.8 Wettability

21

Figure 1.4 A hydrophilic island (blank) in a hydrophobic material (dashed), and a hydrophobic

island (dashed) in a hydrophilic material (blank).

Air can be considered as an extremely superhydrophobic material, since small drops are always spherical, i.e. have θ1 =180◦ . If one of the materials present at the surface is air trapped in the surface grooves, the first term on the right-hand side in Eq. (1.46) is negative, and if f2  f1 and material 2 is hydrophobic, cos θinhomogeneous → −1 according to Eq. (1.46). Thus such a rough surface with the entrapped air in the grooves will behave as superhydrophobic, which is observed for example on Lotus leaves. If the roughness consists of two disparate length scales, minimizing the contact area of deposited particles on the surface, then the surface can also exhibit self-cleaning properties, the so-called Lotus effect. It should be emphasized that both Wenzel and Cassie–Baxter equations (1.45) and (1.46) attribute the value of the apparent contact angles on rough and/or inhomogeneous surfaces to contact area between the drop and substrate and thus imply the importance of surface energy. Gao and McCarthy (2007) in their seminal paper criticized this assumption and thus the entire foundations of the Wenzel and Cassie–Baxter equations (1.45) and (1.46). The latter, for example, implies that an island of a material occupying practically the entire drop footprint but not touching the contact line, could significantly affect the contact angle value, even though the entire contact line is located on the surrounding material (Fig. 1.4, left and right). In a series of brilliantly simple experiments, Gao and McCarthy (2007) showed that this is not true. As a result, they made a point that the Wenzel and Cassie–Baxter equations (1.45) and (1.46) can work only when the overall surface roughness and composition resemble the ones of the areas in direct contact with the contact line. While the above discussion essentially should deal with the idealized equilibrium contact angle θ defined by Young’s equation (1.44), experimentally this angle is often difficult to resolve. More often only an apparent contact angle can be measured, as illustrated in Fig. 1.5.

22

Introduction

Fluid Liquid

θac

θap

Solid Figure 1.5 Illustration of actual (θac ) and apparent contact angle (θap ) on rough surfaces (Tropea et al. 2007). With permission of Springer.

1.9

Rheological Constitutive Equations of Non-Newtonian Fluids and Solids In the present section a brief survey of several generic phenomenological rheological constitutive equations (RCEs) of incompressible non-Newtonian fluids and solids is given. Steady-state simple shear flows were traditionally used for measuring fluid viscosity as the ratio of the shear stress τxy to an imposed shear rate γ˙ . For a number of liquids and gases the ratio μ = τxy /γ˙ appears to be constant in a wide range of γ˙ variation. That permits μ to be considered as a basic rheological parameter of such fluids and these fluids are called Newtonian. For them the tensorial rheological constitutive equation (1.22) was constructed such that it reproduces the relation following from the corresponding simple shear experiments, namely τxy = μγ˙ . From the beginning of the twentieth century a growing number of fluids were found for which the ratio τxy /γ˙ varied with shear rate γ˙ , even in simple shear flow. The corresponding relation in many cases could be approximated in a certain range of variation of γ˙ as τxy = K(γ˙ )n .

(1.47)

In Eq. (1.47) two basic rheological parameters appear: the consistency and flow behavior indexes, K and n, respectively. In this case the variable shear viscosity μsh = τxy /γ˙ is not a constant but depends on the flow field μsh = K(γ˙ )n−1 .

(1.48)

Such fluids are known as non-Newtonian fluids. Following the phenomenological approach, a tensorial RCE for the deviatoric stress tensor of such fluids was designed to reproduce Eq. (1.47) in simple shear flows. Namely, τ = 2K[2tr(D 2 )](n−1)/2 D

(1.49)

with D being the rate-of-strain tensor, and the stress tensor given by σ = −pI + τ .

(1.50)

Here and hereinafter I denotes the unit tensor. Equation (1.49) corresponds to the so-called Ostwald–de Waele or power-law fluids. For n < 1 the shear viscosity of such fluids decreases as the shear rate γ˙ increases [see Eq. (1.48)] and they are called pseudoplastic or shear-thinning fluids. If n > 1, the shear viscosity of a power-law fluid increases as γ˙ increases, and such fluids are called dilatant

1.9 Rheological Constitutive Equations

23

or shear-thickening fluids. The pseudoplastic behavior is found much more frequently than the dilatant one. In the case of n = 1 and K = μ, the RCE of the power-law fluids (1.49) reduces to the Newtonian RCE (1.22). Several non-Newtonian fluids, e.g. carbopol solutions or clay suspensions reveal the presence of the yield stress τ0 in simple shear flows, i.e. the behavior which corresponds to the following phenomenological expression for the shear stress τxy = τ0 + K(γ˙ )n

(1.51)

which generalizes Eq. (1.47) even further. Equation (1.51) involves three rheological parameters, the yield stress in shear τ0 , and the consistency and flow behavior indexes, K and n, respectively. This model is termed as the Herschel–Bulkley law. The Bingham plastics RCE corresponds to a particular case of Eq. (1.51), namely, n = 1 and K = μ. When Bingham plastics flow in simple shear flow, i.e. at τxy > τ0 , the shear stress in them becomes τxy = τ0 + μγ˙ .

(1.52)

Moreover, in most cases τxy is sufficiently large compared to τ0 , and the Bingham plastics flow behavior approximately reduces to the one of Newtonian viscous fluids with τxy = μγ˙ . The simplest tensorial RCE for the Herschel–Bulkley fluids is (Macosco 1994)   τ0 (n−1)/2 D for IIτ > τ02 , + K|II | (1.53) τ =2 2D |II2D |1/2 where τ0 , K and n are the basic (invariant, i.e. flow-independent) rheological parameters, and IIX represents the second invariant of a tensor X. Equation (1.53) is designed to reproduce Eq. (1.51) in steady-state simple shear flows. The tensorial RCEs for the power-law and Herschel–Bulkley fluids can describe relatively accurately flow curves measured in steady-state simple shear flow, which allows determination of their basic rheological parameters K, n and τ0 . However beyond that, these equations are quite limited at describing any other flow of the same fluid with the same set of the rheological parameters (Yarin et al. 2014). It is well known that many polymer solutions and melts with flexible macromolecules reveal a shear-thinning flow behavior in steady-state simple shear flow, which implies that they obey the power-law RCE (1.47)–(1.49) with n < 1. However, in sufficiently strong uniaxial elongational flows, where a thread of polymer solution is stretched under the action of the capillary pressure (Stelter et al. 2000) or in a clamp machine stretching a polymer melt thread, the effective elongational viscosity does not reveal the shear-thinning pattern. On the contrary, the elongational viscosity can increase then with the stretching rate, which reveals “shear-thickening” with n > 1 for the same fluid. This contradiction shows that neither the power-law RCE (1.47)–(1.49), nor the Herschel–Bulkley RCE (1.51)–(1.53) are capable of describing rheological behavior of solutions and melts of polymers with flexible macromolecules. It should be emphasized that the simple shear and uniaxial elongational flows correspond to the two opposite kinematical cases, with all other possible flows being a combination of them both. Only when flow behavior of a fluid can be

24

Introduction

described by the same rheological constitutive equation with the same set of the basic rheological parameters in the two utmost kinematics limits, can this RCE be assumed to be universally valid and probably applicable to the other flows with the intermediate kinematics. Newtonian fluids (gases, water, glycerol, different oils, etc.) reveal such an example. In particular, those of them which are viscous enough to make a thread for the uniaxial elongation experiments driven by capillary thinning under the action of surface tension, closely follow there the predictions of the Newton–Stokes RCE and reveal the same viscosity as the one measured in the simple shear experiment (McKinley and Tripathi 2000). There are a few examples of validity of the power-law RCE (1.49) with an invariant set of the rheological parameters K and n for the same fluid in simple shear and uniaxial elongational flows (suspensions of γ -Fe2 O3 particles in oil, some aqueous suspensions of clays, gelled propellant simulants and gypsum slurries; Yarin 1993, Yarin et al. 2004, Sinha-Ray et al. 2011), albeit such examples are rare. For the yield-stress fluids, Tiwari et al. (2009) tested the Herschel–Bulkley model (1.53) simultaneously against the data for the simple shear flow and uniaxial elongational flows of concentrated suspensions of carbon nanotubes in castor oil or in blends of n-decane with castor oil. They showed that RCE (1.53) is incapable of describing the shear and elongational data of the same material with the same set of parameters; hence the Herschel–Bulkley model can be considered only as a very crude approximation in that case. Overall, the inability of an RCE to describe the flow behavior of the same fluid with a single invariant set of the rheological parameters in steady-state and uniaxial elongation experiments is an alarm that such an RCE is flawed, even though it seemingly “works” in the simple shear. It should be emphasized that in the case of drop impact onto a solid wall, liquid elements undergo strong azimuthal stretching in the spreading liquid lamella in addition to shear and a need to search for an adequate uniformly valid non-Newtonian RCE is acute. The failure of the power-law or the Herschel–Bulkley RCEs (1.49) or (1.53) to describe simultaneously simple shear and uniaxial elongation of solutions and melts of flexible polymers was recognized long ago and triggered an extensive search for another RCE which will be uniformly valid. The early experiments (see, for example, Bird et al. 1987) revealed that polymeric liquids possess elasticity in addition to viscosity, i.e. they are viscoelastic liquids. Therefore, the inelastic power-law or the Herschel–Bulkley RCEs (1.49) or (1.53) are inherently flawed, and miss the most important physical phenomenon in the rheological behavior of polymer solutions and melts. The physical origins of the elasticity of polymer solutions and melts are related to the tendency of flexible polymer macromolecules to change their conformation due to the inevitable thermal fluctuations (de Gennes 1979, Doi and Edwards 1986, Yarin et al. 2014). In brief, elasticity of polymeric liquids stems from the tendency of polymer macromolecules to coil, or their tendency to reorient themselves via reptations in concentrated solutions and melts. Both tendencies are thermodynamic in nature and, according to the second law of thermodynamics, release the internal structure of polymeric liquids from any order imposed by their deformation due to flow. Here we concentrate on the phenomenological description of viscoelasticity.

1.9 Rheological Constitutive Equations

25

The elastic effects are related to a material “memory.” For example, a stretched spring always “remembers” that it was shorter, and resists stretching via the internal elastic stresses. The rheological response of the linearly elastic solid materials is described by Hooke’s law (Landau and Lifshitz 1970) σ =2Gε+I

2GνP trε 1 − 2νP

(1.54)

where the two rheological parameters, G and νP , are the shear modulus and Poisson’s ratio; note that instead of G, Young’s modulus E can be used, since G = E/[2(1 + νP )]. The strain tensor ε = (∇u + ∇uT )/2, where u is the displacement vector. In the incompressible linearly elastic solids νP =1/2, trε = 0, and the isotropic, second term on the right-hand side in Eq. (1.54) reduces to pressure, i.e. the stress is given by Eq. (1.50) with the deviatoric stress tensor τ =2Gε.

(1.55)

It should be emphasized that the linearly elastic Hookean bodies have an infinite “memory,” which means that they “remember” their unloaded reference states forever, and always tend to return to them (i.e. resist stretching forever). The “memory” of a Hookean elastic solid is embedded in the strain tensor ε, which links a current loaded configuration to an unloaded reference state in the past. Description of “memory” effects in viscoelastic liquids undergoing large nonlinear deformations involves additional kinematic tensors which act between reference and current states of material. Neither Newtonian fluids (1.22), nor power-law or Herschel– Bulkley fluids (1.49) or (1.53) possess any “memory,” and accordingly, the only kinematic tensor required to describe their rheological behavior is the rate-of-stress tensor D =(∇v + ∇vT )/2, where the tensor gradient of velocity ∇v is fully determined by the flow field at the current moment of time. Therefore, to formulate phenomenological RCEs of viscoelastic fluids, there is a need for additional kinematic tensors. They are introduced as follows. Consider an infinitesimally small material element represented by the vector dX (τ ) corresponding to a reference configuration at time τ (any time moment in the past). It is emphasized that the material element is embedded in the matrix consisting of the same material. Let the element undergo deformation, which is not necessarily small, during a finite time interval t − τ , with t being the current time. Then, the material element acquires a new configuration corresponding to vector dX (t ). The link of the current and reference configurations is produced by the gradient-of-deformation tensor Fτ (t ) dX (t ) = F τ (t ) · dX (τ ).

(1.56)

The tensor Fτ (t ) is asymmetric and cannot be directly related to a symmetric deviatoric stress tensor, which we are heading to construct. A symmetric Green tensor Bτ (t ) is thus introduced as Bτ (t ) = Fτ (t ) · [Fτ (t )] T

(1.57)

26

Introduction

with the notation [Fτ (t )] T used for the transposed tensor. Note that Fτ (τ ) = Bτ (τ ) = I. The gradient-of-deformation tensor Fτ (t ) evolves in time according to the following equation (Astarita and Marrucci 1974) DFτ (t ) (1.58) = ∇v(t ) · Fτ (t ) Dt with D/Dt being the material time derivative, and ∇v(t ) being the tensor gradient of velocity at time t. Equations (1.57) and (1.58) reveal the following differential equation describing the evolution of the Green tensor in time DBτ (t ) =∇v(t ) · Bτ (t ) + Bτ (t ) · [∇v(t )] T . (1.59) Dt The Green tensor, by its introduction in Eq. (1.57), is associated with transformations of lengths of linear material elements or areas of material platelets from a reference to an actual current configuration (Astarita and Marrucci 1974). The elastic stresses are determined by such transformations. Accordingly, a nonlinear incompressible neoHookean elastic body possesses the following deviatoric stress tensor τ = G[Bτ (t ) − I].

(1.60)

The relation between the linear Eq. (1.55) and the nonlinear one, (1.60), can be illustrated considering small deformations. For example, in the case of an axisymmetric cylindrical thread of a neo-Hookean body (1.60), which undergoes uniaxial elongation along its axis Ox with a constant rate of stretching ε, ˙ Eq. (1.59) yields the axial and radial components of the Green tensor Bxx and Byy in the following form Bxx = exp[2ε(t ˙ − τ )],

Byy = exp[−ε(t ˙ − τ )].

(1.61)

Here y is the radial coordinate. Since the strains are small, εxx = ε(t ˙ − τ )  1. Then, Eqs. (1.61) are linearized to yield Bxx = 1 + 2εxx ,

Byy = 1 − εxx .

(1.62)

Then, the axial and radial stresses are found from Eqs. (1.50), (1.60) and (1.62) as σxx = −p + 2Gεxx ,

σyy = −p − Gεxx

(1.63)

which is exactly the result following from the linear Hooke’s law for an incompressible elastic body, Eqs. (1.50) and (1.55). In distinction from the Hookean or neo-Hookean elastic solids, polymeric liquids possess fading “memory,” which means that stresses in them relax in time, even though they are deformed (Astarita and Marrucci 1974, Bird et al. 1987, Yarin 1993, Yarin et al. 2014). Accordingly, in a certain time, the stretching forces can be removed and a polymeric liquid will remain stretched forever, for example a stretched polymeric thread will stay stretched. In the framework of the phenomenological approach it is natural to assume that any reference configuration of material in the past τ is forgotten at the current moment t with

1.9 Rheological Constitutive Equations

27

the exponential fading memory, exp[−(t − τ )/θ ], with θ being a physical parameter of material, its relaxation time. Accordingly, the RCE of the neo-Hookean body (1.60) is modified to yield

t G[Bξ (t ) − I] exp[−(t − ξ )/θ ]dξ (1.64) τ (t ) = −∞ t −∞ exp[−(t − ξ )/θ ]dξ where the dummy variable is denoted ξ . Due to dimensional considerations, viscosity μ = Gθ , which shows that a viscoelastic material emerges from the combination of the elastic and relaxation effects, while Eq. (1.64) yields the following integral RCE  μ t [Bξ (t ) − I] exp[−(t − ξ )/θ ]dξ . (1.65) τ (t ) = 2 θ −∞ By differentiating this equation by t and using the kinematic Eq. (1.59), an equivalent differential RCE of viscoelastic liquid is obtained τ (t ) 2μ Dτ (t ) = ∇v(t ) · τ (t ) + τ (t ) · [∇v(t )] T + D(t )− Dt θ θ

(1.66)

where all the terms on the right-hand side depend only on the current time moment t. Note that the equivalence of the integral and differential RCEs (1.65) and (1.66) was established by Lodge (1964). Equation (1.66) is called the Upper-Convected Maxwell model, UCM, and the origin of this name can be found in the monographs of Astarita and Marrucci (1974) and Bird et al. (1987). The phenomenological approach to viscoelasticity outlined in the present section is inevitably rather arbitrary. Not only the Green tensor can be linked to the transformation of lengths of linear material elements or areas of material platelets under deformations. The Cauchy tensor Cτ (t ) = [Fτ (t )] T · Fτ (t )

(1.67)

is also linked to such transformations and is symmetric, albeit being different from the Green tensor. Therefore, the viscoelastic RCEs, Eqs. (1.65) and (1.66) are not unique, and an alternative formulation based on the Cauchy tensor is also possible. A phenomenological RCE can be supported to some extent by a plausible micro-mechanical model, and a better comparison with the available experimental data than an alternative RCE. Multiple RCEs of viscoelastic polymeric liquids are discussed from different perspectives in the monographs of Astarita and Marrucci (1974), Bird et al. (1987) and Larson (1988). The Deborah number De compares the elastic relaxation time θ with the characteristic flow time τf , i.e. De=θ /τf , see Eq. (1.7) in Section 1.2 in this chapter. For slow processes, in the so-called weak flows, the Deborah number De → 0, and the Upper-Convected Maxwell model (1.66) reduces to the viscous Newtonian fluid with τ = 2μD given by Eq. (1.22). It is easy to see that in the other limit of the strong flows with De → ∞, the Upper-Convected Maxwell model (1.66) reduces to the neo-Hookean elasticity described by Eq. (1.60).

28

Introduction

The elastic forces in polymeric liquids are associated with the tendency of flexible macromolecular coils to return to their most disordered most probable configuration, namely to a macromolecular coil. Any deformation of macromolecules due to the macroscopic flow distorts macromolecular coils. Since these distorted conformations are less probable on their own, they correspond to a reduced entropy S. As a result, the distortions of macromolecular configurations due to flow increase the Helmholtz free energy F = U − T S, where U is the internal energy and T is temperature. It should be emphasized that the internal energy of polymeric liquids is practically constant. Therefore, when the distorted macromolecules resist deformations, which is expressed in the tendency to diminish the free energy F , they tend to increase the entropy, and thus, the corresponding restoring elastic force associated with dF , is rooted in the entropy change −dS. The latter means that the elasticity of polymeric liquids is of entropic origin (de Gennes 1979, Doi and Edwards 1986). In addition, the intermolecular friction in flowing polymeric liquids or friction between macromolecules and a solvent are responsible for the viscosity of polymeric liquids (de Gennes 1979, Doi and Edwards 1986). Those are the physical origins of the Upper-Convected Maxwell model (1.65) and (1.66). The physical foundations of the rheological behavior of metals are completely different from those of the polymeric liquids. From a microscopic point of view metals possess an ordered crystalline atomic structure which also includes defects and dislocations. For small deformations of a metal from its unloaded state, the crystalline structure is slightly distorted, which causes an increase in the internal energy U , while the entropy practically does not change. Therefore, the Helmholtz free energy F , given the thermodynamic tendency to diminish it, i.e. the elastic resistance to small deformations, is associated with the changes in the internal energy dU . That is the physical origin of Hooke’s law (1.54). When deformations become larger, irreversible processes in metals are associated with creation and motion of the new and pre-existing dislocations through the atomic arrays (Landau and Lifshitz 1970). Such irreversible processes cause macroscopic flows of metals and are associated with plasticity. They are strain-rate dependent, in distinction from the elasticity, which is rate independent. Dissipative plastic flows of metals (the viscoplastic processes) cause stress relaxation in metals. Even though the physical foundation of the rheological response of viscoelastic polymeric liquids and the elastic-viscoplastic metals are completely different, formally both groups of materials possess certain elasticity and viscosity/plasticity. Therefore, it might be expected that on the phenomenological level, there might be a certain similarity between RCEs of viscoelastic polymeric liquids and the elastic-viscoplastic metals. Indeed, Rubin and Yarin (1993, 1995) revealed that the Upper-Convected Maxwell model (1.66) is formally close to the RCE of incompressible elastic-viscoplastic metals which can be rearranged to the following form 2 Dτ (t ) = ∇v(t ) · τ (t ) + τ (t ) · [∇v(t )] T − [τ (t ) : D(t )]I + 2GD(t )−τ (t ). Dt 3 (1.68) Here G is the same shear modulus as in Hooke’s RCE for an incompressible elastic solid (1.55), and the elastic-viscoplastic response  is rate-dependent. For large slow

1.10 Instabilities and Small Perturbations

29

deformations of the elastic-viscoplastic metals Eq. (1.68) reduces to G τ =2 D 

(1.69)

which is reminiscent of Eq. (1.22) for the Newtonian viscous fluids. However,  is not a constant for metals, but is rate-dependent. In the case of the so-called perfect plasticity model it is chosen such that Eq. (1.69) takes the following form

  Y 2 1/2 D (1.70) τ = 3 (D : D)1/2 where Y is the yield stress for uniaxial tension considered to be a constant physical material parameter. It should be emphasized that the plastic rheological response associated with Eq. (1.70) is completely different from that of a seemingly similar-looking RCE of the Ostwald–de Waele or power-law fluid (1.49). Any impact or collision or any other problem of fluid or solid mechanics is solved using the momentum balance equation (1.14) (or its static counterpart) with the corresponding rheological constitutive equation for the deviatoric stress τ . In the case of the simplest rheological models, like the inviscid fluid [Eq. (1.15)], or viscous Newtonian fluid [Eq. (1.22)], the algebraic RCE for τ is substituted in Eq. (1.14), and then the Euler equation (1.16) or the Navier–Stokes equations (1.24) arise. Similarly, in the case of the linearly elastic solids Hooke’s law (1.54) is substituted in the momentum balance Eq. (1.11), which results in the differential equations of the theory of elasticity (Landau and Lifshitz 1970). However, in the case of viscoelastic or elastic-viscoplastic rheological models, the deviatoric stress τ is described by partial differential equations [Eqs. (1.66) or (1.68)], which are solved simultaneously with the momentum balance (1.14), with the components of tensor τ considered as independent variables.

1.10

Instabilities and Small Perturbations: Rayleigh Capillary Instability, Bending Instability, Kelvin–Helmholtz Instability, Rayleigh–Taylor Instability Many formal solutions of the equations of mechanics, in general, and hydrodynamics, in particular, are not realizable in reality, even though the boundary and initial conditions are practically the same as those considered theoretically. The real conditions are always slightly different than the idealized ones which are assumed in the theory. The difference results from small perturbations which are always present in reality. Only those solutions which are practically not disrupted by these small perturbations have a chance to survive and resemble reality. For example, all solutions for laminar flows are formally valid at any values of the Reynolds number. However, in reality they are only observed at sufficiently small Reynolds numbers, and are replaced by totally different flows at Reynolds numbers above certain critical threshold values. At these critical Reynolds numbers laminar flows become unstable and are replaced by turbulent flows.

30

Introduction

(a)

(b)

(c)

Figure 1.6 (a) Rim instability on top of a crown resulting from drop impact. Relatively regular

undulations issue short jets which, in turn, issue individual secondary droplets [Courtesy of Harold & Ester Edgerton Foundation and Palm Press, Inc.]. (b) Irregular jetting from top of a crown after drop impact on a thin liquid layer on a wall. Yarin and Weiss (1995). [Courtesy of Cambridge University Press]. (c) Irregular jetting from a rim on a film: relatively long jets break up into secondary droplets [Courtesy of M. O. Kornfeld].

When stability to small perturbations is studied, in distinction from global stability, linearization of the stability problem is admissible, and neglecting the higher order perturbations is possible. Solutions of linearized equations for perturbations allow superposition of individual modes, which do not interact and exchange energy. The relative simplicity of the linear analysis of small perturbations makes it a popular tool and explains why multiple important results have been obtained in this manner. Several of these generic results are also important in the context of drop impact onto solid walls and liquid surfaces. In particular, the rim surrounding the top of a crown resulting from drop impact onto a thin liquid film is developing visible undulations (Fig. 1.6a). Many of these undulations issue upward jets seen in Figs. 1.6a–c, and these jets, in turn, break up into tiny secondary droplets. This fascinating cascade of hydrodynamic instabilities deserves explanation. In addition, the formation of tiny secondary droplets results in secondary fine atomization (see Section 8.2), which might facilitate fuel combustion in Diesel engines, or be detrimental due to coolant losses in spray cooling. Some of the relevant generic instabilities are briefly discussed in the present section.

1.10 Instabilities and Small Perturbations

1.10.1

31

Rayleigh Capillary Instability The foundations of stability theory of free liquid jets were laid down in the seminal work of Lord Rayleigh (1878). This theory implies that liquid jets break up into droplets due to the instability of an infinite liquid thread of circular cross-section of radius a0 under the action of surface tension. Rayleigh’s result can be obtained as a solution of the continuity and Euler equations, (1.10) and (1.16) (neglecting gravity, a = g = 0) in cylindrical coordinates r, θ and z, after linearizing them for small perturbations and accounting for the linearized boundary conditions at the jet surface (Lamb 1959). On an infinite thread, small perturbations inevitably depend on time t and the azimuthal and axial coordinates θ and z as exp(γ t − ikz + isθ ), and the characteristic equation has the form  σ χ Is (χ ) (1 − s2 − χ 2 ) (1.71) γ =± ρa30 Is (χ ) where γ is the perturbation increment (the growth rate), χ = 2π a0 / is the dimensionless wavenumber, with  being the wavelength, s is the azimuthal wavenumber, Is (χ ) are the modified Bessel functions of the s-th order, and i is the imaginary unit. In the case of the axisymmetric perturbations s = 0, and the radicand in Eq. (1.71) is positive for χ < 1. Then, one of the values of the perturbation increment γ is real and positive. This means that a solution for perturbations exponentially increasing in time exists. Therefore, liquid threads are unstable to axisymmetric perturbations with the wavelength  > 2π a0 (equivalent to χ < 1). The unstable perturbations reveal a system of periodic waves standing on a thread with the amplitudes growing exponentially in time. Rayleigh’s result means that the thread (and jet) configuration, which does not correspond to a minimum of the surface energy, is unstable. As a result of growth of the longwave (χ < 1) perturbations, the surface area and energy diminish, which results in the capillary breakup of a jet into droplets [this explanation of the capillary breakup is due to Plateau (1873)]. Note that the absolute minimum of surface energy would correspond to a liquid sphere, whereas a local minimum is given by a succession of liquid spheres (droplets) emerging from a cylinder (jet/thread). According to Eq. (1.71), the increment maximum γ∗ corresponds to the wavenumber χ∗ = 0.698. Rayleigh assumed that jet breakup is driven by the perturbation with the corresponding wavelength ∗ = 2π a0 /χ∗ ≈ 9a0 as in the case of an infinite liquid thread with the same cross-sectional diameter. The comparison of Rayleigh’s spectrum (1.71) with the experimental data shown in Fig. 1.7 is excellent. If one assumes that during the entire process of capillary breakup of liquid jets the linear stability analysis is accurate enough even at the stage when perturbation amplitudes become finite, it is possible to evaluate the breakup time T during which the amplitude becomes equal to the unperturbed radius of the jet, and the breakup length of the jet L T =

ln(a0 /δ0 ) , γ∗

L = V0 T.

(1.72)

In Eq. (1.72) δ0 is the amplitude of the initial perturbation which is unknown, as a rule, and ln(a0 /δ0 ) is taken in the range 10–14 to fit the experimental data for L; V0 is the velocity of the unperturbed jet.

32

Introduction

Figure 1.7 Dimensionless growth rate versus wavenumber. Rayleigh’s result (1.71) is shown by

the curve. Open symbols show the data of Donnelly and Glaberson (1966), filled symbols the data of Cline and Anthony (1978) (Yarin 1993). Reprinted with the permission of Pearson Education, Inc.

In the case of non-axisymmetric perturbations (s ≥ 1), the increment determined by Eq. (1.71) will be imaginary for all wavelengths and a liquid thread (jet) remains stable. In this case the perturbations represent traveling waves. Rayleigh’s theory of capillary breakup was extended to the case of viscous fluids in the work of Weber (1931). He solved the linearized Navier–Stokes equations (1.10) and (1.24) (with a = g = 0) for the case of small perturbations of an infinite liquid thread. In the most important case of longwave axisymmetric perturbations his result for the perturbation growth rate reduces to the following dimensionless expression  3 9 2 4 Ohχ 2 γ μa0 2 = − Ohχ + Oh χ + (1 − χ 2 ) (1.73) σ 2 4 2 which generalizes Rayleigh’s result (1.71) for the case of viscous fluids.  The fastest growth rate corresponds to χ∗ = [2(1 + 3 Oh/2)]−1/2 with the Ohnesorge number Oh = μ2 /(ρσ a0 ). Note that the Ohnesorge number of Eq. (1.1) in Section 1.2 in this chapter is the square root of the present one, with the drop diameter D standing instead of the cross-sectional radius a0 . The shortwave axisymmetric perturbations with χ > 1 always diminish for any finite viscosity and are unimportant in the framework of the capillary instability. The comparison of the predictions of the Rayleigh–Weber theory with the experimental data for the breakup length (1.72) shows a good agreement in the linear section BC of the experimental curve L = L(V0 ) in Fig. 1.8. Section AB corresponds to dripping of relatively short jets at low exit velocity, which cannot be considered similarly to the “infinite” threads. Section CD and the appearance of the maximum are associated with the dynamic effect of the surrounding gas (e.g. air) on the growth of the axisymmetric capillary perturbations at sufficiently high exit velocities V0 . A detailed review of

1.10 Instabilities and Small Perturbations

33

Figure 1.8 Empirical dependence of Newtonian jet length on its exit velocity (Yarin 1993).

Reprinted with the permission of Pearson Education, Inc.

the capillary instability of liquid jets and different factors that affect it can be found in Ashgriz and Yarin (2011). Beyond the maximum D in Fig. 1.8, low-viscosity liquid jets typically either atomize due to the Kelvin–Helmholtz instability discussed below, or become turbulent, which is equivalent to a significant growth of the initial perturbation level δ0 as velocity increases. As a result, such jets become shorter up to the velocity corresponding to point E. At higher velocities, the eddy viscosity significantly increases and jets of low-viscosity fluids can be stabilized and become longer (section EF in Fig. 1.8) until ultimately atomization prevails. Liquid jets with sufficiently high viscosity do not become turbulent beyond point D. However, due to the interaction with the surrounding gas, the leading perturbation mode changes and amplified bending waves appear, resulting in section DE in Fig. 1.8. The bending instability is discussed briefly in the following subsection.

1.10.2

Bending Instability Thin liquid jets demonstrate not only capillary breakup but also some other regular longwave forms of instability and breakup, e.g. bending instability of jets moving in gas with relatively high speed V0 . Theoretical investigations of the dynamics of the bending instability due to small perturbations of liquid jets rapidly moving in gas were begun by Weber (1931) and Debye and Daen (1959). This led to a coupled problem of the dynamic interaction of gas flow with a bending liquid jet. The linear stability analysis of the bending perturbations on an infinite inviscid jet can be carried out using the continuity equation (1.10) and the linearized version of the Euler equation (1.16) with a = g = 0. Debye and Daen (1959) obtained an expression for the perturbation growth rate. Neglecting several minor terms, this expression takes the form  γ =χ −





ρgV02 K1 (χ )I1 (χ ) σ χ I1 (χ ) . − ρa20 K1 (χ )I1 (χ ) ρa30 I1 (χ )

(1.74)

34

Introduction

In addition to the previously introduced notation, the gas density is denoted by ρg ; I1 and K1 denote the modified Bessel functions. In the case of bending perturbations the surface tension is a stabilizing factor, since bending results in an increase of the jet surface area [I1 (χ ) > 0, K1 (χ ) < 0 for any χ ]. Above a threshold value of the relative gas velocity V0 , the first (positive) term in the radicand in Eq. (1.74) acquires a larger magnitude than the second term, which corresponds to the onset of the bending instability and an exponential growth of the bending perturbations. The bending instability is determined by the pressure distribution in gas over the jet surface: the gas pressure on convex surface elements is lower than on the concave ones. In the case of highly viscous jets, bending perturbations are expected to have long wavelengths and for small perturbations a linearized version of the quasi-onedimensional equations of the jet dynamics can be employed (Yarin 1993). The characteristic equation for the growth rate of small bending perturbations of highly viscous slender “infinite” liquid jets moving in gas reads   ρgV02 3 μχ 4 σ χ 2 = 0. γ + − (1.75) γ2 + 4 ρa20 ρa20 ρa30 Both planar and three-dimensional (helical) small bending perturbations increase with the growth rate predicted by Eq. (1.75) if the relative velocity of gas flow  (1.76) V0 > σ /(ρga0 ) i.e. when the dynamic action of gas can overbear the resistance of surface tension to the growth of bending perturbations. Then, according to Eq. (1.75), there is always a real positive solution for γ , with a maximum in the longwave range 0 < χ < 1 corresponding to the fastest growing mode. In comparison, the growth rate of the axisymmetric capillary perturbations is much smaller than that of the bending perturbations for sufficiently viscous liquids when the inequality μ2

1 ρa20 ρgV02

(1.77)

holds. In this case deformations of the jet due to the capillary Rayleigh–Weber instability can be neglected during bending. It is worth noting that in the case of inviscid liquids when μ=0, Eq. (1.75) coincides with the longwave limit (χ → 0) of Eq. (1.74). The breakup length of jets in the case of the aerodynamically driven bending instability is determined by the following expression (Yarin 1993) 1/3  3μρa20V03 (1.78) L = ln(ma0 /δ0 ) (ρgV02 − σ /a0 )2 where m =2–4. The value of the factor m is chosen in agreement with the experimental data and the energy estimates, which show that as the bending perturbation amplitude reaches the value of the order of a few cross-sectional radii, the jet is almost immediately squeezed by the gas pressure difference at its surface. It should be emphasized that Eq. (1.78) predicts a decrease in the jet breakup length as the jet velocity V0 increases, which agrees with the experimental data (see section DE in Fig. 1.8). A detailed review

1.10 Instabilities and Small Perturbations

35

of different aspects of the bending instability of liquid jets rapidly moving in gas can be found in Yarin (2011).

1.10.3

Kelvin–Helmholtz Instability Relative motion of two immiscible fluids along their common interface can cause another type of instability, known as the Kelvin–Helmholtz instability. Consider two layers of inviscid fluids with densities ρ1 and ρ2 . The layers are assumed to be infinitely long in the x-direction, with x being the coordinate along the unperturbed interface. The thicknesses of the layers are h1 and h2 and their absolute unperturbed velocities are V1 and V2 , respectively. Gravity is directed normal to the unperturbed interface from layer 2 to layer 1 against the y-axis normally to the x-axis. If the interface is perturbed by small perturbations y = ζ (x, t ) = δ 0 exp(γ t + ikx) [the exponential form is expected to be a solution in the case of small perturbations of the infinite layers; see the subsection on the Rayleigh capillary instability; δ0 is the initial amplitude], the corresponding perturbed potential flow can be found as a solution of the continuity and linearized Euler equations (1.10) and (1.16), respectively. Such a problem readily reduces to solving the Laplace equation (1.17) and using the linearized Bernoulli integral (1.18) to find the perturbed pressure field. After using the kinematic and dynamic boundary conditions at the interface between the two layers and the conditions that the outer boundaries of the layers are unperturbed (either being in contact with solid walls, or at y = ±∞), the characteristic equation relating the increment γ with the wavenumber k is found as (Lamb 1959, Kochin et al. 1964, Landau and Lifshitz 1987, Chandrasekhar 1981) ρ1 (c − V1 )2 coth(kh1 ) + ρ2 (c − V2 )2 coth(kh2 ) =

(ρ1 − ρ2 )g k

(1.79)

where c = iγ /k. The effect of surface tension is neglected, since perturbation waves are assumed to be sufficiently long gravitational waves. If both layers are sufficiently deep compared to the perturbation wavelength, coth(kh1 ) = 1 and coth(kh2 ) = 1, the solutions of Eq. (1.79) read

 ρ1V1 + ρ2V2 ρ1 ρ2 (V1 − V2 )2 g(ρ1 − ρ2 ) −i γ =k ± − . (1.80) (ρ1 + ρ2 )2 k(ρ1 + ρ2 ) ρ1 + ρ2 The solution (1.80) shows that the relative velocity of the layers is a destabilizing factor. On the other hand, gravity is a stabilizing factor, and if the lower fluid layer is denser than the upper one (ρ1 > ρ2 ) and the inequality   g ρ12 −ρ 22 2 (V1 − V2 ) < (1.81) kρ1 ρ2 holds, the instability is suppressed. Then, both solutions (1.80) for γ are imaginary, the waves are neutrally stable and correspond to traveling waves of a constant amplitude. If smaller scale motions of free liquid jets and drops are of interest, then surface tension effects (capillary pressure) should also be included. Then, Eq. (1.80) is replaced

36

Introduction

by (Kochin et al. 1964, Landau and Lifshitz 1987, Chandrasekhar 1981)



σk ρ1 ρ2 (V1 − V2 )2 g(ρ1 −ρ 2 ) ρ1V1 + ρ2V2 − γ =k ± − −i . (ρ1 + ρ2 )2 k(ρ1 + ρ2 ) ρ1 + ρ2 ρ1 + ρ2

(1.82)

For small-scale motions gravity becomes of no consequence and can be neglected. Also, typically liquid (1) is in contact with gas (2), and ρ1 ρ2 . In this case it is also convenient to associate the x-axis with the unperturbed moving liquid, i.e. to take V1 =0 and denote the relative gas velocity V2 as V . Then, the expression (1.82) reduces to  γ =k

ρ2V 2 − σ k . ρ1

(1.83)

Only one solution for γ with a positive square root is considered, since this solution will correspond to the Kelvin–Helmholtz instability. Surface tension will dominate for very short waves (with very large wavenumbers k), for which capillary pressure is larger than the dynamic pressure of gas, i.e. σ k > ρ2V 2 . Surface tension tends to minimize the surface area and energy by counteracting surface ripple formation. Then γ , given by Eq. (1.83), is imaginary, and we are still dealing with traveling waves of constant amplitude. However, when the dynamic pressure of gas dominates and ρ2V 2 > σ k

(1.84)

the increment (1.83) becomes real and positive. Then, the small perturbations correspond to standing waves with exponentially growing amplitudes. Their growth is sustained by transfer of mechanical energy from the gas flow to the liquid, which in particular allows an increase in surface area and energy, contrary to surface tension counteraction. This is the Kelvin–Helmholtz instability per se. The fastest growing wave of the Kelvin–Helmholtz instability corresponds to the maximum of γ in Eq. (1.83). Its wavenumber, wavelength and the growth rate are given by the following expressions k∗ =

2ρ2V 2 , 3σ

∗ =

3π σ , ρ2V 2

 γ∗ =

4ρ23V 6 27ρσ 2

1/2 .

(1.85)

The higher the relative velocity, the shorter are the perturbation wavelengths which grow. They might be much shorter than even the unperturbed radii of capillary jets rapidly moving in gas, i.e. ∗  a0 . Then, curvature of the unperturbed liquid surface can be neglected, and the above theory becomes fully applicable to jets. This is the case of the atomization of low-viscosity liquid jets and it corresponds to section DE in Fig. 1.8. Droplets resulting from atomization are much smaller than those resulting from the Rayleigh capillary instability (∗ ≈ 9a0 ). As an example, for a water jet moving in gas with velocity V = 100 m/s, one finds ∗ ∼ 10−2 cm and γ∗ ∼ 105 s−1 (also, see Section 8.1 in Chapter 8).

1.11 Total Mechanical Energy of Deforming Bodies

1.10.4

37

Rayleigh–Taylor Instability The interface of two adjacent fluid layers can be unstable even without relative motion when V1 = V2 = 0 and when a denser layer is on the top, i.e. ρ2 > ρ1 . Then, the real positive solution of the characteristic equation for small perturbations found from Eq. (1.80) takes the following form for sufficiently thick liquid layers  gk(ρ2 − ρ1 ) . (1.86) γ = (ρ1 + ρ2 ) In this case small perturbations are standing waves of growing amplitude. The energy for the growing perturbations is supplied by gravity. The instability of this type is called the Rayleigh–Taylor instability (Chandrasekhar 1981). The effect of surface tension is stabilizing, and accordingly, Eq. (1.82) yields, instead of Eq. (1.86), the following expression for the perturbation growth rate  gk(ρ2 − ρ1 ) σ k3 . (1.87) − γ = (ρ1 + ρ2 ) ρ1 + ρ2 The effect of viscosity can also be included (Chandrasekhar 1981). If liquid occupies domain 2, while gas occupies domain 1, then ρ = ρ2 ρ1 , and the fastest growing mode of the Rayleigh–Taylor instability corresponding to the spectrum (1.87) is given by   1/4  ρg 1/2 2 ρg3 2π , ∗ = , γ∗ = . (1.88) k∗ = 3σ (ρg/3σ )1/2 3 3σ The Rayleigh–Taylor instability is associated not only with acceleration due to gravity directed from a denser to lighter fluid. It emerges in response to the acceleration of any origin directed from a denser to lighter fluid, in particular from liquid to gas. Therefore, any liquid layer accelerated normally to its plane by gas pressure difference on its surfaces is subjected to the instability of this type (Yarin 1993). In the context of drops impacting onto a wall, any liquid element accelerating toward the surrounding gas is subjected to the Rayleigh–Taylor instability, and the explanations of fingering based on this type of instability were proposed in the literature (for the review, see Yarin 2006).

1.11

Total Mechanical Energy of Deforming Bodies: Where Is It Lost? The total mechanical energy of a system, experiencing deformation, reduces in time due to dissipative energy losses. The energy loss of a system can be rather significant. Its estimation could help to solve many important problems. In this section we discuss the energy balance approach using as an example a binary drop collision. In Jiang et al. (1992) the estimated energy loss during binary drop collisions is approximately 50% of the initial energy, which is the sum of the kinetic and surface energies of the colliding drops. This energy loss is attributed to viscous dissipation. It is surprising that the ratio of the lost energy to the initial energy of the drops, estimated in

38

Introduction

Jiang et al. (1992), does not depend on the drop viscosity. Numerous models of drop collisions are based on the consideration of the energy balance of the system in which the viscous dissipation is estimated from the assumed flow in the deforming drops. Ashgriz and Poo (1990), Qian and Law (1997), Ko and Ryou (2005) and Munnannur and Reitz (2007) used the energy balance approach for the description of the breakup process, Brazier-Smith et al. (1972) for the description of drop coalescence, Pan et al. (2009) for the estimation of the maximum spreading diameter. One common issue in all these models and generally in all the models based on the energy balance approach is that for a correct estimation of the dissipated energy one requires a correct velocity field. This means that the solution has to be known in order to correctly formulate the energy balance of the system, but not vice versa. In the modeling of drop collision problems the edge effects are the main contributors to the energy dissipation, but they are usually neglected in the energy analysis. To illustrate the problem, let us consider the simplest flow for which a correct solution of Taylor (1959) is well known, namely rim formation at the edge of a free, stationary liquid sheet of a uniform thickness h. Capillary forces lead to the formation of a propagating growing rim at the edge of the sheet. If x(t ) is the rim position and w(t ) is its volume per unit length, the mass and momentum balances yield  1/2 2σ dx = , x = L − U t, w = hU t (1.89) U ≡ dt ρh where U is the rim velocity in the x-direction. These equations correspond to the initial conditions x(0) = L and w(0) = 0. Equation (1.89) is obtained in (Taylor 1959) by balancing the capillary force applied to the rim, −2σ , with the inertia of the flow entering the rim, ρhU 2 . The average rim radius a(t ) can be estimated from geometrical considerations as a = (w/π )1/2 . The total mechanical energy of the system, which includes the sheet and the rim, is the sum of their kinetic and surface energies E = 2σ x + 2π σ a + ρwU 2 /2.

(1.90)

The final expression for the total energy E can be obtained using Eq. (1.89) as 1/4   3 1/2 32hπ 2 σ 5 2σ E = 2Lσ + t 1/2 − t. (1.91) ρ ρh The total energy decreases in time and its rate does not depend on the liquid viscosity, which is in accordance with the results of Jiang et al. (1992). The lost energy is spent on the generation of the internal flow in the rim. It is dissipated in the viscous boundary layer in the sheet near the rim and is dissipated in the rim. One can estimate the energy dissipated in the viscous boundary layer appearing in the free sheet near the rim, propagating with the constant velocity U . The length of this boundary layer is of order lBL ∼ μ/(ρU ). The scale for the viscous dissipation in the layer is  ∼ μ(U/lBL )2 . The total energy loss per unit width of the sheet is therefore EBL = lBL ht ∼ ρhU 3t, and does not depend on viscosity. It can be shown that for the

1.12 References

39

characteristic velocity U defined in Eq. (1.89) the lost energy EBL has the same form as the last term on the right-hand side of Eq. (1.91). When the rim radius a is much larger than the thickness of the viscous boundary layer inside the rim and the rate of its change a˙ ≡ da/ dt  U , the thickness of the boundary layer can again be estimated as lBL and its length along the rim surface as 2π a. The total energy increase of the rim radius can be therefore esti lost due to the 2 ˙ BL ) dt ∼ ρh3/2U 5/2t 1/2 . It can be shown with the help mated as Erim ∼ μ 2π alBL (a/l of Eq. (1.89) that the energy Erim has the same form as the second term on the right-hand side of Eq. (1.91), which also does not depend on viscosity. Let us estimate now whether the terms in Eq. (1.91) are significant in the case of binary drop collision. Denote the dominant last term on the right-hand side of Eq. (1.91) as Elost . The ratio e of the energy loss to the initial kinetic energy of both drops is e≈

6π Dmax Elost . π ρD30V02

(1.92)

The typical time of drop deformation in the experiments of Roisman et al. (2012) can be estimated as t ≈ 0.3D/V0 We1/2 , while the maximum spreading diameter is Dmax ≈ 0.3DWe1/2 . The scale for the average film thickness is estimated in Roisman et al. (2012) as h ≈ 0.39D3 /(V02t 2 ). With these values the ratio of the lost energy to the initial kinetic energy can be estimated as e ≈ 0.4. This ratio does not depend on the parameters of impact and it is comparable with the experimental result from Jiang et al. (1992). The ratio e is definitely not small enough to be neglected, as is usually done in the existing models of drop collision. Since the rim propagation is an important element which determines the deformation dynamics of the colliding drops, any energy balance considerations have to account for the significant energy losses related to the rim. Unfortunately, such energy losses have never been correctly accounted for in the existing models.

1.12

References Ashgriz, N. and Poo, J. Y. (1990). Coalescence and separation in binary collisions of liquid drops, J. Fluid Mech. 221: 183–204. Ashgriz, N. and Yarin, A. (2011). Capillary instability of free liquid jets, in N. Ashgriz (ed.), Springer Handbook of Atomization and Sprays: Theory and Applications, Springer, Heidelberg, chapter 1, pp. 3–53. Astarita, G. and Marrucci, G. (1974). Principles of Non-Newtonian Fluid Mechanics, McGrawHill, New York. Backman, M. E. and Goldsmith, W. (1978). The mechanics of penetration of projectiles into targets, Int. J. Eng. Sci. 16: 1–99. Barenblatt, G. I. (1987). Dimensional Analysis, Gordon and Breach Science Publisher, New York. Barenblatt, G. I. (2000). Scaling, Self-similarity, and Intermediate Asymptotics, Cambridge University Press. Barenblatt, G. I. (2014). Flow, Deformation and Fracture, Cambridge University Press. Batchelor, G. K. (2002). An Introduction to Fluid Dynamics, Cambridge University Press.

40

Introduction

Bird, R. B., Armstrong, R. C. and Hassager, O. (1987). Dynamics of Polymeric Liquids, Vol. 1. Fluid Mechanics, John Wiley & Sons Inc., New York. Birkhoff, G., MacDougall, D. P., Pugh, E. M. and Taylor, G. I. (1948). Explosives with lined cavities, J. Appl. Phys. 19: 563–582. Blake, T. D. (1993). Dynamic contact angles and wetting kinetics, in J. C. Berg (ed.), Wettability, Marcel Dekker, New York, pp. 251–309. Brazier-Smith, P., Jennings, S. and Latham, J. (1972). The interaction of falling water drops: coalescence, Proc. R. Soc. London Ser. A-Math. 326: 393–408. Bridgman, P. W. (1931). Dimensional Analysis, Yale University Press, New Haven. Bussmann, M., Mostaghimi, J. and Chandra, S. (1999). On a three-dimensional volume tracking model of droplet impact, Phys. Fluids 11: 1406–1417. Butt, H.-J., Graf, K. and Kappl, M. (2013). Physics and Chemistry of Interfaces, John Wiley & Sons, Weinheim. Chandrasekhar, S. (1981). Hydrodynamic and Hydromagnetic Stability, Dover Publications, New York. Chen, L., Bonaccurso, E. and Shanahan, M. E. R. (2013). Inertial to viscoelastic transition in early drop spreading on soft surfaces, Langmuir 29: 1893–1898. Cline, H. E. and Anthony, T. R. (1978). The effect of harmonics on the capillary instability of liquid jets, J. Appl. Phys. 49: 3203–3208. Cox, R. G. (1986). The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow, J. Fluid Mech. 168: 169–194. de Gennes, P.-G. (1979). Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca. de Gennes, P.-G. (1985). Wetting: statics and dynamics, Rev. Mod. Phys. 57: 827–863. de Gennes, P.-G., Brochard-Wyart, F. and Quéré, D. (2004). Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves, Springer, New York. Debye, P. and Daen, J. (1959). Stability considerations on nonviscous jets exhibiting surface or body tension, Phys. Fluids 2: 416–421. Doi, M. and Edwards, S. F. (1986). The Theory of Polymer Dynamics, Clarendon Press, Oxford. Donnelly, R. J. and Glaberson, W. (1966). Experiments on the capillary instability of a liquid jet, Proc. R. Soc. London Ser. A-Math. 290: 547–556. Dror, Y., Salalha, W., Avrahami, R., Zussman, E., Yarin, A. L., Dersch, R., Greiner, A. and Wendorff, J. H. (2007). One-step production of polymeric micro-tubes via co-electrospinning, Small 3: 1064–1073. Dussan V, E. B. (1979). On the spreading of liquids on solid surfaces: static and dynamic contact lines, Annu. Rev. Fluid Mech. 11: 371–400. Dussan V, E. B. and Davis, S. H. (1974). On the motion of a fluid-fluid interface along a solid surface, J. Fluid Mech. 65: 71–95. Edgerton, H. E. and Killian, J. R. (1954). Flash!: Seeing the Unseen by Ultra High-speed Photography, Branford, Boston. Gao, L. and McCarthy, T. J. (2007). How Wenzel and Cassie were wrong, Langmuir 23: 3762– 3765. Happel, J. and Brenner, H. (1991). Low Reynolds Number Hydrodynamics, Kluwer, Dordrecht. Hoffman, R. L. (1975). A study of the advancing interface. I. Interface shape in liquid-gas systems, J. Colloid Interface Sci. 50: 228–241. Jiang, Y., Umemura, A. and Law, C. (1992). An experimental investigation on the collision behaviour of hydrocarbon droplets, J. Fluid Mech. 234: 171–190.

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Josserand, C. and Thoroddsen, S. (2016). Drop impact on a solid surface, Annu. Rev. Fluid Mech. 48: 365–391. Kim, S. and Karilla, S. (2005). Microhydrodynamics. Principles and Selected Applications, Dover Publications, New York. Kistler, S. F. (1993). Hydrodynamics of wetting, in J. C. Berg (ed.), Wettability, Marcel Dekker, New York, pp. 311–429. Ko, G. H. and Ryou, H. S. (2005). Modeling of droplet collision-induced breakup process, Int. J. Multiph. Flow 31: 723–738. Kochin, N. E., Kibel, I. A. and Rose, N. V. (1964). Theoretical Hydrodynamics, Interscience Publishers, New York. Lamb, H. (1959). Hydrodynamics, Cambridge University Press. Landau, L. D. and Lifshitz, E. M. (1970). Theory of Elasticity, Pergamon Press, Oxford. Landau, L. D. and Lifshitz, E. M. (1987). Fluid Mechanics, Pergamon Press, New York. Larson, R. (1988). Constitutive Equations for Polymer Melts and Solutions, Buttersworths, New York. Lavrentiev, M. A. (1957). Shaped-charge jets and the principles of their work, Usp. Mat. Nauk 12(N4): 41–56. (in Russian). Lodge, A. (1964). Elastic Liquids, Academic Press, London. Loitsyanskii, L. G. (1966). Mechanics of Liquids and Gases, Pergamon Press, Oxford. Lundgren, T. S. (1989). A free surface vortex method with weak viscous effects, in R. E. Caflisch (ed.), Mathematical Aspects of Vortex Dynamics, Pergamon Press, Philadelphia, pp. 68–79. Macosco, C. W. (1994). Rheology – Principles, Measurements and Applications, John Wiley & Sons, New York. McKinley, G. H. and Tripathi, A. (2000). How to extract the Newtonian viscosity from capillary breakup measurements in a filament rheometer, J. Rheol. 44: 653–670. Munnannur, A. and Reitz, R. D. (2007). A new predictive model for fragmenting and nonfragmenting binary droplet collisions, Int. J. Multiph. Flow 33: 873–896. Munroe, C. E. (1900). The applications of explosives, Appleton’s Popular Science Monthly 56: 300–312, 444–455. Oron, A., Davis, S. H. and Bankoff, S. G. (1997). Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69: 931–980. Pan, K.-L., Chou, P.-C. and Tseng, Y.-J. (2009). Binary droplet collision at high Weber number, Phys. Rev. E 80: 036301. Plateau, J. (1873). Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires, Gauthier Villars, Paris. Pozrikidis, C. (1992). Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press. Prandtl, L. (1952). Essentials of Fluid Dynamics, Hafner, New York. Qian, J. and Law, C. (1997). Regimes of coalescence and separation in droplet collision, J. Fluid Mech. 331: 59–80. Lord Rayleigh (1878). On the instability of jets, Proc. London Math. Soc. 10: 4–13. Renardy, Y., Popinet, S., Duchemin, L., Renardy, M., Zaleski, S., Josserand, C., DrumrightClarke, M. A., Richard, D., Clanet, C. and Quéré, D. (2003). Pyramidal and toroidal water drops after impact on a solid surface, J. Fluid Mech. 484: 69–83. Reznik, S. N. and Yarin, A. L. (2002a). Spreading of a viscous drop due to gravity and capillarity on a horizontal or an inclined dry wall, Phys. Fluids 14: 118–132.

42

Introduction

Reznik, S. N. and Yarin, A. L. (2002b). Spreading of an axisymmetric viscous drop due to gravity and capillarity on a dry horizontal wall, Int. J. Multiph. Flow 28: 1437–1457. Rieber, M. and Frohn, A. (1999). A numerical study on the mechanism of splashing, Int. J. Heat Fluid Flow 20: 455–461. Ristenpart, W. D., McCalla, P. M., Roy, R. V. and Stone, H. A. (2006). Coalescence of spreading droplets on a wettable substrate, Phys. Rev. Lett. 97: 064501. Roisman, I. V., Planchette, C., Lorenceau, E. and Brenn, G. (2012). Binary collisions of drops of immiscible liquids, J. Fluid Mech. 690: 512–535. Rosenberg, Z. and Dekel, E. (2012). Terminal Ballistics, Springer, Berlin. Rubin, M. B. and Yarin, A. L. (1993, 1995). On the relationship between phenomenological models for elastic-viscoplastic metals and polymeric liquids, J. Non-Newton. Fluid Mech. 50: 79– 88. Corrigendum, J. Non-Newton. Fluid Mech. 57: 321. Schlichting, H. (1968). Boundary-Layer Theory, McGraw-Hill, New York. Schönecker, C., Baier, T. and Hardt, S. (2014). Influence of the enclosed fluid on the flow over a microstructured surface in the Cassie state, J. Fluid Mech. 740: 168–195. Sedov, L. I. (1993). Similarity and Dimensional Methods in Mechanics, CRC Press, Boca Raton. Sinha-Ray, S., Srikar, R., Lee, C. C., Li, A. and Yarin, A. L. (2011). Shear and elongational rheology of gypsum slurries, Appl. Rheol. 21: 63071. Stelter, M., Brenn, G., Yarin, A. L., Singh, R. P. and Durst, F. (2000). Validation and application of a novel elongational device for polymer solutions, J. Rheol. 44: 595–616. Tadmor, Z. and Gogos, C. G. (2013). Principles of Polymer Processing, John Wiley & Sons, New York. Taylor, G. I. (1959). The dynamics of thin sheets of fluid II. Waves on fluid sheets, Proc. R. Soc. London Ser. A-Math. 253: 296–312. Thoroddsen, S. T., Etoh, T. G. and Takehara, K. (2008). High-speed imaging of drops and bubbles, Annu. Rev. Fluid Mech. 40: 257–285. Tikhonov, A. N. and Samarskii, A. A. (1990). Equations of Mathematical Physics, Dover Publications, New York. Tiwari, M. K., Bazilevsky, A. V., Yarin, A. L. and Megaridis, C. M. (2009). Elongational and shear rheology of carbon nanotube suspensions, Rheol. Acta 48: 597–609. Tropea, C., Yarin, A. L. and Foss, J. F. (2007). Springer Handbook of Experimental Fluid Mechanics, Springer, Heidelberg. van Dyke, M. (1964). Perturbation Methods in Fluid Mechanics, Academic Press, New York. Weber, C. (1931). Zum Zerfall eines Flüssigkeitsstrahles, Z. Angew. Math. und Mech. 11: 136– 154. Weiss, D. A. and Yarin, A. L. (1999). Single drop impact onto liquid films: neck distortion, jetting, tiny bubble entrainment, and crown formation, J. Fluid Mech. 385: 229–254. Worthington, A. M. (1908). A Study of Splashes, Longmans, Green, and Company, London. Yarin, A. L. (1993). Free Liquid Jets and Films: Hydrodynamics and Rheology, Longman Scientific & Technical and John Wiley & Sons, Harlow and New York. Yarin, A. L. (2006). Drop impact dynamics: splashing, spreading, receding, bouncing …, Annu. Rev. Fluid Mech. 38: 159–192. Yarin, A. L. (2011). Bending and buckling instabilities of free liquid jets: Experiments and general quasi-one-dimensional model, in N. Ashgriz (ed.), Springer Handbook of Atomization and Sprays: Theory and Applications, Springer, Heidelberg, chapter 2, pp. 55–73. Yarin, A. L., Pourdeyhimi, B. and Ramakrishna, S. (2014). Fundamentals and Applications of Micro- and Nanofibers, Cambridge University Press.

1.12 References

43

Yarin, A. L. and Weiss, D. A. (1995). Impact of drops on solid surfaces: self-similar capillary waves, and splashing as a new type of kinematic discontinuity, J. Fluid Mech. 283: 141–173. Yarin, A. L., Zussman, E., Theron, A., Rahimi, S., Sobe, Z. and Hasan, D. (2004). Elongational behavior of gelled propellant simulants, J. Rheol. 48: 101–116. Yarin, L. P. (2012). The Pi-Theorem: Applications to Fluid Mechanics and Heat and Mass Transfer, Springer, Heidelberg. Zenit, R. and Hunt, M. L. (1998). The impulsive motion of a liquid resulting from a particle collision, J. Fluid Mech. 375: 345–361.

2

Selected Basic Flows and Forces

Penetration of solid bodies into liquids can be frequently treated in the framework of inviscid or potential flow hydrodynamics. In addition, collisions and penetration of solid bodies into solids in many cases, especially at the ordnance and ultra-ordnance velocities, can be effectively reduced to potential flows of ideal liquids possessing only inertia, since the stresses involved are significantly higher than the elastic and plastic stresses (see Section 1.1 in Chapter 1). Therefore, the present chapter is mostly devoted to several questions traditional to inviscid or potential flow hydrodynamics, which are either directly relevant in the context of collisions and penetration of solid bodies into liquids, or as simplified models useful for solid–solid collisions. Section 2.1 is devoted to the inviscid film flows on planar and curved surfaces. In the inertia-dominated regime characteristic of drop impact onto a thin liquid film on a wall, such flows give rise to kinematic discontinuities considered in Section 2.2 and associated with crown formation, which is considered in Section 6.7 in Chapter 6. The potential flow about an ovoid of Rankine discussed in Section 2.3 will be also employed in Chapter 13 in the case of projectile penetration into armor. The flow about an expanding and translating sphere outlined in Section 2.3 is also important in a purely hydrodynamic or rigid-projectile penetration context. Flows past axisymmetric bodies of revolution discussed in Section 2.4 are also important in the context of the projectile penetration. Transient motions of solid bodies in liquids inevitably involve deceleration associated with the added masses discussed in Section 2.5. A potential flow with separation about a blunt body (a plate moving normally to itself) is covered in Section 2.6 using the hodograph method of complex analysis to predict the shape drag. Friction drag associated with viscous effects is also discussed in Section 2.6. Finally, the dynamics of a rim bounding a free liquid film, for example a crown formed due to drop impact, is discussed in Section 2.7.

2.1

Inviscid Flow in a Thin Film on a Wall

2.1.1

Flat Planar Substrate In the present section the quasi-one-dimensional theory of Yarin and Weiss (1995) (see Section 1.7 in Chapter 1) is generalized to the case of a two-dimensional plane film. We consider the flow in a thin liquid film on a wall. The effects of the velocity component normal to the wall are secondary and of gravity are neglected. Following the simplified

2.1 Inviscid Flow in a Thin Film on a Wall

45

quasi-two-dimensional approach, the continuity equation of the film is ∂h + ∇ · (hV ) = 0, ∂t

(2.1)

where h is the film thickness, t is the time, ∇ is the two-dimensional gradient operator in the plane parallel to the wall and V is the average velocity vector over the film thickness and parallel to the wall. The momentum equation of the film represents the balance between the inertial forces and surface tension. The effect of surface tension includes two components. The first of these is the pressure distribution in the film. The inertial effects in the direction normal to the wall are negligibly small, and the average pressure over the film thickness is approximated by the local capillary pressure pσ given as pσ = σ κ,

(2.2)

where σ is surface tension and κ is the local curvature of the free surface of the film. The second component is the surface tension force applied to the free surface of the film, z = h, and directed tangentially to this surface. However, the free surface (of zero mass) can be excluded from the momentum balance of the film and the second component can be substituted by the distribution of the capillary pressure −pσ n over the surface z = h (with n being the local unit normal vector). Accordingly, the momentum balance equation takes the following form     ∂hV + ∇ · hV ⊗ V = −∇(hpσ ) + pσ ∇h, (2.3) ρ ∂t where ⊗ denotes the diadic product. Note that Eq. (2.3), written for the planar or axisymmetric cases, is identical to the corresponding momentum equations given in Yarin and Weiss (1995), see Eqs. (1.35) and (1.37) in Section 1.7 in Chapter 1. Equation (2.3) can be reduced to the following form using Eq. (2.1)   ∂V + (V · ∇)V = −∇(pσ ). (2.4) ρ ∂t It can be shown that in cases when |∇h|  1 the surface-tension-related term in Eqs. (2.4) and (2.2) can be linearized and the momentum balance can be written in dimensionless form as

∂V 1 2 + (V · ∇)V = − ∇(∇ h), (2.5) ρ ∂t We where the initial normal drop velocity, V0 , is used as a velocity scale, its initial diameter, D, is used as a length scale and ∇ 2 is the Laplace operator in two dimensions in the plane parallel to the wall. The Weber number in the cases corresponding to the crown formation and splash is much larger than unity [We 1; cf. Eq. (1.1) in Section 1.2 in Chapter 1]. Therefore, in the first-order approximation the capillary effects in the momentum balance equation,

46

Selected Basic Flows and Forces

the right-hand side of Eq. (2.5), vanish and Eq. (2.4) takes the form DV = 0, Dt

(2.6)

where D/Dt denotes the material time derivative. The solution of Eq. (2.6) in parametric form is therefore, V = F (ζ ),

x = F (ζ )t + ζ,

(2.7)

where x is the current radius-vector of a material element, and function F yields the initial value of the velocity vector V at the initial radius-vector ζ of this element. The continuity equation (2.1) can be rewritten in a more convenient form as Dh = −h(∇ · V ). Dt

(2.8)

The solution of Eq. (2.8) using Eq. (2.7) yields the following expression for the thickness of the film h(ζ ) =

h0 (ζ ) , 1 + (∇ζ · F )t + det(∇ζ F )t 2

(2.9)

where ∇ζ ≡ ∂/∂ζ denotes the gradient operator at the initial instant of time in the field determined by the radius-vector ζ, and ∇ζ · F is, accordingly, the divergence of the vector F ; h0 (ζ ) is the initial distribution of the film thickness. If in some ζ-region det(∇ζ F ) is negative, the denominator on the right-hand side of Eq. (2.9) can vanish at some positive time instant t and, as noted in Yarin and Weiss (1995) and Whitham (1974), the solution produces a kinematic discontinuity. This kinematic discontinuity will be analyzed in Section 6.2 in Chapter 6.

2.1.2

Arbitrary Shape of Substrate In the case of a curved surface the analysis is more complicated. But it can be significantly simplified by considering the film of a thickness much smaller than the local radius of curvature of the substrate. In this case the influence of the centrifugal forces on the pressure gradient in the film can be neglected in comparison with the inertial effects along the substrate. The effect of the Laplace pressure can also be assumed to be small (definitely, only if the Weber number based on the typical radius of curvature of the substrate is much higher than unity). The only force applied to a material element in this flow is the centrifugal force directed along the local vector normal to the surface, n. This force does not influence the magnitude of the velocity vector. Therefore, even on a curved surface a material element moves with a velocity of a constant magnitude. The second conclusion from the force analysis is that the unit vector normal to the surface, n, coincides with the unit vector normal to the trajectory of the material element. In other words, a material element in a thin film flow on a rigid surface moves with a constant velocity along a geodesic line (Weinstock 1974).

2.1 Inviscid Flow in a Thin Film on a Wall

47

Let us define the surface and the thin film flow on this surface in the parametric form. The surface of the rigid substrate can be defined as x = X (ξ , ψ )

(2.10)

where ξ and ψ are independent parameters, x is the position vector. The motion of a material element on the surface can be determined through ξ = (t ),

ψ = (t ).

(2.11)

The radius-vector r of this material element is therefore r(t ) = X[(t ), (t )].

(2.12)

The velocity and the acceleration of the material element can be written in the following form Dr = Xξ t + Xψ t Dt Du = A + Xξ tt + Xψ tt a= Dt

u=

(2.13) (2.14)

where A = Xξ ξ t2 + Xψψ t2 + 2Xξ ψ t t .

(2.15)

Here and hereinafter subscripts denote the corresponding partial differentiation. The condition for the free motion along a geodesic line can be expressed in the following form a · Xξ = 0,

a · Xψ = 0

(2.16)

since both vectors, Xξ and Xψ , are tangential to the surface x = X. The set of Eqs. (2.16) leads to the following system of ordinary differential equations for the functions (t ) and (t ) which determine the trajectory of the material element Xξ L{ψψ} − Xψ L{ξ ψ} L{ξ ψ}2 − L{ψψ} L{ξ ξ } Xψ L{ξ ξ } − Xξ L{ξ ψ} tt = A · L{ξ ψ}2 − L{ψψ} L{ξ ξ } tt = A ·

(2.17) (2.18)

at X = r, where operator L{i j} is defined as L{i j} = Xi · X j .

(2.19)

Equations (2.17) and (2.18) should be solved subject to the initial conditions  = ξ0 ,

 = ψ0 ,

t = u0 ,

t = v0

at t = 0.

(2.20)

Since the trajectories of various material elements in the flow under consideration are independent of each other and are determined only by the initial conditions, the flow on the curved surface can now be completely described in the Lagrangian system using the functions (ξ0 , ψ0 , t ), (ξ0 , ψ0 , t ), t (ξ0 , ψ0 , t ) and t (ξ0 , ψ0 , t ) obtained by the integration of Eqs. (2.17) and (2.18).

48

Selected Basic Flows and Forces

Figure 2.1 Sketch of a two dimensional substrate.

Consider now an initial distribution of the thickness of the film on the surface, h0 (ξ0 , ψ0 ). The initial position of a material element is denoted as r0 = X (ξ0 , ψ0 ). The area of the surface element can be determined as    ∂r0 ∂r0   dξ0 dψ0 ,  × (2.21) dS0 =  ∂ξ0 ∂ψ0  whereas the element of volume of the film is dV0 = h0 dS0 . At the time instant t > 0 the area covered by the same material element is    ∂r ∂r   dξ0 dψ0 ,  × dS =  ∂ξ0 ∂ψ0 

(2.22)

whereas the corresponding element of volume of the film is dV = h dS. The incompressibility condition, dV = dV0 , yields the following expression for the film thickness: h(ξ0 , ψ0 , t ) = h0 (ξ0 , ψ0 )

(∂r0 /∂ξ0 ) × (∂r0 /∂ψ0 ) . (∂r/∂ξ0 ) × (∂r/∂ψ0 )

(2.23)

Equations (2.17), (2.18) and (2.23) determine the solution for the film flow on an arbitrary surface. For some simple shapes explicit analytical solutions for the velocity field and film thickness can be obtained. Several examples are considered below.

2.1.3

Film Flow on a Two-dimensional Structured Substrate Consider a Cartesian coordinate system {x, y, z} and a simple example of a film flow on a two-dimensional surface of a rigid substrate X = {x, y, Z(x)}, as shown in Fig. 2.1. Consider y and  x dZ (2.24) 1 + Z 2 dx, Z = s= dx 0 as the defining parameters (with s being the arc length along the surface in the plane y = const.). Denote v = Dy/Dt as the y component of the film velocity and u = Ds/Dt as its s component. In this case the equations of motion (2.17) and (2.18) can be significantly simplified, Du = 0, Dt

Dv =0 Dt

(2.25)

2.1 Inviscid Flow in a Thin Film on a Wall

49

y

Figure 2.2 Sketch of a surface of revolution.

and easily solved subject to the initial conditions s = s0 ,

y = y0 ,

u = u0 (s0 , y0 ),

v = v0 (s0 , y0 )

at t = 0.

(2.26)

The solution is s(s0 , y0 , t ) = s0 + u0 (s0 , y0 )t, u(s0 , y0 , t ) = u0 (s0 , y0 ),

y(s0 , y0 , t ) = y0 + v0 (s0 , y0 )t

(2.27)

v(s0 , y0 , t ) = v0 (s0 , y0 ).

(2.28)

The corresponding velocity field is   uZ u . , v, √ u= √ 1 + Z 2 1 + Z 2

(2.29)

Consider now the initial distribution of the film thickness h = h0 (s0 , y0 )

at t = 0.

(2.30)

The solution for the distribution of the film thickness can be obtained with the help of Eq. (2.23) in the following form h(s0 , y0 , t ) =

1+



∂u0 ∂s0

h0 (s0 , y0 )   ∂u0 ∂v0 0 t + + ∂v − ∂y0 ∂s0 ∂y0

∂u0 ∂v0 ∂y0 ∂s0



t2

.

(2.31)

In the case of a film flow on a flat surface (s0 = x0 , s = x), the solution is reduced to the expressions for the velocity field and film thickness determined by Eqs. (2.7) and (2.9).

2.1.4

Film Flow on a Surface of Revolution Consider now another example: film flow on a surface of revolution r = R(z). The cylindrical coordinates system {r, ϕ, z} is defined in Fig. 2.2. The coordinate s which is the length on the surface along its generatrix is found as  z dR 1 + R 2 dz, R = s= . (2.32) dz 0

50

Selected Basic Flows and Forces

The equations of motion on the surface of revolution can be obtained using Eqs. (2.17) and (2.18) with the parameters s and ϕ RR Du =√ v2 Dt 1 + R 2

(2.33)

Dv 2R =− √ vu Dt R 1 + R 2

(2.34)

where u=

Ds , Dt

v=

Dϕ . Dt

(2.35)

This set of differential equations must be solved subject to the initial conditions z = z0 ,

R = R0 = R(z0 ),

ϕ = ϕ0 ,

u = u0 (s0 , ϕ0 ),

s = s0 = s(z0 ), v = v0 (s0 , ϕ0 ),

at t = 0.

(2.36)

The solution of Eqs. (2.33) and (2.34) yields the expression for the velocity field on the surface:  v 2 R4 v0 R20 (2.37) v = 2 , u = ± u20 + v02 R20 − 0 2 0 . R R The sign of the velocity u initially coincides with the sign of u0 . Then, the sign changes when a material element reaches the minimum radius R on the surface Rmin = 

|v0 |R20 u20 + v02 R20

.

(2.38)

The velocity vector can be written in the cylindrical coordinate system with the base vectors {er , eϕ , ez } in the form: uR er uez u= √ + Rveϕ + √ . 2 1+R 1 + R 2

(2.39)

The trajectory of the material element can be obtained by the integration of the velocity field, Eq. (2.37):  s dζ  (2.40) t= s0 ± u2 + v 2 R2 − v 2 R4 /R(ζ )2 0 0 0 0 0  ϕ = ϕ0 +

s s0



v0 R20 dζ

± R(s)2 u20 + v02 R20 − v02 R40 /R(ζ )2

.

(2.41)

The distribution of the film thickness on the surface of revolution can be generally obtained using the expression (2.23). The problem can be simplified significantly if the flow is symmetric relative to a certain plane, say x = 0. In this case the flow field and the film thickness in the vicinity of the symmetry plane can be obtained from Eqs. (2.40),

2.1 Inviscid Flow in a Thin Film on a Wall

z

(a)

(b)

51

z u

u0

r X( , )

Ωt

Rs

r0

y

x

y

x

Figure 2.3 Sketch of a spherical substrate. (a) Definition of the coordinate system; (b) motion of a

material element along the geodesic line.

(2.41) and (2.23) as s = s0 + u0t h(s) =

 R(s) 1 +

∂u0 t ∂s0

(2.42)

h0 (s0 )R(s0 )  ,

s −2 R(s )2 dζ 1 + u−1 ∂v /∂ϕ R(ζ ) 0 0 0 0 s0

(2.43)

at ϕ = ± π /2.

2.1.5

Film Flow on a Spherical Surface Consider an inviscid film flow on a spherical surface of radius Rs . The spherical coordinate system {r, ϑ, ϕ} is defined in Fig. 2.3 where ϑ is the zenith angle and ϕ is the azimuth angle. A radius-vector of a point on a spherical surface is determined by X = Rs (sin ϑ cos ϕex + sin ϑ sin ϕey + cos ϑez ). It is well known that geodesic lines on a sphere are great circles. Therefore, a material element with the initial conditions r = r0 , and u = u0 at t = 0 moves in a plane formed by the vectors r0 and u0 . Condition (2.16) leads to the conclusion that the angular velocity  of the radius-vector r in the plane {r0 , u0 } is constant. Since r0 · u0 = 0 the position of a material element can be obtained in the following form r = r0 cos t + u0 −1 sin t,

=

u0  = const. Rs

(2.44)

and its velocity is expressed by u = −r0  sin t + u0 cos t.

(2.45)

The distribution for the film thickness can be obtained by substituting the solution (2.44)–(2.45) in Eq. (2.23). In the framework with the parameters ϑ and ϕ this

52

Selected Basic Flows and Forces

equation yields h(ϑ0 , ϕ0 , t ) =

h0 (ϑ0 , ϕ0 )R2s

 −1  ∂r ∂r    . sin ϑ0  × ∂ϑ0 ∂ϕ0 

(2.46)

Equation (2.46) yields the analytical expression for h(ϑ0 , ϕ0 , t ). The explicit form of this expression is cumbersome and is not included here. Interested readers can obtain it by substituting Eq. (2.44) in Eq. (2.46). In the case of a flow which is symmetric relative to the plane x = 0 the distribution of the film thickness at the generatrix ϕ = ±π /2 can be derived by linearizing the flow given by Eq. (2.44) and accounting for the symmetry conditions in the following form ϑ = ϑ0 + ω(ϑ0 )t h(ϑ, t ) =

h0 (ϑ0 )|ω(ϑ0 ) sin ϑ0 | , |ω(ϑ0 ) sin ϑ + ∂v0 /∂ϕ0 sin ϑ0 sin ωt||1 + ∂ω/∂ϑ0t|

(2.47) (2.48)

at ϕ = ±π /2, where v0 ≡ Dϕ/Dt is the initial angular velocity in the azimuthal direction, and ω(ϑ0 ) ≡ Dϑ/Dt is the initial angular velocity in the zenith direction. This expression is relevant to the solution of a flow generated by drop impact onto a spherical target, since it is always possible to choose the coordinate system in which the plane x = 0 includes the initial trajectory of the drop just before impact. In the particular case of the axisymmetric flows (generated, for example, by a coaxial drop impact) the expression for the film thickness given by Eqs. (2.47) and (2.48) can be reduced to the form obtained in Bakshi et al. (2007) as h(ϑ, t ) = h0 (ϑ0 )

sin ϑ0 , sin ϑ[1 + ( dω/ dϑ0 )t]

(2.49)

which is valid at any ϕ.

2.2

Propagation of Kinematic Discontinuity The appearance of the kinematic discontinuity leads to the formation of the uprising free liquid sheet. Several regions can be distinguished in the flow produced by the kinematic discontinuity (see Fig. 2.4). Regions 1 and 2 are the regions of the wall film separated by the base of the uprising sheet. At the interface XB between these two regions the thickness of the film jumps from h1 to h2 and the velocity from V1 to V2 . This interface, which is the base of the uprising sheet (region 3 in Fig. 2.4) is treated in the theory of Yarin and Weiss (1995) as a kinematic discontinuity. The thickness and the height of this kinematic discontinuity is of order of the film thicknesses h1 and h2 , and the rate of change of the mass of the discontinuity and the inertial effects associated with the acceleration of the kinematic discontinuity are neglected. Denote by U the magnitude of the velocity of the discontinuity normal to its front. In the plane parallel to the wall consider a Cartesian coordinate system with the base unit vectors {en , eτ } normal and tangent to the discontinuity front. The mass balance and the momentum equation of the kinematic discontinuity in the plane parallel to the wall can

53

2.2 Propagation of Kinematic Discontinuity

Figure 2.4 Sketch of the uprising sheet produced by an inertia-dominated film flow on the wall.

be written in the form h1 (V1 − U en ) · en − h2 (V2 − U en ) · en = Q

(2.50)

h1 [(V1 − U en ) · en ](V1 − U en ) − h2 [(V2 − U en ) · en ](V2 − U en ) = Q(Vd − U en )

(2.51)

where Vd is the velocity of the liquid at the discontinuity front and Q is the specific volume flux into the discontinuity [also called in Yarin and Weiss (1995) the sink strength at the discontinuity]. If the viscous forces are negligibly small in comparison with the inertial forces, this velocity is equal to the center-of-mass velocity Vd =

V1 h1 + V2 h2 . h1 + h2

(2.52)

From Eqs. (2.50), (2.51) and (2.52) we arrive at U =

1 (Vn1 + Vn2 ), 2

Q=

1 (h1 + h2 )(Vn1 − Vn2 ). 2

(2.53)

The shape of the discontinuity front can be defined in the parametric form as X = XB (ξ , t )

(2.54)

where X is the radius-vector, t is the time and ξ is a Lagrangian parameter. It can be shown that the line defined as ∂XB (ξ , t ) V1 (XB , t ) + V2 (XB , t ) = ∂t 2

(2.55)

moves normally to the discontinuity with the velocity U given in Eq. (2.53). Equation (2.55) is the differential equation of the motion of the kinematic discontinuity, which can be integrated for given velocity fields V1 (X, t ) and V2 (X, t ).

54

Selected Basic Flows and Forces

V

e e

V

V

Figure 2.5 Sketch of the formation of the upward sheet (crown) issued from the kinematic

discontinuity (Roisman and Tropea 2002). Reproduced with permission.

2.2.1

Formation of a Liquid Sheet If the strength of the sink Q determined in Eq. (2.53) is positive, the interaction of two liquid flows on the wall results in an uprising liquid sheet at the kinematic discontinuity, as shown in Fig. 2.5. A similar geometry of the splash was considered in Peregrine (1981). In this work a steady-state, one-dimensional model of the splash is developed. In the present model of the sheet jetting, the general case is analyzed, when the components of the film velocity parallel to the kinematic discontinuity do not vanish. Moreover, the surface tension effects are taken into account in the momentum equation. Consider now the Cartesian coordinate system {e n , e τ , e z } moving with the discontinuity front with the velocity Vcs = U e n +

Vτ 1 + Vτ 2 eτ . 2

(2.56)

Here the base unit vectors e n and e τ are normal and tangent to the kinematic discontinuity, the base unit vector e z is normal to the wall. The velocity U , defined in Eq. (2.53), is the velocity of propagation of the discontinuity in the normal, e n , direction, Vτ 1 and Vτ 2 are the tangential components of the velocity of the liquid in the films with the thicknesses h1 and h2 on both sides of the discontinuity, respectively. The velocities of the liquid on the wall on the two sides of the discontinuity in this coordinate system are V1rel =

Vn1 − Vn2 Vτ 1 − Vτ 2 en + eτ = −V2rel . 2 2

(2.57)

Denote by α the angle of inclination of the issued sheet to the wall in the coordinate system {e n , e τ , e z }. Thus, the unit vector eS in the direction of the uprising sheet can be

2.2 Propagation of Kinematic Discontinuity

55

defined as eS = e n cos α + e z sin α.

(2.58)

Assuming the pressure to be constant, neglecting the mass accumulated inside the kinematic discontinuity and the inertial effects of this mass, the mass balance, the axial momentum balance in the normal and tangential directions, and the energy equation at the base of the crown can be written in the following form −h1 V1rel · e n + hB VBrel · eJ + h2 V2rel · e n = 0

(2.59)

P · e n P · e τ V1rel · V1rel V1rel · V1rel

= 2σ cos α

(2.60)

=0

(2.61)

= =

VBrel V2rel

· ·

VBrel V2rel ,

(2.62) (2.63)

where eJ is the unit vector directed along the uprising liquid jet in Fig. 2.5. Here the parameter P is defined as P = −ρ h1 (V1rel · e n )V1rel + ρ hB (VBrel · eJ )VBrel + ρ h2 (V2rel · e n )V2rel ,

(2.64)

where hB is the thickness of the crown at its base, α is the inclination angle at the base and VBrel is the velocity of the liquid in the inclined sheet at the base relative to the coordinate system {e n , e τ , e z }. The mass balance (2.59) expresses the fact that the sum of the volume fluxes of the liquid entering into the kinematic discontinuity must vanish, i.e. to be propelled off the wall. The momentum equations (2.60) and (2.61) express the balance of the inertial and surface tension forces in the plane parallel to the wall. The Bernoulli equations (2.62) and (2.63) use the fact that the pressure p0 in the films of thicknesses h1 and h2 , as well as in the crown outside the kinematic discontinuity, is equal to the constant pressure in the surrounding air. Note that the choice of the coordinate system {e n , e τ , e z } moving with the velocity (2.56) was partially motivated by the fact that the condition (2.59) would be satisfied automatically. Note, also, that the uprising sheet appears at the kinematic discontinuity because the pressure inside this discontinuity differs from p0 . Moreover, the vertical reaction of the wall per unit length of the discontinuity, Fp , associated with this pressure, can be estimated using the momentum balance equation in the z-direction Fp = ρ hB (VBrel · eJ )(VBrel · ez ) − 2σ sin α.

(2.65)

Equations (2.59)–(2.63) are similar to the corresponding relations derived in Peregrine (1981). However, the non-stationary inertial effects are neglected in the analysis of Peregrine (1981) and, as a result, his theory predicts a steady-state motion of the base of the crown. This result is not confirmed by the experimental observations expressed by Eq. (6.12) in Section 6.2 in Chapter 6. In the present analysis only the terms associated with the acceleration of the kinematic discontinuity itself are neglected. The width and the height of this region is of the order of h, which is the characteristic thickness of the liquid on the wall. The condition that the

56

Selected Basic Flows and Forces

non-stationary terms inside the kinematic discontinuity can be neglected is, therefore dVcs . (2.66) dt The condition (2.66) applied to the case of the normal impact of a single drop onto a stationary liquid film of thickness h f yields with the help of Eq. (6.12), the condition h f /RB  1, where RB is the radius of the discontinuity base. Therefore, the inertial effects of the liquid inside the kinematic discontinuity can indeed be neglected in the remote asymptotic solution considered in the present section. The velocity vector VBrel must be parallel to the sheet. This means that this velocity can be written in the form: ρh(V rel )2 ρh2

rel VBrel = VJrel B eS + Vτ B eτ

where the unit vector eS is defined in Eq. (2.58). The solution of the system of equations (2.59)–(2.63) with the help of relation (2.57) yields the following expressions for the velocity of the liquid in the sheet at the base of the crown, its thickness and the inclination angle in the moving coordinate system {e n , e τ , e z }: h1 − h2 rel V h1 + h2 τ 1    rel 2 4h1 h2  rel 2 1/2 rel V VJ B = Vn1 + (h1 + h2 )2 τ 1 (h1 + h2 )Vn1rel hB = VJrel B 2  (h1 − h2 ) Vn1rel . cos α = (h1 + h2 )Vn1rel VJrel B − 2σ /ρ Vτrel B =

(2.67) (2.68) (2.69) (2.70)

The final expressions for the velocity of the sheet at the base in the laboratory coordinate system {en , eτ , ez } and its thickness are obtained using Eqs. (2.53), (2.56) and (2.67)–(2.70) V1 h1 + V2 h2 2σ S cos α en + sin α ez + h1 + h2 ρQ 2(h1 + h2 ) (h1 + h2 )2 (Vn1 − Vn2 ) hB = S where the function S and the angle α are defined as  S = (h1 + h2 )2 (Vn1 − Vn2 )2 + 4h1 h2 (Vτ 1 − Vτ 2 )2   (h1 − h2 ) (Vn1 − Vn2 )2 α = arccos (Vn1 − Vn2 )S − 8σ /ρ VB =

(2.71) (2.72)

(2.73) (2.74)

and Q is determined in Eq. (2.53). The absolute value of the argument of the arccosine on the right-hand side of Eq. (2.74) should be smaller than unity. Otherwise, the solution for α does not exist and the crown is not formed. It can be shown that the solution for the angle α exists always in the asymptotic case σ = 0. Consider for simplicity the case σ > 0 for an axisymmetric

2.2 Propagation of Kinematic Discontinuity

57

geometry of the impact (Vτ 1 = Vτ 2 = 0). Equations (2.72) and (2.74) can be reduced to the following form 

hB = h1 + h2

(h1 − h2 )(Vn1 − Vn2 )2 α = arccos hB (Vn1 − Vn2 )2 − 8σ /ρ

 (2.75)

and the necessary condition for the crown formation is 8σ /ρ < min(h1 , h2 )(Vn1 − Vn2 )2 . The latter condition is not satisfied after an impact with a low Weber number. In this case, the inertial forces are weak in comparison with the surface tension forces, preventing crown formation. In the present section only impacts with high Weber numbers are considered, when the inertia plays a dominant role in the force balance. Expressions (2.71) and (2.72) together with the position of the crown base XB obtained by the integration of Eq. (2.55) can be used as the initial conditions for the determination of the shape of the ejected sheet (the crown). The dynamic equations of propagation of the sheet are given below.

2.2.2

The Shape and Thickness of a Free Sheet The sheet is assumed to be thin. The radius-vector corresponding to the median surface of the sheet (see Fig. 2.4) can be defined in the Lagrangian form for any instant of time t as XS (ξ , tB , t ), where ξ is a Lagrangian parameter, tB is the time instant at which the material element located at XS was ejected from the wall (at the base of the crown) such that XS (ξ , tB , tB ) = XB (ξ , tB ).

(2.76)

Parameter ξ is associated with the position of a given material element along the front of the kinematic discontinuity XB . The equation of motion of the material element of the sheet in its general form reads ρhS ∂AS

∂2 XS (ξ , tB , t ) = ∂T ∂t 2

(2.77)

where hS is the local thickness of the sheet, ∂AS is the element of the sheet surface, ∂T is the total force applied to the given element of the sheet, including in general the capillary, viscous, body forces and the gas drag force. The details of the force ∂T applied to an element of a free-moving sheet, as well as the quasi-two-dimensional equations of motion of free liquid films, can be found in Yarin (1993). The effect of the surface tension consists of the capillary pressure in the film and of the δ-functional surface tension force acting at the two free surfaces bounding the considered element of the area ∂AS . The effect of the capillary pressure can be neglected in a free thin film due to the cancelation of the effects of the neighboring convex and concave sides. The δ-functional surface tension force at each free side of the sheet can be expressed by a resulting force σ κ∂AS nS , where κ is the local curvature of the sheet, nS is the unit normal vector directed toward the center of curvature.

58

Selected Basic Flows and Forces

In the present analysis the effect of viscosity is neglected and the momentum balance equation of the rising sheet takes the form 2σ κ ∂2 XS (ξ , tB , t ) = nS − g ez ∂t 2 ρhS

(2.78)

where g is the gravity acceleration. Consider now the thickness hS (ξ , tB , t ) of the sheet. Conservation of mass of an element of the crown yields hB (ξ , tB )∂AS (ξ , tB , tB ) = hS (ξ , tB , t )∂AS (ξ , tB , t ) where the element of the area of the sheet, ∂AS , can be determined as    ∂XS ∂XS   ∂ξ ∂tB . × ∂AS =  ∂ξ ∂tB 

(2.79)

(2.80)

Equations (2.79) and (2.80) yield the following expression for the thickness of the sheet |[∂XS (ξ , tB , t )/∂ξ ] × [∂XS (ξ , tB , t )/∂tB ]||t=tB . (2.81) hS (ξ , tB , t ) = hB (ξ , tB ) |[∂XS (ξ , tB , t )/∂ξ ] × [∂XS (ξ , tB , t )/∂tB ]| In order to calculate the shape of the sheet and its thickness, Eq. (2.78) must be integrated with the help of Eq. (2.81) subject to the initial conditions at the base of the crown ∂XS = VB (ξ , tB ), hS = hB (ξ , tB ). (2.82) t = tB : XS = XB (ξ , tB ), ∂t

2.3

External Irrotational Flows About Blunt Bodies In many situations, especially when the Reynolds number is high enough, the external flow about a body can be approximated by an irrotational (potential) flow, as shown in Sections 1.1 and 1.4 in Chapter 1. Moreover, this is true not only for collisions of solid bodies with liquids but also for collisions of solid bodies (especially metals) at the ordnance and ultra-ordnance velocities. Since the Laplace equation (1.17) is linear, any potential flow can be described as a superposition of several potential flows. Several well-known elementary potential flows are generated by the following potentials φ = U ez · r, u = U ez , uniform flow in the z-direction  θ , u= ez × er , potential line vortex along the z-axis φ= 2π 2π r m φ = m ln r, u = er , a line source or sink r m m φ = , u = 2 er , a point source or sink r r mez · r m φ=− , u = 3 (3[ez · er ]er − ez ), a 3D dipole, r3 r

(2.83) (2.84) (2.85) (2.86) (2.87)

2.3 External Irrotational Flows About Blunt Bodies

59

Figure 2.6 Sketch of the cavity and definition of the fixed and moving reference frames.

where m is the strength of a source, sink or dipole,  is the circulation of the potential vortex flow, r is the relative position vector r ≡ x − X0 ,

(2.88)

with X0 being the position vector of the origin of the coordinate system. The superposition principle is the fundamental aspect of various methods of computing the external potential flows, like the method of images (for flows near a flat wall) and the method of singularity distribution (Kochin et al. 1964, Batchelor 2002, Rouse 1964).

2.3.1

Flow Past an Expanding and Translating Sphere This flow is important in the context of an impact of a solid sphere onto a liquid layer covered in Part IV, and in the context of high-speed solid-to-solid impact covered in Part V. Consider a reference frame fixed at the impact point O with the horizontal x-axis and the vertical z-axis (the impact axis) directed along the gravity acceleration, as in Fig. 2.6. Consider also a moving spherical coordinate system {r, θ , ϕ} with the origin at the center O of the spherical cavity, where r is the radial coordinate and θ is the zenith angle. The radius of the crater in the relative spherical reference system is denoted by a(t ) and the position of its center on the symmetry axis by zc (t ). The potential φ rel of a relative irrotational flow about the sphere in the moving coordinate system fixed at the sphere center, satisfying the Laplace equation (1.17) in Section 1.4 in Chapter 1 can be obtained as a superposition of a well-known flow past a solid sphere (found as a superposition of a flow generated by a single dipole and a uniform flow) and the radially expanding flow generated by a sphere expansion, which acts effectively as a point source   a2 a3 rel (2.89) φ = −˙zc r 1 + 3 cos θ − a˙ , 2r r where dot denotes time differentiation. The velocity components of the corresponding relative velocity field urel = ∇φ rel are obtained in the following form     a2 a3 a3 rel U cos θ , u U sin θ , (2.90) = a ˙ − 1 − = 1 + urel r θ r2 r3 2r3

60

Selected Basic Flows and Forces

where U ≡ z˙c is the velocity of the sphere translation. Velocity urel satisfies the following kinematic boundary conditions at the sphere (crater) surface and far from it urel r = a˙

r = a;

at

urel = −U ez

at

r → ∞.

(2.91)

The components of the velocity field in the laboratory framework and the corresponding flow potential are a2 a3 a3 + 3 U cos θ , uθ = 3 U sin θ , 2 r r 2r a2 a3 φ = −U 2 cos θ − a˙ . 2r r

ur = a˙

(2.92) (2.93)

Finally, the pressure field can be expressed using the Bernoulli equation (1.18) with the help of Eq. (1.21) of Section 1.4 in Chapter 1. The pressure distribution at the sphere surface if the gravity is neglected   9 2 aU˙ U2 3a˙2 3 ps 1 − sin θ + = cos θ + + aa¨ + aU ˙ cos θ (2.94) ρ 2 4 2 2 2 is derived under condition that the pressure vanishes far from the sphere. This pressure distribution is useful for the description of the cavitation phenomena or boiling, and also for modeling a cavity expansion produced by a pointwise explosion or drop or solid impact.

2.3.2

Flow Past a Semi-infinite Ovoid of Rankine Consider a flow generated by a moving source of strength m(t ), located at the position z = zc (t ). The potential and the corresponding velocity field are defined by the superposition of the potentials from Eqs. (2.83) and (2.86). Then, the relative velocity field reads u=

m er − z˙c (t )(er cos θ − eθ sin θ ). r2

(2.95)

Consider now the stream function of this flow, defined as ur =

r2

1 ∂ψ , sin θ ∂θ

uθ = −

1 ∂ψ . r sin θ ∂r

(2.96)

Then, Eqs. (2.95) and (2.96) yield ψ =−

r 2U sin2 θ − m(cos θ − 1). 2

(2.97)

The streamline ψ = 0 corresponds to the z-axis and to a surface of revolution   m(1 − cos θ ) 1/2 . r= 2 U sin2 θ

(2.98)

2.4 Flows Past Arbitrary Axisymmetric Bodies of Revolution

61

Figure 2.7 Ovoid of Rankine.

Taking m = R2∞U/4, one can replace m by the radius of the surface of revolution to which it is approaching asymptotically, as R∞ = limθ→π r sin θ at z → −∞. The thus obtained surface of revolution is an ovoid of Rankine, shown in Fig. 2.7 √ 1 − cos θ . (2.99) r = R∞ √ 2 sin θ The tip of the ovoid as θ → 0 is R∞ /2. The pressure field around the ovoid translating with the velocity U = z˙c in the zdirection is R2 U˙ R2 U 2 R4 U 2 p = − ∞ + ∞ 2 cos θ − ∞ 4 , ρ 4r 4r 32r

(2.100)

which yields at the surface of the ovoid 1 + 4 cos θ + 3 cos(2θ ) 2 cos(θ /2) ps = U − U˙ , ρ 16 2

(2.101)

under the conditions of zero pressure far from the ovoid and the effect of gravity being negligibly small.

2.4

Flows Past Arbitrary Axisymmetric Bodies of Revolution The potential φ can be represented as a sum of three potentials, φ (1) , φ (2) and φ (3) , of flows corresponding to a solid body of revolution moving with longitudinal velocity U , transversal velocity V and rotating in the plane of the velocity vector with the components U and V with an angular velocity α, ˙ respectively, as φ = φ (1) + φ (2) + φ (3) .

(2.102)

62

Selected Basic Flows and Forces

The functional forms for these potentials can be deduced using the method of singularities (Kochin et al. 1964, Batchelor 2002, Rouse 1964) via the introduction of a distribution of sources and dipoles along the axis of revolution. A dimensionless distribution of sources q(1) generates potential φ (1) for the longitudinal motion, a distribution of doublets q(2) generates potential φ (2) for the transversal motion and a distribution of doublets q(3) generates potential φ (3) for rotation. The expressions for the components of the potential are  ξL q(1) (η) dη φ (1) = , (2.103) 2 2 1/2 ξA [(ξ − η) + r ]  ξL q(2) (η) dη (2) , (2.104) φ = r cos ϕ 2 2 3/2 ξA [(ξ − η) + r ]  ξL q(3) (η) dη , (2.105) φ (3) = r cos ϕ 2 2 3/2 ξA [(ξ − η) + r ] where q(i) (η) are the strengths of singularities per unit length at the point ξ = η on the body axis, r is the radial coordinate and ϕ is the polar coordinate in the cylindrical coordinate system {r, ϕ, ξ } fixed at the body axis, ξA is the axial coordinate of the body’s tip and ξL is that of the body’s rear end. All the coordinates in this section are rendered dimensionless by the characteristic body radius R∞ used as a length scale, the potential φ (1) is scaled by U R∞ , φ (2) is ˙ scaled by V R∞ and φ (3) is scaled by R2∞ α. The expressions for the functions q(i) can be derived from the conditions of impenetrability of the body’s surface, which yields the following integral equations  ξL  ξL (ξ − η)q(1) (η) dη R2 − = q(1) (η) dη, (2.106) 2 2 1/2 2 ξA [(ξ − η) + R ] ξA  ξL (ξ − η)2 − 2R2 + 3R dR (ξ − η) dξ q(2) dη = −1, (2.107) [(ξ − η)2 + R2 ]5/2 ξA  ξL (ξ − η)2 − 2R2 + 3R dR (ξ − η) dR dξ . (2.108) q(3) dη = ξ + R 2 2 5/2 [(ξ − η) + R ] dξ ξA Here R = R(ξ ) is the known shape of a body of revolution under consideration. These integral equations have to be solved numerically for any specific shape of the solid body of revolution. One of the solutions for an ovoid of Rankine is shown in Fig. 2.8. The form of the ovoid of Rankine is convenient for simulations of the problems related to projectile penetration into armor (see Section 13.6 in Chapter 13) since it is similar to the real projectile form and also since the flow associated with the longitudinal body motion corresponds to a simple analytical solution (see subsection 2.3.2). The numerically predicted dimensionless distribution of sources q(1) (η) in Fig. 2.8 approaches to the delta-function δ(η)/4, which is expected for a single source generating the ovoid of Rankine in longitudinal motion (the analytical solution used as a benchmark case here). Also, for comparison the distributions of sources and doublets corresponding to a cone-nose cylindrical body are shown in Fig. 2.9.

63

2.5 Transient Motion and Added Masses in Inviscid Fluids

1.5

12

(a)

8 q(1)(η)

0.5 R

0 –0.5

4

0 –2

–1.5 –1

0

1

ξ

2

3

4

–1

0.5

(c)

0.5

0

1

0

1

η

2

3

4

2

3

4

(d)

0 q(3)(η)

0.4 q(2)(η)

6

2

–1

0.6

(b)

10

1

0.3 0.2

–0.5 –1 –1.5

0.1 0

–2 –1

0

1

η

2

3

4

–1

η

Figure 2.8 Distributions of singularities corresponding to an ovoid of Rankine. (a) the shape of the body, (b) the distribution of sources q(1) (η), (c) the distribution of doublets q(2) (η), and (d) the distribution of doublets q(3) (η) (Roisman et al. 1997). Reprinted with permission from Elsevier.

The distribution of sources and doublets determines the flow field past a solid body and also on its surface. For example, for a body shaped as an ovoid of Rankine, the instantaneous streamlines in the plane of symmetry of the body and on its surface are shown for a pure longitudinal motion in Fig. 2.10 and for a combined motion which includes the body longitudinal and transversal translation, and rotation in Fig. 2.11.

2.5

Transient Motion in Inviscid Fluids and Forces Associated with the Added Masses Steady-state potential flows about arbitrary bodies without separation always produce no drag force, the result known as d’Alembert’s paradox (Loitsyanskii 1966, Landau and Lifshitz 1987, Batchelor 2002). However, transient not necessarily translational flows about arbitrary bodies result in an inertial drag force. Consider transient not necessarily translational motion of an arbitrary body in an inviscid fluid, which is at rest at infinity following Kirchhoff (1897) [also see Loitsyanskii (1966)]. Denote by the asterisks all

64

Selected Basic Flows and Forces

2

4

1

3

0.5

2

q(1)(η)

R

1.5

5

(a)

0

1

–0.5

0

–0.1 –1.5

–1 –2 –3

–2 –1

1.5

(b)

0

1

ξ

2

3

–1

4

2

(c)

0

1

0

1

η

2

3

4

2

3

4

(d)

1 1 q(3)(η)

q(2)(η)

0 0.5

–1 –2

0 –3 –0.5

–4 –1

0

1

η

2

3

4

–1

η

Figure 2.9 Distributions of singularities corresponding to a cone-nose body. (a) the shape of the

body, (b) the distribution of sources q(1) (η), (c) the distribution of doublets q(2) (η), and (d) the distribution of doublets q(3) (η) (Roisman et al. 1997). Reprinted with permission from Elsevier.

the parameters associated with the body, namely, its velocity vector V ∗ , its angular velocity vector ω ∗ about an instantaneous axis through a certain material element O∗ and the Cartesian coordinate system x∗ , y∗ and z∗ embedded into the body with the origin at O∗ . Accordingly, V ∗ = V0∗ + ω ∗ × r ∗

(2.109)

where V0∗ is the velocity vector of the point O∗ , and r ∗ is the radius-vector of another material element in the body where the velocity V ∗ is calculated. The Cartesian components of vector V0∗ are denoted as u∗0 , v0∗ and w0∗ , while the Cartesian components of vector ω ∗ are denoted as ωx∗ , ωy∗ and ωz∗ . In addition, introduce an absolute Cartesian coordinate system x, y and z, which is at rest. In the general case, at any moment of time one can assume that the movable (relative) and the absolute Cartesian coordinate systems O∗ x∗ y∗ z∗ and Oxyz, respectively, coincide with each other (by choosing an appropriate absolute system Oxyz). Assume fluid motion generated by the body to be potential (irrotational) under the conditions listed in Section 1.4 in Chapter 1. Therefore, the velocity field of fluid motion

65

2.5 Transient Motion and Added Masses in Inviscid Fluids

(a)

(b)

(a)

(c)

–1 –0.5

(c)

2 1 z x 0 0.5

(b)

3 –1 –0.5

0 1

–1 –0.5 0 0.5 x

1

2 1 z 0 x 0.5

3

0 1

–1 –0.5 0 x

0.5

1

Figure 2.10 The instantaneous streamlines relative to

Figure 2.11 The instantaneous streamlines relative to

the body (a) and relative to the target (b) in the plane of symmetry of the body, and the instantaneous streamlines on the surface (c) for longitudinal motion of an ovoid of Rankine. U = 700 m/s, V = 0 m/s, αR ˙ ∞ = 0 m/s (Roisman et al. 1997). Reprinted with permission from Elsevier.

the body (a) and relative to the target (b) in the plane of symmetry of the body, and the instantaneous streamlines on the surface (c) for a combination of the body translation and rotation. U = 700 m/s, V = 110 m/s, αR ˙ ∞ = −20 m/s (Roisman et al. 1997). Reprinted with permission from Elsevier.

relative to the absolute coordinate system is generated by a potential φ(x, y, z, t ) (with t being time), as V = ∇φ, and the potential φ satisfies the Laplace equation (1.17) in Chapter 1. The boundary conditions at the body surface S require that the normal components of the body and fluid velocity be the same, namely at S Vn =

∂φ = Vn∗ = V0∗ · n + (ω ∗ × r ∗ ) · n = u∗0 nx + v0∗ ny + w0∗ nz ∂n + ωx∗ (ynz − zny ) + ωy∗ (znx − xnz ) + ωz∗ (xny − ynx )

(2.110)

where n is the outer unit normal vector at a point of S, and use is made of Eq. (2.109). In addition, for the potential flow, which is at rest at infinity, the second boundary condition reads φ = O(1/R2 ),

where

R2 = x2 + y2 + z2 → ∞.

(2.111)

Since the Laplace equation (1.17) in Chapter 1 is linear, it admits the following superposition φ(x, y, z, t ) = u∗0 (t )φ1 (x, y, z) + v0∗ (t )φ2 (x, y, z) + w0∗ (t )φ3 (x, y, z) + ωx∗ (t )φ4 (x, y, z) + ωy∗ (t )φ5 (x, y, z) + ωz∗ (t )φ6 (x, y, z)

(2.112)

66

Selected Basic Flows and Forces

where each of the potentials φi (x, y, z) satisfies the Laplace equation ∇ 2 φi = 0,

with i = 1, 2, ...6,

(2.113)

and the boundary conditions ∂φ1 ∂φ2 ∂φ3 = nx , = ny , = nz ∂n ∂n ∂n ∂φ4 ∂φ5 ∂φ6 = ynz − zny , = znx − xnz , = xny − ynx ∂n ∂n ∂n φi = O(1/R2 ), where R2 = x2 + y2 + z2 → ∞.

(2.114) (2.115) (2.116)

The factorization of the time and spatial dependencies as in Eq. (2.112) is possible since the Laplace equation governing the flow potential φ does not involve the time derivative. Note also, the fact that the movable (embedded in the body) and the absolute Cartesian coordinate systems O∗ x∗ y∗ z∗ and Oxyz, respectively, coincide with each other at the moment under consideration. The potentials φi essentially correspond to the following motions: the potentials φ1 , φ2 and φ3 correspond to the flows generated by steady-state translational motions of the body under consideration when it moves with velocities equal to unity along the x, y and z axes, respectively, while the potentials φ4 , φ5 and φ6 correspond to the flows generated by steady-state rotational motions of the body under consideration when it rotates with unit angular velocities about the x, y and z axes, respectively. Therefore, these steadystate flows, in principle, can be found using the ordinary methods of the potential flow theory. To find the net pressure force and the net moment of pressure forces acting on a moving body from the fluid side, surround the moving body by a sphere of a large radius R0 and surface S0 and apply the second law of Newton to the fluid volume τ (t ) located between the body surface S and the sphere surface S0 . Denote by Q the momentum of the fluid in volume τ (t ), by F the net pressure force acting from the fluid to the body at its surface S, and by F0 the net pressure force acting from the outside at the surface S0 . The second law of Newton reads DQ = −F + F0 (2.117) Dt which accounts for the fact that the net force acting on the fluid from the body at its surface S is equal to −F according to the third law of Newton. In Eq. (2.117) D/Dt is the material time derivative. In the inviscid fluid the force F0 acting on the fluid at the surface S0 is  F0 = − pn0 dS0 (2.118) S0

where n0 is the outer normal unit vector, with pressure found from the Bernoulli integral (1.18) in Chapter 1 as ∂φ ρV 2 −ρ 2 ∂t with gravity being neglected, and V being the fluid velocity magnitude. p = ρ f (t ) −

(2.119)

2.5 Transient Motion and Added Masses in Inviscid Fluids

67

Since at infinity in the fluid at rest V → 0, φ → 0 and p → p∞ , Eq. (2.119) reveals that f (t ) = p∞ /ρ. Accordingly, Eqs. (2.118) and (2.119) yield   ∂φ ρ F0 = ρ n0 dS0 + V 2 n0 dS0 . (2.120) ∂t 2 S0

S0

By definition the momentum of fluid in volume τ (t ) is   Q = ρV dτ = ρ∇φ dτ. τ

(2.121)

τ

Applying the Gauss theorem to the latter integral, one obtains   Q = −ρ φn dS + ρ φn0 dS0 S

(2.122)

S0

where the minus sign before the first integral results from the fact that n, being the outer unit normal to the body, is the inner unit normal to the fluid volume under consideration. Note that similarly to Eq. (1.21) in Chapter 1 ⎞ ⎛    ∂φ D ⎝ (2.123) ρ φn0 dS0 ⎠ = ρ n0 dS0 + ρVn0 V dS0 Dt ∂t S0

S0

S0

with Vn0 = V · n0 . Accordingly, Eqs. (2.122) and (2.123) yield ⎞ ⎛    D ⎝ ∂φ DQ =− ρφn dS ⎠ + ρ n0 dS0 + ρVn0 V dS0 . Dt Dt ∂t S

S0

(2.124)

S0

Substituting the latter equation into Eq. (2.117) and using Eq. (2.120), one finds the net pressure force acting on a body moving in fluid as ⎞ ⎛    2  V D ⎝ n0 − Vn0 V dS0 . F = ρφn dS ⎠ + ρ (2.125) Dt 2 S

S0

When the radius of the outer sphere R0 → ∞, the integrand in the second integral on the right-hand side in Eq. (2.125) tends to zero as R−6 0 , and thus the entire second . Accordingly, the net pressure force acting at the body integral tends to zero as R−4 0 surface is ⎞ ⎛  D ⎝ F = ρφn dS ⎠ . (2.126) Dt S

Similarly, the net moment of pressure forces acting on a moving body from the fluid side M can be found as ⎞ ⎛  D ⎝ ρφr × n dS ⎠ . (2.127) M= Dt S

68

Selected Basic Flows and Forces

Denote the net momentum and the moment-of-momentum of a solid body moving in liquid as Q∗ and K ∗ , respectively. Then, the equations of the solid-body motion read, respectively (Landau and Lifshitz 1969, Feynman et al. 2006) DQ∗ = F + F ∗, Dt

DK ∗ = M + M∗ Dt

(2.128)

where F ∗ and M ∗ are the net force and moment of force of the non-hydrodynamic origin (if any) acting on the body. Combining Eqs. (2.126)–(2.128), we arrive at D (K ∗ + I ) = M ∗ Dt

D (Q∗ + B) = F ∗ , Dt where

 B=−

(2.129)

 ρφn dS,

I=−

S

ρφr × n dS.

(2.130)

S

Equations (2.129) reveal that motion of a solid body in fluid can be described by the equations of the solid body motion in vacuum if one adds vectors B and I to the net momentum and the moment-of-momentum of the solid body Q∗ and K ∗ , respectively. Consider in detail the net pressure force acting on a body, which is expressed, according to Eqs. (2.126) and the first Eq. (2.130), as F =−

D∗ B DB =− − ω∗ × B Dt Dt

(2.131)

where D∗ /Dt denotes the material time derivative in the frame of reference embedded in the solid body. Similarly, it is possible to show using Eq. (2.127) and the second Eq. (2.130) that the net moment of pressure forces acting on a moving body from the fluid side M =−

D∗ I ∗ − ω ∗ × I ∗ − V0∗ × B Dt

where I∗ = −



(2.132)

ρφr ∗ × n dS.

(2.133)

S

Introduce for convenience the following notations q1 = u∗0 ,

q2 = v0∗ ,

q3 = w0∗ ,

q4 = ωx∗ ,

q5 = ωy∗ ,

q6 = ωz∗

(2.134)

which means that Eq. (2.112) becomes φ=

6 

q jφ j.

(2.135)

j=1

Similarly, denote B1 = Bx ,

B2 = By ,

B3 = Bz ,

B4 = Ix ,

B5 = Iy ,

B6 = Iz .

(2.136)

2.5 Transient Motion and Added Masses in Inviscid Fluids

69

Then, from Eqs. (2.113)–(2.116) and the Eqs. (2.130), (2.135) and (2.136), one obtains  Bi = −ρ

φ S

∂φ dS = −ρ ∂n

  6 j=1

S

 ∂φi dS = λi j q j ∂n j=1 6

q jφ j

(2.137)

for i = 1, 2, . . . 6, where  λi j = −ρ S

∂φi φ j dS. ∂n

(2.138)

The 36 constants λi j (i = 1, 2, . . . 6 and j = 1, 2, . . . 6) are calculated in the frame of reference embedded in the moving solid body, and depend only on the fluid density and the body shape, but not on time, since the potentials φi are time-independent, as shown above. Note also, that it is possible to show that λi j = λ ji (Loitsyanskii 1966). These parameters are called the added masses and they determine the dynamic effect of fluid on transient and not necessarily translational motions of solid bodies in inviscid fluids. For example, if a sphere of radius a is moving along the x∗ axis in fluid with velocity ∗ u0 (t ), in accordance with Eqs. (2.112)–(2.116) the corresponding non-zero potentials described using the spherical coordinate system with the origin at the sphere center read φ1 = −

a3 cos θ , 2 R2

φ=−

a3 cos θ ∗ u (t ) 2 R2 0

(2.139)

where R is the spherical radial coordinate, and θ is the spherical zenith angular coordinate reckoned from the direction of motion. Accordingly, the only non-zero added mass found from the integrals (2.138) is 2π λ11 = −ρ

dϕ 0



π dθ

a3 cos θ 2 a2



a3 2 cos θ 2 a3



2 m a2 sin θ = ρ π a3 = 3 2

(2.140)

0

where ϕ is the azimuthal spherical angular coordinate reckoned about the direction of motion, and m is the fluid mass displaced by the sphere volume. Therefore, in this particular case the sphere motion in fluid along the x∗ axis is described by the equation following from the first Eq. (2.129) as m∗

Du∗0 m Du∗0 = Fx∗ + Fx = Fx∗ − Dt 2 Dt

(2.141)

which yields 

m∗ +

m  Du∗0 = Fx∗ . 2 Dt

(2.142)

The latter shows that the added mass of a solid sphere in such transient translational motion is equal to one half of the mass of fluid which could occupy the sphere volume.

70

Selected Basic Flows and Forces

2.6

Friction and Shape Drag

2.6.1

Blunt Bodies Drag forces acting on blunt bodies moving in fluid with a constant velocity are determined by viscous friction forces (the friction drag) and flow asymmetry between the front and rear surfaces resulting from flow separation (the shape/form drag). Flow separation is still triggered by viscous forces. In the general case of a blunt body moving with a high enough Reynolds number the friction and shape drag cannot be predicted theoretically or numerically and are determined experimentally and lumped into the socalled dimensionless drag coefficient CD . Then, the drag force FD applied to a body moving with a constant absolute velocity u in a fluid flow with the absolute velocity v is presented in the form ρA w|w|, (2.143) 2 where A is the body area projected on a plane normal to the relative velocity vector (called also reference area), w = v − u is the relative velocity. The drag coefficient is determined by the body geometry and orientation in an external flow and by the Reynolds number, based on the material properties, density and viscosity, of the external fluid and velocity magnitude |w|. For a solid spherical particle of diameter D the drag coefficient is accurately described by the following empirical Clift–Gauvin correlation (Clift and Gauvin 1971, Clift et al. 1978)   24 0.42 (1 + 0.15Re0.687 ) + for Re < 2 × 105 (2.144) CD = Re 1 + 42.5/Re1.16 ! "  FD = CD (Re)

Schiller−Naumann

ρD|w| . Re ≡ μ

(2.145)

The term in the square brackets is known as the Schiller–Naumann formula (Schiller and Naumann 1933). Figure 2.12 compares Eq. (2.144) with experimental data and other fits available in the literature. This correlation is valid up to the laminar–turbulent transition in the boundary layer on the sphere, i.e. up to the critical Reynolds number. A very similar behavior of CD is observed for a circular cylinder placed normally to the flow. Non-spherical bodies can be classified either as regularly shaped particles (e.g. ellipsoids, cuboid or cones) or irregularly shaped particles (non-symmetric rough surfaces). Regularly shaped particles can arise in natural processes such as crystallization. Irregularly shaped particles can stem from random coagulation, but can also result from breakup of larger solid objects or particles, e.g. following collision phenomena. Irregular particles pose a significant challenge since their shape characterization is often not well-documented and the surface area or projected area may be difficult or impossible to measure. Moreover, the drag coefficient will also depend on the orientation of the particle. Drag coefficients of irregular particles in a wide range of the so-called shape factor can be found in Yarin (2012).

71

2.6 Friction and Shape Drag

1000

Experimental data Stokes theory Oseen theory White fit Clift & Gauvin fit Schiller-Naumann fit

Laminar Separated attached flow flow

100

10 CD

Laminar separation & turbulent wake

1

0.1 Turbulent separation & turbulent wake

0.01 0.1

1

10

100

1000 Rep

104

105

106

107

Figure 2.12 Drag coefficient for a smooth solid sphere in a wide range of Reynolds number for

incompressible flow conditions: the experimental data versus the Stokes theory for small Re and different empirical fits (Loth 2008). Reprinted with permission from Elsevier.

Loth (2008) has given a very comprehensive overview of drag coefficients for nonspherical particles with comparisons of numerous available correlations (Chien 1994, Haider and Levenspiel 1989, Hölzer and Sommerfeld 2008) to a large body of experimental data. His conclusion is that, while several expressions are quite reliable for many regularly shaped non-spherical particles, for substantially non-spherical shapes, “it is always best to obtain CD based on experiments with that same particle shape and Reynolds number.” Krueger et al. (2015) are equally skeptical about a universal correlation, stating that: “Due to the multitude of possible particle shapes the objective of deriving a single, universally valid drag law appears to be too far-reaching.” Perhaps the most successful of all correlations is that proposed by Ganser (1993), given as  24  0.4305 CD = 1 + 0.1118(ReK1 K2 )0.6567 + ReK1 K2 1 + 3305/ReK1 K2 K2 with

 K1 =

 DV 1 2 1/2 +  − 2.25 3 3 D

for isometric bodies, or    DV 1 2 K1 = ⊥ + 1/2 − 2.25 3 3 D

0.5743

K2 = 101.8148(− log())

0.5743

K2 = 101.8148(− log())

(2.146)

(2.147)

(2.148)

for non-isometric ones, whereby the various geometric parameters used to describe the shape influence are summarized in Table 2.1.

72

Selected Basic Flows and Forces

Table 2.1 Geometric definitions for regular, non-spherical particles (ellipsoids, cones, disks). Volume equivalent diameter

DV =

Particle sphericity

=

Surface area of particle

S

Crosswise sphericity

⊥ =

 3

6Vp π

π DV2 S π DV2 4A

More recently Dioguardi and Mele (2015) proposed a new shape-dependent drag correlation for non-spherical rough particles, valid in the 0.03–10000 Reynolds number range. This correlation, as with most other correlations, incorporates the projected area of the particle towards the flow direction (A), underlying the fact that the shape drag strongly dominates the friction drag for most blunt bodies, typically in the 80%/20% ratio.

2.6.2

The Rayleigh Formula for the Shape Drag of a Plate Moving in Liquid with Flow Separation at Its Edges It is instructive to calculate the shape drag force in the case where it is possible, for a plate moving in liquid normally to itself with flow separation. In this case the planar problem can be solved by applying the hodograph method developed by Helmholtz and Kirchhoff (Birkhoff and Zarantonello 1957, Lamb 1959, Gurevich 1966, Loitsyanskii 1966). The flow structure is sketched in Fig. 2.13, where the physical complex plane z = x + iy is depicted, with x and y being the Cartesian coordinates centered at the plate center, and i being the imaginary unit. The flow velocity at infinity at point C is horizontal and its magnitude is V∞ . The flow separates at the plate edges B1 and B2 . The plate width is equal to h. The flow is assumed to be inviscid and irrotational. The flow potential φ is equal to −∞ at C, and can always

Figure 2.13 Flow about a plate with separation at the edges.

2.6 Friction and Shape Drag

73

Figure 2.14 (a) The complex potential plane χ . (b) The hodograph plane Z. (c) The complex

√ plane ω = 1/ χ.

be made equal to zero at the stagnation point A by choosing the additive constant. At infinity at C1 and C2 φ = ∞. On the other hand, the stream function ψ varies in the range from −∞ to 0 below the x-axis, and from 0 to ∞ above the x-axis. Therefore, the flow portrait in the complex potential χ = φ + iψ plane looks as is shown in Fig. 2.14a. Introduce also the hodograph plane as the plane of the inverse conjugate velocity V , namely, Z = dz/ dχ = 1/V . Denote the polar angle corresponding to the velocity direction by θ , and the velocity magnitude |V |. Then, Z = eiθ /|V |. On the free streamlines B1 C1 and B2 C2 pressure is constant and equal to p∞ , which is the pressure at the stagnation zone behind the plate (and also at infinity at point C). Therefore, from the steady Bernoulli equation |V | = V∞ at the free streamlines. Accordingly, the image of the free streamlines in the hodograph plane Z is the right-hand side of a circle of radius 1/V∞ (see Fig. 2.14b). On the other hand, the flow direction over the plate surface AB1 or AB2 in the physical plane (Fig. 2.13) is known, and there θ = π /2 or −π /2, respectively. Therefore, in the hodograph plane Z, the corresponding images are located on the vertical axis sections with θ = ±π /2 (see Fig. 2.14b). One is interested to establish the relation of the hodograph plane Z and the complex potential χ . If the function Z = f (χ ) is known, then dz/ dχ = f (χ ), and thus the complex potential χ = χ (z) could be found by integration. To achieve this goal, we √ introduce an additional complex plane ω = 1/ χ . It is easy to see that in this plane the flow field corresponds to the lower half-plane, as shown in Fig. 2.14c. Note that at point B1 in the physical plane z in Fig. 2.13 the flow potential φ = c, where c is an unknown positive real number, and ψ=0, since the streamline CAB1 and CAB2 is taken as the zero streamline. Hence, in the ω-plane points B1 and B2 correspond, respectively, √ √ to 1/ c and −1/ c (Fig. 2.14c). The flow in the physical plane z generates the corresponding “flow” in the hodograph plane Z, and this is the “flow” about a cylinder of

74

Selected Basic Flows and Forces

radius 1/V∞ with a certain velocity iM at infinity (Y = −∞), since it is directed along the vertical axis. It is emphasized that the “flow” about the right-hand side of the cylinder can be analytically continued to the left-hand side. The magnitude of that velocity M is a real number to be found. On the other hand, such a flow about a cylinder corresponds to the “complex potential” ω. Therefore, ω and Z are related as in the ordinary case of a flow about a circular cylinder, namely (Loitsyanskii 1966)     1 1 2 iM = −iM Z − 2 , (2.149) ω = iMZ + V∞ Z V∞ Z where the overbar denotes complex conjugate. √ At points B1 in the ω- and Z-planes, ω = 1/ c and Z = i/V∞ , respectively. Then, √ Eq. (2.149) yields M = V∞ /(2 c), and solving Eq. (2.149) for Z, we obtain √ √ 1 − cω2 i c ω+ . (2.150) Z= V∞ V∞ Note that the plus sign before the second term on the right-hand side in Eq. (2.150) is chosen to have ω = 0 and Z = i/V∞ at point C, as follows from the physical situation √ depicted in Fig. 2.13. Since Z = dz/ dχ and ω = 1/ χ, Eq. (2.150) can be re-written as √ √ i c−χ + c dz = . (2.151) √ dχ V∞ χ Integrating the latter equation from the stagnation point and accounting for the boundary condition that at z = 0 the complex potential χ = 0 (see Figs. 2.13 and 2.14a), one finds     c − 2χ c i  √ + 2 cχ . χ (c − χ ) + arccos (2.152) z= V∞ 2 c According to Figs. 2.13 and 2.14a, z = ih/2 corresponds to χ = c, which allows one to find c using Eq. (2.152). Thus, c=

hV∞ . π +4

(2.153)

The drag force FD acting on the plate is found as  h/2  h/2  2  FD = 2 V∞ − |V |2 dy (p − p∞ )dy = ρ 0

(2.154)

0

where the steady-state Bernoulli equation is used, and ρ is the liquid density. At the wetted surface of the plate AB1 the stream function ψ = 0, and thus χ = φ. Also, x = 0, and thus z = iy there. Therefore, Eq. (2.151) yields on AB1 √ √ 1 c−φ+ c dy = . (2.155) √ dφ V∞ φ Note also that on AB1 the x-component of the flow velocity V is equal to u = 0. On the other hand, the y-component of the flow velocity V is equal to v = dφ/ dy.

2.7 Dynamics of a Rim Bounding a Free Liquid Sheet

75

Therefore, on AB1 the velocity magnitude |V | = v = dφ/ dy. Then, using Eqs. (2.154) and (2.155), we find the drag force as  c√ c−φ dφ = π ρV∞ c. (2.156) FD = 2ρV∞ √ φ 0 Substituting Eq. (2.153) into Eq. (2.156), we arrive at the Rayleigh formula for the drag force acting on a plate FD =

π 2 hρV∞ 4+π

(2.157)

with the corresponding drag coefficient CD being given by CD =

π FD = ≈ 0.44. 2 (1/2)ρV∞ 2h 4+π

(2.158)

It is instructive to compare this theoretical result with the experimental value (Prandtl and Tietjens 1957) CD ≈ 1 (in our notation). The theory significantly underestimates the drag force, since it assumes that pressure behind the plate is equal to p∞ , whereas in reality it is much less than that. This fact cannot be taken into account by the method of Kirchhoff. Still, it is remarkable that the quadratic dependence of the drag force on velocity V∞ is predicted theoretically.

2.7

Dynamics of a Rim Bounding a Free Liquid Sheet At the free edges of a free, stationary, uniform, liquid sheet the surface tension can be balanced only by the inertia of the liquid. Capillary forces are responsible for the emergence of a free rim propagating toward the liquid sheet with a finite velocity (Taylor 1959, Culick 1960)   2σ 1/2 (2.159) V = ρhS where ρ and σ are the liquid density and surface tension, hS is the thickness of the free sheet and V is the relative steady rim velocity in the direction normal to the rim centerline. Expression (2.159) is valid for low-viscosity liquids (Savva and Bush 2009); also see Section 1.11 in Chapter 1. The theoretical model of Entov et al. (1986) and Yarin (1993) for an arbitrary stationary rim bounding a planar sheet accounts for the flow and internal stresses in the rim and in the free sheet. Clanet and Villermaux (2002) have recently applied the rim dynamics equations to describe the stationary shape of the rim that arises when a radially expanding liquid sheet collides with an obstacle. Many physical phenomena can be explained by the propagation of the rim, among them the spreading and receding of a drop impacting onto a dry partially wettable substrate (Roisman et al. 2002, Rozhkov et al. 2002, Bartolo et al. 2005), dewetting of a dry substrate (Brochard-Wyart et al. 1987, Brochard-Wyart and de Gennes 1997),

76

Selected Basic Flows and Forces

y’

nS

FS

z’

XR

XR

XR

Figure 2.15 Element dζ of a free rim. (a) Geometry of the rim, (b) definition of the coordinate system, (c) the surface forces applied to the rim element (Roisman 2010). Reproduced with permission.

aerodynamic drop deformation by a shock wave (Hsiang and Faeth 1992), drop binary collisions (Ashgriz and Poo 1990, Brenn et al. 2001, Roisman 2004) and the interaction of two jets (Bush and Hasha 2004). In some cases the rim becomes unstable, which leads to the emergence of finger-like jets which subsequently break up into drops (see Fig. 1.6 in Section 1.10 in Chapter 1). The jets appear usually in the plane of the film. One of the most important phenomena related to the rim instability is liquid atomization by a fan spray sheet (Clark and Dombrowski 1972) or pressure swirl atomizers and secondary atomization by spray/wall or drop/substrate interaction (Yarin and Weiss 1995).

2.7.1

Definition of coordinate system and geometry of the rim centerline Consider a median surface of a free sheet in parametric form as x = XS (ξ, t ) and the average velocity in the sheet as uS (ξ, t ), where ξ is a two-component vector, and t is time. Define also a unit normal vector NS to the surface x = XS (ξ, t ). Consider also a free rim bounding a free sheet (Fig. 2.15). Here the centerline of the rim is defined in the parametric form as x = XR (ζ , t ), where ζ is a parameter associated with a material element moving with the rim. The area of the rim cross-section S is denoted by A(ζ , t ), the thickness of the liquid sheet at the rim location is hSR (ζ , t ). Consider then a coordinate system {s , y , z } with its origin at the rim centerline and the corresponding unit vectors {τ , n, b}, where τ is the unit tangent vector to the curve x = XR , n is the unit principal normal, b is the unit binormal vector as shown in Fig. 2.15b. Denote a unit vector nS = τ × NS normal to the rim centerline and parallel to the sheet surface. Any radius-vector x can be presented in the form x = XR + r, where r is the radius-vector in the moving system {τ , n, b} embedded in the rim. The length of the element dζ of the rim at the centerline is λ dζ , where the rim stretching parameter λ is defined as λ = |XR,ζ | with differentiation by ζ denoted in the subscript. The unit base vectors can now be defined in the form τ = λ−1 XR,ζ ,

n = τ,ζ |τ,ζ |−1 ,

b = τ × n.

(2.160)

2.7 Dynamics of a Rim Bounding a Free Liquid Sheet

77

The relationship between the base vectors and the local rim curvature, κ, and geometric torsion, τ , of the centerline can be obtained from the Serret–Frenet formulas τ,ζ = λκn,

n,ζ = λ(−κτ + τ b),

b,ζ = −λτ n.

(2.161)

Denoting the center-of-mass of the rim volume element G and the center-of-mass of the rim cross-section by C, the position of G is determined as r = rG , whereas the position of the center-of-mass C of the cross-section S is r = 0 by definition. For an element of the rim cross-section dS, an element of the rim volume is given by λ(1 − κy ) dS dζ such that the positions of the elements C and G are defined through

(y n + z b) dS ≡0 (2.162) rC = S A

(y n + z b) λ(1 − κy ) dS dζ κ = − (Ib n − Ibn b) rG = S (2.163) ) dS dζ A λ(1 − κy S where

 Ib =

y 2 dS, S

 Ibn =

y z dS,

S

 A=

dS,

(2.164)

S

and Ib and Ibn are the moments of inertia of the rim cross-section; Ibn vanishes for a symmetrical rim cross-section. The relative velocity w(r, ζ , t ) in the cross-section S is associated with rim stretching, deformation and swirl induced by the flow entering from the liquid sheet. Therefore, the absolute velocity v in S can be expressed as v =V +Ω×r+w

(2.165)

where V (ζ , t ) = XR,t ,

Ω = (n,t · b)τ − (τ,t · b)n + (τ,t · n)b

(2.166)

are the velocity of the centerline and the angular velocity in the cross-section S. The components of the angular velocity are obtained from Eqs. (2.160), (2.161) and (2.166) as n = −λ−1 Vb,ζ − τ Vn , b = λ−1 Vn,ζ + κVτ − τVb $ # τ = λ−1Vb,ζ ,ζ + λ−1 [λτVn ],ζ + λκτVτ − λτ 2Vb .

(2.167) (2.168)

The values of n and b are the same as obtained by Entov and Yarin (1984) for a free liquid jet.

2.7.2

Rim Balance Equations In order to formulate the rim balance equations the surface determining an element of the rim is subdivided into four regions (Fig. 2.15c). Regions 1 and 2 are the crosssections of the rim, region 3 is the interface between the rim and the sheet and region 4 is the free surface of the rim. The unit vectors τ1 and τ2 are normal to the cross-sections 1 and 2, whereas the unit vector nS is normal to area 3.

78

Selected Basic Flows and Forces

The mass balance is written in the Lagrangian form as (λA),t = hSR λ(V − uSR ) · nS (1 − κy S )

(2.169)

where uSR is the average velocity of the liquid in the liquid sheet at the position of the rim under consideration, y S is the coordinate of the mouth of the sheet entering the rim. The forces F1 and F2 (corresponding to the stretching of the rim) are applied to the surfaces 1 and 2, and the force FS is applied to the surface 3 from the sheet. The force FS consists of the surface tension (acting from the two free surfaces of the liquid sheet) and of the force associated with the viscous stresses in the liquid sheet FS = (2σ nS + hSR σS · nS )λ dζ (1 − κy S )

(2.170)

where σS is the average stress tensor in the liquid sheet. The total force F applied to the cross-section of the rim can be subdivided into the longitudinal, Pτ , and shear, Q, components. The momentum balance for an element of the rim can be conveniently expressed in the coordinate system moving with the center-of-mass G ρA(V,t + rG,tt ) = (2σ nS + hSR σS · nS )(1 − κy S ) − ρhSR [(uSR − V ) · nS ] (uSR − V )(1 − κy S ) + λ−1 F,ζ . The angular momentum of an element of the rim is defined as  dL = l λ dζ , with l = (1 − κy ) x × ρ v dS

(2.171)

(2.172)

S

with l being the angular momentum of the rim per unit length. The moment-of-momentum balance equation for the rim is written relative to the center-of-mass G of the element of the rim as l,t + λ,t λ−1 l = λ−1 MG,ζ + τ × Q + ρhS (1 − κy S )[(uSR − V ) · nS ][xGL × (uSR − V )]

(2.173)

where MG is the moment of the stresses relative to the center-of-mass, and xGL = rS − rG is the radius-vector of the mouth of the sheet in contact with the rim. It can be shown that the rate of change of the angular momentum of the rim element relative to G can be expressed with the help of Eq. (2.172) as  l,t = (1 − κy )xG × {wG,t + 2Ω × wG + Ω × (Ω × xG ) + Ω,t × xG + XG,tt }ρ dS S = (1 − κy )xG × {wG,t + 2Ω × wG }ρ dS + ρΩ × (IG · ΩG ) + ρIG · Ω,t S

(2.174) where XG = XR + rG is the radius-vector of the center-of-mass of the rim element, xG = r − rG is the radius-vector with the origin at point G, wG = w − rG,t is the relative velocity in the rim in the coordinate system fixed at the point G. The time derivatives rG,t and wG,t are taken in the accelerating and rotating coordinate system fixed at G. The

79

2.7 Dynamics of a Rim Bounding a Free Liquid Sheet

term IG ρλ dζ is the moment of inertia tensor of the rim element, defined by  IG = (1 − κy )[(xG · xG )I − xG ⊗ xG ] dS

(2.175)

S

where I is the unit tensor, symbol ⊗ denotes the diadic product. Note also that the volume integral of xG in the rim element vanishes due to the definition of the center-ofmass.

2.7.3

Force Components and Moments of Stresses Acting in the Rim Cross-section Approximating the rim cross-section by a circle of radius a(ζ , t ), it is natural to describe the stresses and the forces associated with the flow in the rim by the stresses which appear in a free circular cylinder of the same radius. If the effect of the acceleration of the rim centerline on the shape of its cross-section is small, the internal stresses in the rim can be estimated from the quasi-one-dimensional theory of the dynamics of free liquid jets (Entov and Yarin 1984, Yarin 1993). For a circular jet cross-section of radius a the diagonal components of the rate-of-strain tensor DJ are obtained there in the following form   3 DJ τ τ = −2δ 1 + y κ 2   $ #  (2.176) − z τ b − λ−1 n,ζ − κτ + y λ−1 b,ζ + τ n DJ τ τ DJ nn = DJ bb = − (2.177) 2 where 1 (2.178) δ = − (λ−1Vτ,ζ − κVn ). 2 Equation (2.177) follows from the continuity equation. The stresses in a Newtonian jet (rim) are given as in Eq. (1.23) in Chapter 1: σ = −pI + 2μD,

(2.179)

and are then obtained from the boundary conditions at the free surface as σJ nn = σJ bb = −σ [H − κa−1 y (1 + λ−2 a2,ζ )−3/2 ] H ≡ a−1 (1 + λ−2 a2,ζ )−1/2 − (1 + λ−2 a2,ζ )−3/2 λ−1 [λ−1 a,ζ ],ζ

(2.180) (2.181)

where p is the pressure, and H is the double mean curvature of the surface. In addition to the stresses associated with the rim stretching and bending, the stresses in the rim are influenced also by the flow generated by the flux entering the rim from the sheet. The volumetric flow rate of this flow per unit length of the rim is W = hSR (V − uSR ) · nS (1 − κy S ).

(2.182)

This flow leads to an additional growth of the rim radius a and the corresponding stresses are shear-free at the rim surface. To estimate these stresses, the linearized relative flow in the rim cross-section associated with the flow rate W is approximated by its

80

Selected Basic Flows and Forces

radial expansion wW =

W (y n + z b). 2π a2

(2.183)

The corresponding average components of the strain rate tensor are DW nn = DW bb =

W , 2π a2

DW τ τ = 0.

(2.184)

Finally, the rate-of-strain tensor in the rim is presented as D = DJ + DW .

(2.185)

Moreover, if one considers the rim dynamics, the terms associated with the velocity acceleration in the n and b directions are significant and cannot always be neglected. These terms do not appear in expressions (2.180). Since the shear stresses in the rim and the inertial effects associated with the radial velocity relative to the rim axis are assumed to be small and thus neglected, one expects that ∂σnn = −ρV,t · n, ∂y

∂σbb = −ρV,t · b ∂z

(2.186)

which is inconsistent with expressions (2.180). In fact, the gradients of the stresses in the rim resulting from the rim acceleration lead to deformation of the rim cross-section. Therefore, Eq. (2.180) is not applicable in such cases. Note that if the Bond number, Bo =

ρa2 |V,t − (V,t · τ )τ | σ

(2.187)

is small, the deformation of the rim cross-section is also negligibly small and its shape is accurately approximated by a circle and the average pressure in the rim is not affected significantly. Here the term [V,t − (V,t · τ )τ ] expresses the projection of the material acceleration on the plane {n, b}. In the present first-order approximation of the stresses in the rim the deformation of the rim cross-section is neglected σnn = σbb =

≡ −σ H − ρV,t · (y n + z b).

(2.188)

The stresses in the τ direction are determined using Eqs. (2.176), (2.179), (2.184) and (2.185) in the following form στ τ = −σ H − ρV,t · (y n + z b) + 3μDJ τ τ −

μW . A

(2.189)

The total longitudinal force Pτ acting in the jet (rim) cross-section consists of the force component associated with the internal stress στ τ and surface tension. It can be expressed in the form  −1/2  . (2.190) P = στ τ dS + 2π σ a 1 + λ−2 a2,ζ S

81

2.7 Dynamics of a Rim Bounding a Free Liquid Sheet

The expression for the force P applied to the rim of circular cross-section can then be derived from Eqs. (2.190) using (2.188) and (2.189) as    −1/2 −3/2 −1  −1   λ a,ζ ,ζ − 6μδ − μW. + σ 1 + λ−2 a2,ζ λ P = A σ a−1 1 + λ−2 a2,ζ (2.191) The moment of stresses M acting in the rim cross-section relative to the rim centerline is estimated from   M ≡ στ τ (r × τ ) dS = στ τ (z n − y b) dS. (2.192) S

S

The expression for M is obtained with the help of Eqs. (2.188) and (2.189) in the following form   (2.193) Mn = 3μI λ−1 n,ζ − τ b + κτ + ρI (V,t · b)   3 3 Mb = 3μI λ−1 b,ζ + τ n + κλ−1Vτ,ζ + κ 2Vn − ρI (V,t · n) (2.194) 2 2 where I = Ib = In = π a4 /4, Ibn = 0. Finally, the moment of stresses relative to the center-of-mass G is determined as MG = M − rG × P .

2.7.4

(2.195)

When a Circular Rim Does Not Appear The conditions at the free edge of a free liquid sheet do not always lead to the formation of a rim of a nearly circular cross-section, considered in this section. The deformations of the rim by the viscous stresses become significant if they are comparable with the capillary pressure. The ratio of the viscous to the capillary stresses is described by the Ohnesorge number Oh (see Section 1.2 in Chapter 1). Therefore, the analysis presented here is valid only for the cases when Oh  1 (Savva and Bush 2009). Moreover, a rim is not formed if the viscous stresses in the film are balanced by the surface tension. The normal stress component in the free sheet in the planar case considered here is σSyy = 4μS [see Eq. (6.91) in subsection 6.6.1 in Chapter 6]. The critical velocity gradient in the sheet at which a rim will not be formed can be obtained from the force balance σSyy h + 2σ = 0. This critical velocity gradient can thus be estimated as −σ /(μh). Several examples of sheet-like flow not accompanied by rim formation can be found in Debrégeas et al. (1995) and Roth et al. (2005). These studies are devoted to the investigation of hole growth in thin very viscous free films. The Ohnesorge number in these studies is of order Oh ∼ 103 , far outside the range of validity of the present theory. The rate of hole growth in Debrégeas et al. (1995) and Roth et al. (2005) is estimated using the energy balance. This solution can be, however, obtained also considering force balance at the hole surface. The axisymmetric creeping flow in the viscous film is v = ˙ (RR/r)e r , where R is the radius of the hole, r is the radial coordinate in the sheet plane,

82

Selected Basic Flows and Forces

er is the unit vector in the radial direction. The stress tensor in this sheet flow field is σrr = −p − 2μ

˙ RR , r2

σϕϕ = −p + 2μ

˙ RR , r2

σzz = −p

(2.196)

where p is the pressure in the film, and ϕ denotes the angular direction (polar coordinates). Since the axial stress at the film surface vanishes (thus p = 0) the expression for ˙ The radial stress at the hole surface r = R can be obtained in the form σrr = −2μR/R. force balance at the hole surface (σrr h + 2σ = 0) yields the differential equation for the hole radius ˙ Rh + σ = 0. (2.197) −μ R The solution of this equation is R = R0 exp[σ t/(μh)], where R0 is the initial hole radius. Such exponential film growth has been observed in the experiments of Debrégeas et al. (1995) and Roth et al. (2005). The result shows that flows at very high Ohnesorge numbers can be modeled without description of the rim propagation. The theory of the rim dynamics developed in the present section is, therefore, inapplicable to such flows.

2.8

References Ashgriz, N. and Poo, J. Y. (1990). Coalescence and separation in binary collision of liquid drops, J. Fluid Mech. 221: 183–204. Bakshi, S., Roisman, I. V. and Tropea, C. (2007). Investigations on the impact of a drop onto a small spherical target, Phys. Fluids 19: 032102. Bartolo, D., Josserand, C. and Bonn, D. (2005). Retraction dynamics of aqueous drops upon impact on non-wetting surfaces, J. Fluid Mech. 545: 329–338. Batchelor, G. K. (2002). An Introduction to Fluid Dynamics, Cambridge University Press. Birkhoff, G. and Zarantonello, E. (1957). Jets, Wakes, and Cavities, Academic Press, New York. Brenn, G., Valkovska, D. and Danov, K. D. (2001). The formation of satellite droplets by unstable binary drop collisions, Phys. Fluids 13: 2463. Brochard-Wyart, F. and de Gennes, P.-G. (1997). Shocks in an inertial dewetting process, C. R. Acad. Sc. Paris 324-IIb: 257–260. Brochard-Wyart, F., Di Meglio, J. and Quéré, D. (1987). Dewetting. growth of dry regions from a film covering a flat solid or a fiber, C. R. Acad. Sc. Paris 304-II-11: 553–558. Bush, J. W. and Hasha, A. E. (2004). On the collision of laminar jets: fluid chains and fishbones, J. Fluid Mech. 511: 285–310. Chien, S.-F. (1994). Settling velocity of irregularly shaped particles, SPE Drill. Complet. 9: 281– 289. Clanet, C. and Villermaux, E. (2002). Life of a smooth liquid sheet, J. Fluid Mech. 462: 307– 340. Clark, C. and Dombrowski, N. (1972). On the formation of drops from the rims of fan spray sheets, J. Aerosol Sci. 3: 173–183.

2.8 References

83

Clift, R. and Gauvin, W. H. (1971). The motion of particles in turbulent gas streams, Brit. Chem. Eng. 16: 229. Clift, R., Grace, J. and Weber, M. (1978). Bubbles, Drops, and Particles, Academic Press, New York. Culick, F. (1960). Comments on a ruptured soap film, J. Appl. Phys. 31: 1128–1129. Debrégeas, G., Martin, P. and Brochard-Wyart, F. (1995). Viscous bursting of suspended films, Phys. Rev. Lett. 75: 3886–3889. Dioguardi, F. and Mele, D. (2015). A new shape dependent drag correlation formula for nonspherical rough particles. Experiments and results, Powder Technol. 277: 222–230. Entov, V. M. and Yarin, A. L. (1984). The dynamics of thin liquid jets in air, J. Fluid Mech. 140: 91–111. Entov, V. M., Rozhkov, A. N., Feizkhanov, U. F. and Yarin, A. L. (1986). Dynamics of liquid films. Plane films with free rims, J. Appl. Mech. Tech. Phys. 27: 41–47. Feynman, R. P., Leighton, R. B. and Sands, M. (2006). The Feynman Lectures on Physics, Pearson/Addison-Wesley, San Francisco. Ganser, G. H. (1993). A rational approach to drag prediction of spherical and nonspherical particles, Powder Technol. 77: 143–152. Gurevich, M. I. (1966). The Theory of Jets in an Ideal Fluid, Pergamon Press, Oxford. Haider, A. and Levenspiel, O. (1989). Drag coefficient and terminal velocity of spherical and nonspherical particles, Powder Technol. 58: 63–70. Hölzer, A. and Sommerfeld, M. (2008). New simple correlation formula for the drag coefficient of non-spherical particles, Powder Technol. 184: 361–365. Hsiang, L.-P. and Faeth, G. M. (1992). Near-limit drop deformation and secondary breakup, Int. J. Multiph. Flow 18: 635–652. Kirchhoff, G. (1897). Vorlesungen über Mechanik, Band 1 von Vorlesungen über mathematische Physik, edited by W. Wien, fourth edn, B. G. Teubner, Leipzig. Kochin, N. E., Kibel, I. A. and Rose, N. V. (1964). Theoretical Hydrodynamics, Interscience Publishers, New York. Krueger, B., Wirtz, S. and Scherer, V. (2015). Measurement of drag coefficients of non-spherical particles with a camera-based method, Powder Technol. 278: 157–170. Lamb, H. (1959). Hydrodynamics, Cambridge University Press. Landau, L. D. and Lifshitz, E. M. (1969). Mechanics, Pergamon Press, New York. Landau, L. D. and Lifshitz, E. M. (1987). Fluid Mechanics, Pergamon Press, New York. Loitsyanskii, L. G. (1966). Mechanics of Liquids and Gases, Pergamon Press, Oxford. Loth, E. (2008). Drag of non-spherical solid particles of regular and irregular shape, Powder Technol. 182: 342–353. Peregrine, D. H. (1981). The fascination of fluid mechanics, J. Fluid Mech. 106: 59–80. Prandtl, L. and Tietjens, O. K. G. (1957). Applied Hydro- and Aeromechanics: Based on Lectures of L. Prandtl, Dover Publications, New York. Roisman, I. V. (2004). Dynamics of inertia dominated binary drop collisions, Phys. Fluids 16: 3438–3449. Roisman, I. V. (2010). On the instability of a free viscous rim, J. Fluid Mech. 661: 206–228. Roisman, I. V., Rioboo, R. and Tropea, C. (2002). Normal impact of a liquid drop on a dry surface: model for spreading and receding, Proc. R. Soc. London Ser. A-Math. 458: 1411–1430. Roisman, I. V. and Tropea, C. (2002). Impact of a drop onto a wetted wall: description of crown formation and propagation, J. Fluid Mech. 472: 373–397.

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Roisman, I. V., Yarin, A. L. and Rubin, M. B. (1997). Oblique penetration of a rigid projectile into an elastic-plastic target, Int. J. Impact Eng. 19: 769–795. Roth, C. B., Deh, B., Nickel, B. G. and Dutcher, J. R. (2005). Evidence of convective constraint release during hole growth in freely standing polystyrene films at low temperatures, Phys. Rev. E 72: 021802. Rouse, H. (1964). Advanced Mechanics of Fluids, John Wiley & Sons, New York. Rozhkov, A., Prunet-Foch, B. and Vignes-Adler, M. (2002). Impact of water drops on small targets, Phys. Fluids 14: 3485–3501. Savva, N. and Bush, J. W. M. (2009). Viscous sheet retraction, J. Fluid Mech. 626: 211–240. Schiller, L. and Naumann, A. (1933). Über die grundlegenden Berechnungen bei der Schwerkraftaufbereitung, Z. Ver. Dtsch. Ing. 77: 318–320. Taylor, G. I. (1959). The dynamics of thin sheets of fluid II. Waves on fluid sheets, Proc. R. Soc. London Ser. A-Math. 253: 296–312. Weinstock, R. (1974). Calculus of Variations, with Applications to Physics and Engineering, Dover Publications, New York. Whitham, G. B. (1974). Linear and Nonlinear Waves, John Wiley & Sons, New York. Yarin, A. L. (1993). Free Liquid Jets and Films: Hydrodynamics and Rheology, Longman & John Wiley & Sons, Harlow, New York. Yarin, A. L. and Weiss, D. A. (1995). Impact of drops on solid surfaces: self-similar capillary waves, and splashing as a new type of kinematic discontinuity, J. Fluid Mech. 283: 141–173. Yarin, L. P. (2012). The Pi-Theorem: Applications to Fluid Mechanics and Heat and Mass Transfer, Springer, Heidelberg.

Part I

Collision of Liquid Jets and Drops with a Dry Solid Wall

3

Jet Impact onto a Solid Wall

The oblique impact of an inviscid planar jet onto a plane wall dealt with in Section 3.1 is one of the classical problems associated with collision phenomena. It has significant ramifications not only for liquid–solid impacts but also for such solid–solid impacts as in the dynamics of shaped-charge (Munroe) jets discussed in Chapter 13. Moreover, as the material of Section 3.1 shows, such problems reveal the tremendous power of the mathematical methods rooted in the complex analysis associated with the theory of analytic functions. Normal impact of axisymmetric liquid jets is treated in Section 3.2, while the associate phenomenon of hydraulic jump is covered in Section 3.3.

3.1

Normal and Inclined Impact of Inviscid Planar Jets onto a Plane Wall An important model impact flow is depicted in Fig. 3.1. A steady-state planar jet of an ideal liquid impinges onto a plane AC. The jet possesses two free surfaces DA and D1 C which are in contact with an inviscid gas, which implies that the free surfaces are subjected to a constant atmospheric pressure. The jet has a uniform velocity at infinity (at the cross-section DD1 ), which will be used as the velocity scale. It delivers a known total flux q0 (rendered dimensionless with the velocity at infinity and a chosen length scale). Also, the angle α of the jet inclination at infinity (at the cross-section DD1 ) to the x-axis is given. Neither the shape of the free surfaces, nor the shape of the separating streamline γ are known in advance; they should be determined by the solution of the problem. Also, the partition of the total flux q0 between the two parts of the jet to the left and to the right from γ , q and q , respectively, is unknown beforehand. Obviously, q + q = q0 . The flow is considered to be potential, i.e. irrotational. The flow potential φ varies from −∞ to +∞ in the direction shown in Fig. 3.1; the value of φ at the stagnation point B (B1 ) can always be taken as zero, since potential is determined up to a certain additive. A stream function ψ is related to φ by the Cauchy–Riemann conditions, which allows one to introduce a complex potential χ = φ + iψ, where i is the imaginary unit and χ is a complex function of the complex variable z = x + iy. The values of the stream function ψ are constant along streamlines and are fully determined by the flux between these streamlines and the streamline coming to the stagnation point B (B1 ). In particular, at the free surfaces DA and D1 C the values of ψ are ψ = q and ψ = −q , respectively. Accordingly, the image of the flow domain in

88

Jet Impact onto a Solid Wall

φ = +∞

D

D1

ψ=0

A A

p∞

γ

y φ = −∞

z

ψ = −iq ′

ψ=q p∞

θ=α

C

φ=0

φ = −∞

C x

B B1

Figure 3.1 Sketch of a planar impinging jet.

the plane of the complex potential χ takes the form shown in Fig. 3.2. It is a horizontal strip from ψ = −iq to ψ = iq with a cut at −∞ < φ < 0. The points corresponding to those in Fig. 3.1 have the same notations. It should be emphasized that the separating streamline γ corresponds to the semi-axis 0 < φ < ∞. The free-jet problems of the type considered in the present section are solved effectively by using the hodograph method dating back to the works of Helmholtz and Kirchhoff (cf. subsection 2.6.2 in Chapter 2). Below the Joukowski modification of this method is used, which employs the modified complex hodograph plane, namely, Z = ln(1/V ), where V = dχ / dz is the conjugate velocity (Birkhoff and Zarantonello 1957, Lamb 1959, Kochin et al. 1964, Gurevich 1966). It should be emphasized that by its definition, Z = − ln |V | + iθ , where |V | is the velocity magnitude, and θ is the angle of inclination of the velocity vector to the x-axis in Fig. 3.1. In any flow restricted by straight solid lines and free surfaces, the plane of Z, which is called the modified hodograph plane, has a polygonal shape. Indeed, since the free surfaces DA and D1 C are isobaric, from the Bernoulli equation in steady-state, without gravity [Eq. (1.18) in Section 1.4 in Chapter 1], it follows that they carry the velocity magnitude of the jet at infinity, i.e. in terms of the dimensionless velocity there we have |V | ≡ 1 and

χ

ψ

A

iq

A C

B B1

C

–iq ′

D γ φ χ=0

D1

Figure 3.2 The complex potential plane corresponding to the physical flow plane of Fig. 3.1.

89

3.1 Normal and Inclined Impact of Inviscid Planar Jets

Y=θ iπ

Z A



B

D D1

–ln1 = 0

0 C

B1

X = –ln |V|

Figure 3.3 The modified hodograph plane corresponding to the physical flow plane of Fig. 3.1.

thus, ln |V | ≡ 0. Then, they correspond to the two sections of the segment of the vertical axis X = ln |V | ≡ 0 and 0 ≤ θ ≤ π in the modified hodograph plane of the present flow shown in Fig. 3.3. It should be emphasized that the velocity direction obviously changes over the free surfaces and thus, all values of θ from the interval 0 ≤ θ ≤ π on the vertical axis in Fig. 3.3 emerge. On the other hand, the velocity direction over the wall in Fig. 3.1 is the same as that of the wall. Therefore, θ = 0 on B1 C and θ = π on BA. Accordingly, the two horizontal straight lines conclude the “triangle” B1 CAB in Fig. 3.3. Note also, that the location of point D (D1 ) corresponding to the oncoming jet at infinity in Fig. 3.1 is known on the modified hodograph plane, since both the velocity magnitude |V | = 1 and its direction θ = α are known there; cf. Fig. 3.1. The potential φ decreases in the flow direction and thus 0 ≤ θ ≤ π corresponds to the flow toward the wall in Fig. 3.1 instead of the flow from the wall. If one were able to establish a relation between the modified hodograph and complex potential planes Z and χ , respectively, it would mean that the equation dχ / dz = f (χ ) is established (where f (•) is a function). Then, by integrating this equation, one could find the complex potential χ = (z) (with (•) being a function), which solves the problem. The required relation is established via an intermediary, auxiliary complex plane ω. Here it is chosen as the upper half-plane shown in Fig. 3.4. One can always request correspondence of any three points of the boundaries of the domains mapped onto the upper half-plane ω. Here we choose that point C is mapped to the point ω = −1, point D (D1 ) to the point ω = a = (q − q )/(q + q ), and point A to the point ω = 1. Then, the general Schwarz–Christoffel formula of complex analysis (Henrici 1974) provides us with the following functions mapping the polygonal domains in the planes χ (in Fig. 3.2) and Z (in Fig. 3.3) onto the upper half-plane ω: χ=

q q + q q ln(ω − 1) + ln(ω + 1) − ln(ω − a), π π π ω = ξ+iη

B1

C –1

D1D a

A 1

B

Figure 3.4 The auxiliary upper half-plane ω.

ξ

(3.1)

90

Jet Impact onto a Solid Wall

for χ , and ω = − cosh Z,

(3.2)

for Z. According to Eq. (3.1), the point χ = 0 of the χ -plane (the stagnation point) appears to be mapped to the point ω = ∞ of the ω-plane, as it is shown in Fig. 3.4. Then, Eq. (3.2) can be also written as  (3.3) eZ = −ω − ω2 − 1, with the minus sign before the square root chosen to guarantee that Z → ∞ as ω → ∞, i.e. that the stagnation point in the Z-plane (Z = ∞, since V = 0) is mapped correctly to the stagnation point ω = ∞ of the ω-plane. Equation (3.3) links the Z-value of point D (D1 ) corresponding to the jet at infinity, where the jet direction is known, Z = iα, with the image of that point in the ω-plane, ω = a, (see Figs. 3.3 and 3.4), which yields cos α = −a = −

q − q . q + q

(3.4)

The latter equation, together with the known total flux q0 , q0 = q + q , allows us to find the fluxes in the separating branches of the jet as q=

q0 (1 − cos α) , 2

q =

q0 (1 + cos α) . 2

(3.5)

According to Eqs. (3.1) and (3.2), the relation between the χ - and Z-planes is given by χ=

q q + q q ln(1 + cosh Z) + ln(1 − cosh Z) − ln(a + cosh Z) − iq . π π π

(3.6)

The asymptotic behavior of the separation streamline γ near the stagnation point B (B1 ) where φ → 0, χ → 0 and Z → ∞ (see Fig. 3.1) is found from Eqs. (3.1)–(3.3) as     x dχ y dχ ≈− ≈ 0, v = − ≈ , x ≈ 0 y ≈ 4A1/2 φ 1/2 , u =  dz 8A dz 8A (3.7) 2 where A = q0 sin α/(2π ), and {•} and {•} denote real and imaginary parts, respectively. In Eqs. (3.7) we explicitly use the fact that χ ≡ φ at the separating streamline γ , where ψ ≡ 0 and thus the complex potential is real. Note also, that according to the first two Eqs. (3.7) the separating streamline γ is always normal to the x-axis, irrespective of the inclination angle of the jet. To find the configuration of the separation streamline γ , Eq. (3.6) is differentiated by z. Then, accounting for the fact that χ ≡ φ at the separating streamline γ , we arrive at the equation determining the conjugate velocity at γ dV = F (V ), dφ

(3.8)

3.2 Normal Impact of Axisymmetric Impinging Jet

91

Figure 3.5 Predicted configurations of the separating streamline γ (ψ = 0) for two angles of jet

inclination: α = 150◦ , and α = 135◦ .

where the complex function F is given by the following expression F (V ) = −

V {q/[π (1+cosh Z)]−q /[π (1−cosh Z)]−(q+q )/[π (a+cosh Z)]} sinh Z

.

(3.9) Accordingly, the separating streamline γ is found by the numerical integration of the following system of four scalar equations     1 1 dv dy du dx = = =  {F } , = −{F }. (3.10) , , dφ dφ dφ dφ V V The integration begins from the asymptotic values of the four unknowns given by Eqs. (3.7) at a small φ = φ0  1. The predicted configurations of the separating streamline γ are shown in Fig. 3.5.

3.2

Normal Impact of Axisymmetric Impinging Jet The impact of axisymmetric jets has been studied rather intensively over many years due to the relevance of this phenomenon to various technologies, like material coating and cutting (Taylor 1966, Mann and Arya 2002, Leach et al. 1966, Brook and Summers 1969), cooling (Mozumder et al. 2006, Mudawar 2001, Pavlova and Amitay 2006, Wolf et al. 1993), cleaning, food processing (Ovadia and Walker 1998, Sarkar et al. 2004), liquid jet atomization (Ibrahim and Przekwas 1991, Ryan et al. 1995), etc. Several theoretical studies of this phenomenon can be found in the literature, including the inviscid solutions (Ryhming 1974, Rubel 1980, Scholtz and Trass 1970, Phares et al. 2000), relevant for the cases of high Reynolds number and are valid outside of the viscous boundary layer (cf. Fig. 3.6). Ryhming (1974) have used an integral method of Belotserkovskii (1957) for obtaining an approximate solution for jet impact onto an axisymmetric crater in a wall, produced by the erosion by an impacting jet. Phares et al. (2000) applied the separation of variables and presented the solution as a series for the

92

Jet Impact onto a Solid Wall

Figure 3.6 Sketch of normal impact of a circular liquid jet onto a solid flat wall at high Reynolds

number. The typical regions of the flow are: I – flow near the stagnation point, II – film flow with developing viscous boundary layer, III – fully developed wall film flow and IV – hydraulic jump, considered in Section 3.3.

stream function. They were able to compute the flows generated by impacts of jets with different distributions of the injection velocity over the cross-sectional jet radius, and also for different distances H between the injector and the wall (cf. Fig. 3.6). In the case of normal impact onto a flat substrate these solutions coincide if the distance from the orifice to the wall exceeds eight jet radii. The dimensionless pressure, scaled by ρUJ2 /2, produced by a steady normal jet impact in this case is shown in Fig. 3.7, where the experimental data from Bradbury (1972) and Giralt et al. (1977) are compared. The jet radius a and the centerline velocity UJ have to be taken near the stagnation point. These values account for the slight spreading and deceleration of the jet after injection. The radial velocity of liquid in the jet near the wall can be estimated

Figure 3.7 Dimensionless pressure produced by a steady normal impact of a circular jet onto a

solid flat substrate. The height of the jet is H/D > 8, where D is the injector diameter. The experimental data are from Bradbury (1972) and Giralt et al. (1977).

3.2 Normal Impact of Axisymmetric Impinging Jet

93

Figure 3.8 Dimensionless radial velocity Ur at the wall during a steady normal impact of a circular jet onto a solid flat substrate for various ratios of H/D. The height of the jet is H/D > 8. The velocity is estimated from the experimental data Giralt et al. (1977) for the wall pressure, shown in Fig. 3.7.

√ from the Bernoulli equation (1.18) in Section 1.4 in Chapter 1, as Ur /UJ = 1 − p/p0 . This velocity, estimated from the pressure data (Giralt et al. 1977) is shown in Fig. 3.8 for various ratios of the distance between the injector and the wall, H , to the injector diameter D. The best fit for the velocity data in the form of the Boltzmann function is Ur /UJ ≈ 1 −

2 , 1 + exp(1.84r/a)

H/D > 8.

(3.11)

For relatively large radii, the radial velocity is uniform and is equal to the impingement velocity. The steady film thickness can then be approximately estimated from the mass balance as h ≈ a2 /(2r), if the effect of the viscous boundary layer is small.

3.2.1

Viscous Boundary Layer Near the Stagnation Point of Impinging Jet Near the stagnation point the radial velocity outside the boundary layer can be approximated by a linear function Ur ≈ γ˙ r, where the constant γ˙ can be determined from the existing data, e.g. in Fig. 3.8. In this region the solution is determined by the dimensionless similarity variable √ z γ˙ (3.12) ξ= √ , ν such that the radial and axial velocity components are taken in the form  ur = Ur f (ξ ), uz = γ˙ νg(ξ )

(3.13)

where ν is the kinematic viscosity of fluid. The laminar boundary layer equations, discussed in Section 1.6 in Chapter 1, can be written in a cylindrical coordinate system {r, ϕ, z} in the axisymmetric flow near a flat

94

Jet Impact onto a Solid Wall

wall in the following form ∂rur ∂ruz + = 0, ∂r ∂z ∂ 2 ur ∂ur ∂ur dUr + uz = Ur +ν 2 . ur ∂r ∂z dr ∂z

(3.14) (3.15)

Substitution of Eqs. (3.13) into Eqs. (3.14) and (3.15) accounting for Eq. (3.12) yields the following system of two ordinary differential equations f (ξ ) = −g (ξ )/2



2

2g − 2gg + g − 4 = 0

(3.16) (3.17)

which has to be solved subject to the boundary conditions f (0) = 0,

g(0) = 0,

f (∞) = 1.

(3.18)

Numerical integration of Eqs. (3.16) and (3.17) yields f (0) ≈ 1.31194. For the impacting jets with H/D > 8 the value of γ˙ ≈ 0.79UJ /a can be estimated from Fig. 3.8. The velocity gradient at the wall near the stagnation point for such impinging jets is therefore  3 1/2 U γ˙ 3/2 r ∂ur , at z = 0, r < a. (3.19) ≈ 1.31 √ ≈ 0.92r 3J ∂z a ν ν Expression (3.19) for the velocity gradient at the wall can be used for the estimation of the shear stresses and heat fluxes at the wall near the stagnation point of an impinging jet.

3.2.2

Viscous Boundary Layer Far from the Stagnation Point At a distance from the stagnation point r a the velocity of the liquid outside the boundary layer is equal to the jet impact velocity UJ . In this case the components of the velocity field in the boundary layer can be taken in the form  νUJ g(ξ ), (3.20) ur = UJ f (ξ ), uz = r where the dimensionless similarity variable in this region is √ z UJ ξ= √ . νr

(3.21)

Substitution of Eq. (3.21) into the boundary layer equations (3.14) and (3.15) yields the following system of two ordinary differential equations 2 f − ξ f + 2g = 0



2 f + (ξ f − 2g) f = 0

(3.22) (3.23)

which has to be integrated subject to the boundary conditions (3.18). Numerical integration of Eqs. (3.22) and (3.23) yields f (0) ≈ 0.5752.

3.2 Normal Impact of Axisymmetric Impinging Jet

95

The displacement thickness of the boundary layer δ ∗ , associated with the volumetric flux, and the boundary layer thickness δ are defined as √  ∞  ∞ νr 1 ∗ (UJ − ur ) dz = √ [1 − f (ξ )] dξ (3.24) δ = UJ 0 UJ 0  √  δ UJ = 0.99. (3.25) f √ νr Numerical solution yields

√ νr δ ∗ = 0.993 √ , UJ

√ νr δ = 2.84 √ . UJ

(3.26)

Employing the mass balance in the jet to estimate the thickness of the radially spreading film in the outer region, one finds a2 + δ∗. (3.27) 2r This result is valid in the region where the film thickness is much larger than the boundary layer thickness, h δ. The critical radius R at which h = δ can be estimated from Eq. (3.27) as h=

R = 0.42aRe1/3 ,

Re ≡

aUJ . ν

(3.28)

The corresponding film thickness at this radius is estimated from Eq. (3.27) as hR = 1.51aRe−1/3 .

3.2.3

(3.29)

Fully Developed Flow in a Wall Film at r > R For a fully developed flow in an axisymmetric wall film the velocity profile is taken in the form z (3.30) ur = A(r) f (ζ ), ζ = h(r) where the function A(r) is determined from the total mass balance in the impinging jet  h(r) 2 π a UJ = 2π r ur dz, (3.31) 0

which yields A=

a2UJ , 2rhF

 F=

1

f (ζ ) dζ .

(3.32)

0

The axial momentum balance equation expresses the balance of the terms associated with the inertia of the liquid flow and viscous shear stresses at the wall. The integral form of the balance equation in the radial direction reads   h  ∂ur ∂ r , at z = 0. (3.33) u2 dz = −νr ∂r ∂z 0

96

Jet Impact onto a Solid Wall

Figure 3.9 Sketch of a hydraulic jump.

Here the viscous stresses associated with the velocity gradient in the radial direction are assumed to be much smaller than the shear stresses at the wall. Finally, Eq. (3.33), with the help of Eqs. (3.30) and (3.32), yields  1 2 f 2 (ζ ) dζ − 2F νr2 f (0) = 0. (3.34) a UJ [h + rh (r)] 0

The general solution of the latter equation reads

1 2ν f (0) 0 f (ζ ) dζ 2 C h= + r ,

1 r 3a2UJ f 2 (ζ ) dζ

(3.35)

0

where C is an integration constant. The function f (ζ ) has to satisfy the no-slip boundary condition at the wall, f (0) = 0, and the shear-free boundary condition at the film interface, f (1) = 0. The approximation of the function f by a parabola yields 5r2 −1 C + Re . (3.36) r 3a Matching this expression with the conditions (3.28) and (3.29) at the critical radius, finally provides the film thickness distribution in the region r > R h≈

h=

3.3

0.51a2 5r2 −1 + Re . r 3a

(3.37)

Hydraulic Jump At some radius the solution for the wall film, described in Section 3.2, becomes unstable and the film thickness jumps from a small value of the film thickness, h, to a relatively large value, s, as shown in Fig. 3.9. This phenomenon is associated with the effect of gravity. The hydraulic jump divides the film into the supercritical inner region with a thin film and subcritical outer region. In Liu and Lienhard V (1993) the phenomena of hydraulic jumps are subdivided into several types: (i) smooth jump without roller, (ii) jump with a single roller, associated with a vortex generated by a jet entering the subcritical region from the supercritical film, (iii) jump with double roller and (iv) unstable jump with turbulent flow and air

3.3 Hydraulic Jump

97

entrainment. The jump becomes less stable when the relative radius of the jump, rJ /a, decreases. The steady jump conditions are based on the mass and momentum balance equations  h UJ a2 u(z, r) dz = r (3.38) 2 0  h  s (p + ρu2 )|r=r j dz = rs (p + ρu2 )|r=rs dz (3.39) rj 0

0

respectively (Watson 1964, Bouhadef 1978, Liu and Lienhard V 1993). In the simplest case, assuming a uniform velocity in a film cross-section, taking rs ≈ r j , and neglecting the effect of surface tension, one finds 1  s = ( 1 + 8Frh − 1), h 2

Frh =

u2 , gh

(3.40)

where u is the average film velocity at r = r j , and Frh is the Froude number. The extensive experimental data (Olsson and Turkdogan 1966, Ishigai et al. 1977, Nakoryakov et al. 1978, Craik et al. 1981) show that the jump conditions given by Eqs. (3.38) and (3.39) are not universal and in some cases (for example, if the radius corresponding to the jump is less than 10 times the diameter of the impinging jet), they do not agree well with the experimental observations. Some improvement of the predictions of the theory can be achieved by accounting for surface tension (Bush and Aristoff 2003). A typical radius of curvature of the film interface at the subcritical side of the hydraulic jump is approximately half of the thickness s (Liu and Lienhard V 1993). The contribution of capillary effects to the pressure is therefore 2σ /s. Adding this pressure to the corresponding term in Eq. (3.39) and assuming h  r j yields the critical Froude number  √ KKr K 2 Kr (Bo + 4) − Bo  Frh = , (3.41) 2Bo(KKr − 1) where K ≡ s/h, Kr ≡ rs /r j and the subcritical Bond number Bo ≡

ρgs2 . σ

(3.42)

Figure 3.10 depicts the predicted values of the critical Froude number (3.41) in comparison with the experimental data (Liu and Lienhard V 1993) and with the simplified model (3.40) for various ratios of the subcritical and supercritical film thicknesses K = s/h. Accounting for the surface tension improves the agreement significantly for the values K < 100. For higher values of the ratio s/h > 100 the flow in the subcritical region near the hydraulic jump cannot be considered as uniform, as was assumed in the derivation of Eqs. (3.40) and (3.41). The flow can be probably better approximated by a wall jet injected from the supercritical region. In any case, as mentioned in Liu and Lienhard V (1993), in order to develop a reliable theory of hydraulic jump, the viscous and surface tension effects should be accurately accounted for.

98

Jet Impact onto a Solid Wall

Figure 3.10 Evaluation of Eq. (3.41) of Bush and Aristoff (2003) and the simplified Eq. (3.40) of

Watson (1964) for hydraulic jump incipience in comparison with the experimental data of Liu and Lienhard V (1993).

3.4

References Belotserkovskii, O. M. (1957). Flow past a circular cylinder with a detached shock wave, Dokl. Akad. Nauk 113: 509–512. Birkhoff, G. and Zarantonello, E. (1957). Jets, Wakes, and Cavities, Academic Press, New York. Bouhadef, M. (1978). Étalement en couche mince d’un jet liquide cylindrique vertical sur un plan horizontal, Z. Angew. Math. Phys. 29: 157–167. Bradbury, L. (1972). The impact of an axisymmetric jet onto a normal ground, Aeronaut. Quart. 23: 141–147. Brook, N. and Summers, D. A. (1969). The penetration of rock by high-speed water jets, Int. J. Rock Mech. Min. Sci. 6: 249–258. Bush, J. W. M. and Aristoff, J. M. (2003). The influence of surface tension on the circular hydraulic jump, J. Fluid Mech. 489: 229–238. Craik, A. D. D., Latham, R. C., Fawkes, M. J. and Gribbon, P. W. F. (1981). The circular hydraulic jump, J. Fluid Mech. 112: 347–362. Giralt, F., Chia, C.-J. and Trass, O. (1977). Characterization of the impingement region in an axisymmetric turbulent jet, Ind. Eng. Chem. Fundam. 16: 21–28. Gurevich, M. I. (1966). Theory of Jets in Ideal Fluids, Pergamon Press, Oxford. Henrici, P. (1974). Applied and Computational Complex Analysis, Vol. 3, John Wiley & Sons, New York. Ibrahim, E. A. and Przekwas, A. J. (1991). Impinging jets atomization, Phys. Fluids A 3: 2981– 2987. Ishigai, S., Nakanishi, S., Mizuno, M. and Imamura, T. (1977). Heat transfer of the impinging round water jet in the interference zone of film flow along the wall, Bull. JSME 20: 85–92. Kochin, N. E., Kibel, I. A. and Rose, N. V. (1964). Theoretical Hydrodynamics, Interscience Publishers, New York. Lamb, H. (1959). Hydrodynamics, Cambridge University Press. Leach, S. J., Walker, G. L., Smith, A. V., Farmer, I. W. and Taylor, G. (1966). Some aspects of rock cutting by high speed water jets [and discussion], Philos. Trans. R. Soc. Lond. Ser. A-Math. 260: 295–310.

3.4 References

99

Liu, X. and Lienhard V, J. H. (1993). The hydraulic jump in circular jet impingement and in other thin liquid films, Exp. Fluids 15: 108–116. Mann, B. S. and Arya, V. (2002). An experimental study to corelate water jet impingement erosion resistance and properties of metallic materials and coatings, Wear 253: 650–661. Mozumder, A. K., Monde, M., Woodfield, P. L. and Islam, M. A. (2006). Maximum heat flux in relation to quenching of a high temperature surface with liquid jet impingement, Int. J. of Heat Mass Transf. 49: 2877–2888. Mudawar, I. (2001). Assessment of high-heat-flux thermal management schemes, IEEE Trans. Compon. Packaging Technol. 24: 122–141. Nakoryakov, V. E., Pokusaev, B. G. and Troyan, E. N. (1978). Impingement of an axisymmetric liquid jet on a barrier, Int. J. Heat Mass Transf. 21: 1175–1184. Olsson, R. G. and Turkdogan, E. T. (1966). Radial spread of a liquid stream on a horizontal plate, Nature 211: 813–816. Ovadia, D. Z. and Walker, C. E. (1998). Impingement in food processing, Food Technol. 52: 46–50. Pavlova, A. and Amitay, M. (2006). Electronic cooling using synthetic jet impingement, J. Heat Transf.-Trans. ASME 128: 897–907. Phares, D. J., Smedley, G. T. and Flagan, R. C. (2000). The inviscid impingement of a jet with arbitrary velocity profile, Phys. Fluids 12: 2046–2055. Rubel, A. (1980). Computations of jet impingement on a flat surface, AIAA Journal 18: 168–175. Ryan, H. M., Anderson, W. E., Pal, S. and Santoro, R. J. (1995). Atomization characteristics of impinging liquid jets, J. Propul. Power 11: 135–145. Ryhming, I. L. (1974). An approximate solution of the steady jet impact problem, Z. Angew. Math. Phys. 25: 515–531. Sarkar, A., Nitin, N., Karwe, M. V. and Singh, R. P. (2004). Fluid flow and heat transfer in air jet impingement in food processing, J. Food Sci. 69: CRH113–CRH122. Scholtz, M. T. and Trass, O. (1970). Mass transfer in a nonuniform impinging jet: Part I. Stagnation flow-velocity and pressure distribution, AIChE J. 16: 82–90. Taylor, G. (1966). Oblique impact of a jet on a plane surface, Philos. Trans. R. Soc. Lond. Ser. A-Math. 260: 96–100. Watson, E. J. (1964). The radial spread of a liquid jet over a horizontal plane, J. Fluid Mech. 20: 481–499. Wolf, D. H., Incropera, F. P. and Viskanta, R. (1993). Jet impingement boiling, Adv. Heat Transf. 23: 1–132.

4

Drop Impact onto a Dry Solid Wall

Drop spreading after an impact onto a dry rigid wall is covered in Sections 4.1 to 4.4 taking into account the inertial and viscous effects, and liquid compressibility as well as geometric and thermal effects, for example associated with phase transition. In addition, the rim dynamics is considered in Section 4.5. The effect of the target curvature on drop spreading as well as surface encapsulation are dealt with in Section 4.6. Different scenarios accompanying drop impacts onto rigid walls are described in Section 4.7. The effect of the reduced gas pressure on drop impact is outlined in Section 4.8. Nonisothermal drop impacts are discussed in Section 4.9. This is extended to solidification and icing accompanying drop impact in Section 4.10. Drop impact onto a dry wall is an important element of various industrial processes; among them are spray cooling, cleaning, coating, wetting and ink-jet printing. Also, naturally occurring impacts can be of interest, for instance raindrop impacts are studied due to their relevance to soil detachment and erosion (Abuku et al. 2009, Imeson et al. 1981) and plant disease spreading. In Guigon et al. (2008) first steps have been made towards harvesting of the energy of raindrop impacts by transforming it to electricity using a piezoelectric system. Other examples include the impact of high-speed drops leading to the erosion of turbine blades (Li et al. 2008, Zhou et al. 2008) or to the deformation and fraction of rocks (Momber 2004). On the latter, also see Section 1.1 in Chapter 1. The phenomena associated with drop impacts have fascinated many researchers over the years. Recent advances in the theoretical modeling, the appearance of user-friendly, high-speed visualization systems and improvement of numerical methods for simulations of interfacial flows allow the elucidation of the drop impact and spreading on the wall in great detail. Drop impact is also a convenient model process to systematically investigate other physical phenomena, such as the nature of the dynamic contact angle: e.g., in Bayer and Megaridis (2006) contact line dynamics was studied and results explained in terms of hydrodynamic wetting theory and the molecular-kinetic theory of wetting. A comprehensive review of the modeling approaches, experimental results and also numerical investigations of drop impacts can be found in Rein (2002), Yarin (2006), Marengo et al. (2011), Josserand and Thoroddsen (2016). Generally, if the Reynolds and Weber numbers are high enough, the spreading drop transforms into a radially spreading lamella and an almost circular rim appearing due to capillary forces (Taylor 1959) and viscosity (see Fig. 4.1); also see Section 1.10 and 1.11 in Chapter 1, and Section 2.7 in Chapter 2. In the case of drop spreading and receding, the motion of

Drop Impact onto a Dry Solid Wall

101

Figure 4.1 Stages of drop impact onto a dry, rigid, smooth, partially wettable substrate: (a) initial

stage, (b) spreading, (c) receding. (d) vertical liquid jet after the drop receding (Roisman et al. 2002). With permission from The Royal Society.

the rim is also influenced by the wall friction and by the forces associated with wettability, which depend on the dynamic contact angle (Roisman et al. 2002). Drop impact onto a dry substrate is a frequently investigated phenomenon in fluid mechanics because of the numerous industrial applications and since the experimental setups can be relatively simple (Worthington 1876). Various outcomes of drop impact, like splash, deposition or rebound (Levin and Hobbs 1971, Stow and Hadfield 1981, Rioboo et al. 2001) are determined by the Reynolds and Weber numbers (see Section 1.2 in Chapter 1), as well as by substrate properties. Among them are the substrate roughness (Range and Feuillebois 1998), shape (Bakshi et al. 2007), elasticity (Pepper et al. 2008), porosity (Kellay 2005) or local wettability (Bico et al. 1999, Mock et al. 2005). These phenomena can be modeled and understood when the main mechanisms involved in drop spreading are identified and well described. Recently several new phenomena have been the subject of closer investigations. One phenomenon is the appearance of a very thin (of order of several nanometers) gas layer under the drop at the initial stage of drop impact and deformation (Xu et al. 2005, Xu et al. 2006). The formation of this film has been recently described theoretically and numerically (Mani et al. 2010). Similar analysis for collision of two drops has been performed much earlier (Footte 1975). In some cases the gas flow in this layer can probably lead to splash or to the entrapment of a single bubble (Chandra and Avedisian 1991). Moreover, drop impact in some cases leads to the appearance of numerous microbubbles entrapped in the region of the advancing contact line similar to the so-called Mesler entrainment (Esmailizadeh and Mesler 1986, Weiss and Yarin 1999, Thoroddsen et al. 2010, Dhiman and Chandra 2009). Large bubbles can remain trapped during the receding phase of water drop impact on a hydrophobic surface. The collapse of these air cavities may lead to high-speed jet ejection (Bartolo et al. 2006). It should be emphasized that the appearance of the small bubbles on the surface is incommensurate with the theoretical results of Mani et al. (2010) which predict the residual air film between the drop and the substrate. In fact this layer is unstable, since the spreading liquid is in contact with solid. Moreover, it is not clear how such a gas layer can lead to splash, as claimed in Mani et al. (2010). Typical processes of drop impact include initial drop deformation, spreading and receding (if the substrate is hydrophobic), Fig. 4.1. If the Reynolds and Weber numbers are high enough, the flow in the drop can be subdivided into three main regions:

102

Drop Impact onto a Dry Solid Wall

z

liquid film

h(r,t)

r

symmetry plane

Figure 4.2 Sketch of a film spreading on a symmetry plane (Roisman et al. 2009). Reprinted with

the permission of AIP Publishing.

r an inviscid, radially expanding flow in a lamella (Yarin and Weiss 1995, Roisman et al. 2009); r the flow in the rim, bounding the lamella; the rim is formed mainly due to capillary forces (Taylor 1959, Yarin 1993, Roisman et al. 2002); r an expanding viscous (and thermal, if the situation is non-isothermal) boundary layer (Roisman 2009).

4.1

Inviscid Flow on a Wall Generated by Inertia-Dominated Drop Impact The dimensionless parameters governing the dynamics of drop spreading on a dry wall are the impact Weber number, We = ρDV02 /σ , the Reynolds number, Re = DV0 /ν and the dynamic contact angle θ , where D is the initial drop diameter, V0 is the impact velocity, ρ, σ and ν are the liquid density, surface tension and kinematic viscosity, respectively (see Section 1.2 in Chapter 1). If both the Reynolds and Weber numbers are much higher than unity, the effect of the viscosity and surface tension on the flow in the spreading drop is small. Moreover, in this case the influence of the rim (see Section 2.7 in Chapter 2) on the flow in the lamella is also negligibly small. In this case the flow in the lamella and in the rim can be treated separately (Roisman et al. 2009).

4.1.1

Axisymmetric Spreading of a Free Thin Sheet Consider an axisymmetric spreading of a thin free liquid sheet with a cylindrical coordinate system {r, ϕ, z} fixed at the axis of the sheet, Fig. 4.2. Following the quasi-two-dimensional approach of Yarin (1993), the flow in the sheet is determined

4.1 Inviscid Flow on a Wall

103

by the averaged through the sheet cross-section velocity vr (r, t ) in the radial direction, where r is the radial coordinate, and t is the time. The solution for the flow in a spreading axisymmetric film, obtained in subsection 2.1.1, corresponds to a uniform stretching of a sheet, namely vr =

r , t +τ

vz = −

2z t +τ

(4.1)

where τ is a constant determined by the initial stages of drop impact (see below). This correspondence is elucidated in the subsequent subsection 4.1.2. Solution (4.1) has the same form as the remote inviscid asymptotic solution of Yarin and Weiss (1995). Note that here and hereinafter velocity is rendered dimensionless by V0 , coordinates by D and time by D/V0 . Expression (4.1) is relevant only in the intermediate central part of the sheet far from the sheet edge. Note that near the edge the capillary effects and the surface curvature are not negligibly small and have to be taken into account.

4.1.2

Evolution of the Lamella Thickness Far from the Rim Let us write the flow field in the lamella (4.1) in the Lagrangian form taking into account the fact that the velocity for any material point in the sheet is constant in the radial direction, namely, ξ τ r(ξ , t ) = ξ + vr t

vr (ξ , t ) =

(4.2) (4.3)

where ξ is the radial coordinate of a material element at t = 0, which follows from Eq. (2.7) in subsection 2.1.1 in Chapter 2. The equation for the conservation of mass of a material element in the sheet of the infinitesimal initial length dξ takes into account the film stretching during its motion in the radial direction dw = h0 (ξ )ξ dξ = h(ξ , t )r(ξ , t ) dr

(4.4)

where dw is the constant volume of the material element and h0 (ξ ) is the distribution of the sheet thickness at the time instant t = 0. Equation (4.4) yields the following general expression for the distribution of the film thickness, obtained with the help of Eqs. (4.2)–(4.4) h(ξ , t ) = h0 (ξ )

τ2 . (t + τ )2

(4.5)

This equation can be used for the description of the film thickness in parametric form in the framework of the Eulerian representation using the substitution rτ . (4.6) ξ= t +τ Expression (4.6) is derived from the equations of motion (4.2) and (4.3).

104

Drop Impact onto a Dry Solid Wall

At the axis the evolution of the lamella thickness given by Eq. (4.5) yields hC =

η , (t + τ )2

η = h0 (0)τ 2

(4.7)

where the constants τ ≈ 0.25,

η ≈ 0.39

(4.8)

are estimated in Roisman et al. (2009) from the numerical simulations of drop impact. This is the expression derived first in Yarin and Weiss (1995). It should be emphasized that the above-mentioned expressions for the lamella thickness can be obtained directly from Eq. (2.9) in subsection 2.1.1 in Chapter 2. The study of Bakshi et al. (2007) of the axisymmetric drop impact onto a dry spherical target confirms Eq. (4.7) experimentally. The target geometry (its convex shape) in this study allows one to observe the development of the lamella during the entire drop spreading process. Besides the other parameters, the height of the deforming drop at the symmetry axis was measured. It has been shown that at the initial phase of drop deformation the drop height first reduces with an almost constant velocity. The rear part of the drop moves almost as a rigid body. Then, at the dimensionless time instant t ≈ 1/2 the process of drop deformation switches to a new regime and the drop height follows the inverse square dependence on time predicted by the remote asymptotic solution (Yarin and Weiss 1995) given in Eqs. (4.1) and (4.7). The evolution of the drop height depends neither on the impact Weber number, nor the Reynolds number but is rather determined by only the ratio of the drop and target radii. Finally, when h is small enough, the viscous stresses govern the flow in the lamella. These viscous stresses, which become significant at time t = tviscous , lead to a flow damping. The residual film thickness is therefore a function of the Reynolds number only. In Fig. 4.3 the experimental data of Bakshi et al. (2007) and the results of the numerical simulations of Fukai et al. (1995), Šikalo et al. (2005), Mukherjee and Abraham (2007) for the evolution of the drop height hC at the symmetry center are shown as a function of time for various impact conditions, while the intermediate asymptotics is in very good agreement with Eq. (4.7). This is a universal expression for the thickness of the lamella generated by drop impact onto a dry flat substrate (or onto a symmetry plane) valid for all the impact conditions when both the Reynolds number and the Weber number are much larger than unity. As shown in Fig. 4.4, the shape of the central part of the lamella also almost does not depend on the impact conditions, except for the edge region associated with the rim formation. The best fit for the universal dimensionless shape function h0 (ξ ), shown in Fig. 4.4 is h0 ≈ 0.16 + 5.93 exp(−44.4ξ 2 ).

(4.9)

In Fig. 4.5 the distribution of the dimensionless pressure p (scaled by ρV02 ), predicted numerically for drop impact onto a symmetry plane, is shown for various time instants

4.1 Inviscid Flow on a Wall

105

Figure 4.3 Drop impact onto a dry substrate. The experimental data for the evolution of the

dimensionless lamella thickness at the impact axis (scaled by the initial drop diameter D) as a function of the dimensionless time (scaled by the characteristic impact time D/V0 ) (Roisman et al. 2009). Reprinted with the permission of AIP Publishing.

for two impact conditions: We = 761, Re = 83 and We = 1165, Re = 104. The curves corresponding to different impact conditions but to the same time instant practically coincide over most of the wetted part of the substrate, except for a short edge region of the lamella. The scatter in the results is attributed to the influence of the numerically predicted small bubbles at the symmetry plane.

(a)

(b)

Figure 4.4 Drop impact onto a symmetry plane. (a) The results of numerical predictions for the

dimensionless lamella thickness at the time instant t = 1 as a function of the dimensionless radius shown by lines compared with the numerical simulations of drop impact onto a flat rigid substrate shown by symbols. (b) The universal dimensionless shape function h0 (ξ ) defined in Eq. (4.5) (Roisman et al. 2009). Reprinted with the permission of AIP Publishing.

106

Drop Impact onto a Dry Solid Wall

Figure 4.5 Drop impact onto a symmetry plane. Numerical predictions of the dimensionless,

universal pressure distribution near the symmetry plane z = 0 as a function of the radial coordinate r at various time instants. The impact conditions are We = 761, Re = 83, and We = 1165, Re = 104 (Roisman et al. 2009). Reprinted with the permission of AIP Publishing.

The results of the numerical predictions of the pressure pC at the impact center {r = 0, z = 0} are shown in Fig. 4.6. The pressure pC decays nearly exponentially in time. The best fit of the data is pC = 1.7 exp(−3.1t ).

(4.10)

The results shown in Figs. 4.5 and 4.6 are also applicable to the prediction of the pressure distribution at the wall in the case of drop spreading on a smooth rigid substrate if the Weber and Reynolds numbers are high, since the pressure drop through a thin viscous boundary layer near the substrate is negligibly small.

4.2

Flow in a Spreading Viscous Drop, Including Description of Inclined Impact and Thermal Effects Consider a normal drop impact onto a planar rigid wall (see Fig. 4.7a). If the Reynolds and Weber numbers are high, the radially expanding flow in the lamella can be accurately described in the cylindrical coordinate system {r, ϕ, z} with the unit base vectors {er , eϕ , ez } by the remote asymptotic solution of Yarin and Weiss (1995) v0 =

rer − 2zez t +τ

(4.11)

following from Eq. (4.1). In the case of an oblique drop impact, the flow is no longer axisymmetric, since the initial transverse component of the impact velocity initiates a translational motion in the

107

4.2 Flow in a Spreading Viscous Drop

Figure 4.6 Drop impact onto a symmetry plane. Numerical predictions of the dimensionless

pressure pC at the impact point {r = 0, z = 0} as a function of the dimensionless time. The impact conditions are We = 761, Re = 83, and We = 1165, Re = 104, both data sets are shown by the same symbols (Roisman et al. 2009). Reprinted with the permission of AIP Publishing.

x-direction. Consider now a Cartesian coordinate system {x, y, z} with the unit base vectors {ex , ey , ez }, shown in Fig. 4.7b. In this coordinate system the drop impact velocity is presented as V = −V ez + U ex , where V and U are the normal and tangential components of the impact velocity. The inviscid flow in the lamella generated by an oblique drop impact can be obtained as a translation of the axisymmetric flow (4.11) with the velocity U ex (Roisman and Tropea 2002). The resulting flow can be presented in the Cartesian coordinate system as v0 =

(x − xc )ex + yey − 2zez + U ex , t +τ

xc = U t.

(4.12)

z z liquid film

U0

xc

y

r substrate

x (a)

(b)

Figure 4.7 Sketches of an axisymmetric spreading film (a) and of non-axisymmetric spreading

film (b) generated by normal and oblique drop impacts, respectively.

108

Drop Impact onto a Dry Solid Wall

The constant τ is usually much smaller than the characteristic time of drop impact D/V (Roisman et al. 2009) and can be neglected at long times after impact. The remote asymptotic solution for the flow in the lamella is therefore v0 =

(x + U t )ex + yey − 2zez . t

(4.13)

It can be shown that this flow satisfies exactly the continuity and momentum balance equations even if the liquid viscosity is significant. However, this velocity field does not satisfy the no-slip conditions at the wall. In order to determine the solution of the problem which accounts for the wall effects, the full Navier–Stokes equations have to be used.

4.2.1

Problem Formulation Consider now a problem analogous to the Stokes’ first problem, albeit an inclined drop impact onto a solid semi-infinite substrate is dealt with. Also, account for the heat convection in the spreading lamella, heat conduction in the substrate and possible phase transition initiated at their contact region. The possible phase transition phenomena include wall remelting, drop solidification or vaporization. The initial temperatures of the drop, Td0 , and of the substrate, Tw0 , are assumed to be uniform. Let us determine a viscous flow v = uex + vey + wez and the temperature distribution T in the spreading lamella and in the wall, which satisfy the following initial and boundary conditions (v − v 0 ) × ez = 0, v = 0,

T = Td0

at z → ∞ ∀ t > 0, and at t = 0 ∀z > 0

(4.14)

T = Tw0 at z → −∞ ∀ t > 0 and at t = 0 ∀z < 0

(4.15)

where v 0 is the remote asymptotic solution determined in Eq. (4.13). It should be emphasized that only the tangential, x and y components of the velocities v and v 0 must be identical far from the wall surface at z → ∞, since the viscous boundary layer developed near the wall/liquid interface and the propagation of the phase transition front can generate an additional uniform flow in the vertical z-direction. The near-wall matching and jump conditions for the temperature field and for the velocity are determined by the phenomena occurring during the liquid/wall contact. They can be different depending on whether drop spreading is accompanied by phase transition or not. Consider a solid/fluid or fluid/fluid interface z = Z ∗ (t ). The matching conditions express the mass, momentum and energy balance at the interface, the continuity of the shear stresses and temperature and the no-slip condition. The matching conditions are slightly different depending on whether there is no phase transition at the interface z = Z ∗ v = 0,

φq = 0,

(σ · ez ) = 0,

T = 0

(4.16)

4.2 Flow in a Spreading Viscous Drop

109

or the moving interface z = Z ∗ (t ) corresponds to a phase change: φq = m˙ ±L (σ · ez ) = m(v ˙ · ez − Z˙ ∗ )ez

ρ(v · ez − Z˙ ∗ ) = − ∗

T =T ,

(v × ez ) = 0

(4.17) (4.18) (4.19)

where σ is the stress tensor, φq is the heat flux. A jump of a physical quantity x through the interface z = Z ∗ (t ) is defined here by x = x(Z ∗+ ) − x(Z ∗− ). In the absence of the phase transition at the interface z = Z ∗ the temperature T ∗ has to be determined from the solution, while the instantaneous phase transition rate is equal to zero, m˙ = 0. If z = Z ∗ (t ) represents a moving phase transition front, the temperature T ∗ is a thermodynamic property equal to the melting or boiling temperature (depending on whether solidification, melting or evaporation takes place at the interface). The local mass rate of phase transition m˙ in this case is not known a priori and has to be determined from the solution. The value of L is equal to the latent heat of fusion or the latent heat of evaporation per unit mass, depending on which kind of phase transition at the interface z = Z ∗ (t ) is considered. The sign of L in Eq. (4.17) is positive in the case of liquid solidification or negative in the case of the substrate remelting or liquid evaporation. The flow and the temperature fields in the spreading drop have to satisfy the continuity, momentum (the Navier–Stokes) and energy balance equations ∂ρ + ∇ · (ρv) = 0 ∂t ∂v ρ + ρ(v · ∇)v = −∇p + ∇ · (μ[∇v + ∇vT ]) ∂t   ∂T + v · ∇T = ∇ · (k∇T ). ρcv ∂t

(4.20) (4.21) (4.22)

The viscosity, density, specific heat and thermal conductivity: μ = μ(T ), ρ = ρ(T ), cv = cv (T ), k = k(T ), depend on the local temperature, and on the local phase state and material.

4.2.2

Similarity Solution We seek a solution in the following form, determined by the remote asymptotic solution (4.13): √ ν0 (x + U0t )ex + yey (4.23) − 2g(ξ ) √ ez v = f (ξ ) t t T = Tw0 + (Td0 − Tw0 )!(ξ ) (4.24) where the dimensionless similarity variable is defined as in the Stefan problem (Tikhonov and Samarskii 2011) z (4.25) ξ=√ . ν0t

110

Drop Impact onto a Dry Solid Wall

√ Here ν0 is the constant characteristic viscosity of liquid, ν0t is the typical viscous length. For convenience we also define a local dimensionless temperature coefficient of a physical property x as Ax (T ) = (Td0 − Tw0 )

1 dx . x dT

(4.26)

Since we consider variable material properties, which depend on the temperature but also on the local phase state and material, the similarity solution has to be applicable to the entire field, which includes spreading liquid lamella, solid wall and the intermediate region appearing as a result of the phase transition. The velocity field and the temperature distribution have to satisfy the following boundary conditions: f = 1,

∂p ∂p = = 0, ∂x ∂y f = g = 0, ! = 0,

! = 1,

at ξ → ∞

(4.27)

at ξ → −∞.

(4.28)

The z-component of the Navier–Stokes equation (4.21) together with the boundary conditions (4.27) and (4.28) yield the following form for the pressure field in the lamella p = ν0 ρP(ξ )/t + P1 (t ),

(4.29)

where the pressure function P(ξ ) can be evaluated by integrating the ordinary differential equation P + PAρ ! + g + ξ g + 4gg + 2

 ν Aμ g ! + g = 0, ν0

(4.30)

with ν(T ) being the local kinematic viscosity. The prime denotes differentiation in ξ . Since the pressure gradients in the x- and y-directions vanish, the corresponding components of the Navier–Stokes equation (4.21) can be simplified. Then, the continuity equation (4.20), the energy equation (4.22) and the Navier–Stokes equation (4.21) in the x- and y-directions can be reduced to the ordinary differential equations with a single variable – the similarity variable ξ Aρ (ξ + 4g)! = 0 4 Pr ! + Ak ! 2 + (ξ + 4g)! = 0 2 ν f − f + f 2 − (ξ + 4g) − (Aμ f ! + f ) = 0 2 ν0 f − g −

(4.31) (4.32) (4.33)

where Pr(T ) = ν0 ρcv /k is the local Prandtl number. Equations (4.31)–(4.33) form a system of ordinary differential equations for the scaled velocity components, f (ξ ) and g(ξ ), and temperature, !, which can be solved numerically if the material properties and their temperature dependencies are known and the boundary conditions at the wall defined. Several examples of various physical processes which can accompany the lamella spreading are depicted in Fig. 4.8.

4.2 Flow in a Spreading Viscous Drop

liquid film

ξ

ξ

ξ

ξ

liquid film

liquid film ξ=Ξ



vapor layer ξ=0

111

liquid film ξ=Ξ

drop/substrate interface



splat ξ= 0

ξ= 0

ξ= 0 ξ=Ξ

melted layer solid substrate

solid substrate

solid substrate

solid substrate

(a)

(b)

(c)

(d)



Figure 4.8 Examples of various thermodynamic processes accompanying lamella (film) spreading

and the interfaces between different regions: (a) spreading without phase transition, (b) film boiling, (c) film solidification and (d) target melting. Interfaces with phase transition are shown by bold lines (Roisman 2010). Reproduced with permission.

The matching conditions are obtained from Eqs. (4.16) for a solid/fluid or solid/solid interface ξ = 0 without phase transition g = 0,

f = 0,

(k! ) = 0,

! = 0

(4.34)

and the corresponding conditions for a fluid/fluid interface are g = 0,

 f = 0,

(μ f ) = 0,

(k! ) = 0,

! = 0.

(4.35)

The matching and jump conditions for a moving solid/fluid boundary ξ = ∗ with phase transition are obtained from Eqs. (4.17)–(4.19) in the following form   Td0 − Tw0 ∗ =− (k! ) (4.36) ρ 2g + 2 ±Lν0 T∗ != , f = 0. (4.37) Td0 − Tw0 The corresponding matching and jump conditions at a moving fluid/fluid boundary ξ = ∗ with phase transition are   Td0 − Tw0 ∗ =− ρ 2g + (k! ), (μ f ) = 0 (4.38) 2 ±Lν0 T∗ != ,  f = 0. (4.39) Td0 − Tw0 In all the cases under consideration the similarity conditions are satisfied, since the variables x, y, z and t do not explicitly appear in the boundary and interfacial conditions. It should be emphasized that a non-zero velocity field exists, in principle, in the solid regions as well, resulting from the thermal expansion. The velocity field in the solid is determined by the temperature field and the thermal expansion coefficient, as well as by the elastic and plastic properties of the material. These velocities, however, are usually much smaller than the typical drop velocity and can be neglected in the present analysis.

112

Drop Impact onto a Dry Solid Wall

4.2.3

Drop Impact with Solidification and Constant Thermophysical Properties of Materials Consider a particular case of drop impact with solidification, in which the dependencies of the material properties of drop, splat and target on temperature are negligible. Then, the system of Eqs. (4.31)–(4.33) reduces to the following form valid for all the three regions f = g Pr ! + (ξ + 4g)! = 0 2 1 g + 2gg + ξ g + g − g 2 = 0. 2

(4.40) (4.41) (4.42)

We can subdivide the entire domain into three regions: solid target g = g = 0 at ξ ∈ [−∞, 0], splat (g = g = 0) at ξ ∈ [0, ∗ ] and the spreading liquid lamella at ξ ∈ [∗ , ∞] (see Fig. 4.8c). For simplicity we also neglect the change in the density of the drop material during solidification.

The flow and the temperature distribution in the liquid drop For the liquid flow, ξ > ∗ , Eq. (4.42) can be solved numerically subject to the boundary conditions g = g = 0,

at

ξ = ∗ ,

and

g = 1,

at

ξ → ∞.

(4.43)

In Fig. 4.9 the results of the numerical calculations of the scaled velocity components g and f = g are shown for various values of ∗ . Each curve intersects the axis g = 0 (or respectively, f = g = 0) at ξ = ∗ . It can be shown that at ξ → ∞ the scaled velocity g behaves as g → ξ − ∗ − γ (∗ ), where the constant γ depends only on the position of the interface ξ = ∗ and is found from the numerical solution of Eq. (4.42). The value of ∗ + γ (∗ ) corresponds to a uniform vertical flow generated by an expansion of the viscous boundary layer and propagation of the solidification front. In Fig. 4.10 the numerical predictions for γ are shown as a function of ∗ . For convenience this function can be fitted by an exponential function of ∗ and written in the following explicit form γ ≈

0.314 0.290 + . exp[0.351∗ ] exp[0.0599∗ ]

(4.44)

The approximation (4.44) is valid on the interval ∗ ∈ [−3, 15] and is used in the further calculations of the temperature field. A general solution for the temperature field in the liquid drop is determined from Eq. (4.41) as ! = C1 + C2 I (Pr, ∗ , ξ ) √  ξ    χ Pr Pr 2 ∗ I (Pr,  , ξ ) = √ exp − χ − 2Pr g(ζ ) dζ dχ 4 π ∗ ∗

(4.45) (4.46)

4.2 Flow in a Spreading Viscous Drop

113

Figure 4.9 Scaled components of the velocity of the liquid with phase transition at various

positions of the interface ξ = ∗ obtained by the numerical integration of Eq. (4.42) (Roisman 2010). Reproduced with permission.

where C1 and C2 are the integration constants, Pr = νρcv /k is the Prandtl number, and ζ and χ are dummy variables. The temperature at the interface ξ = ∗ is equal to the melting point temperature. Therefore, the scaled temperature distribution in the liquid drop, which satisfies the boundary conditions (4.27), is !l (ξ ) = !∗ + (1 − !∗ )

I (Prl , ∗ , ξ ) , I (Prl , ∗ , ∞)

(4.47)

and subscript l denotes liquid.

Figure 4.10 Parameter γ associated with the additional viscous uniform vertical flow at various

values of ∗ (Roisman 2010). Reproduced with permission.

114

Drop Impact onto a Dry Solid Wall

Temperature distributions in the solid regions: target and splat A general solution of the energy balance equation in the solid regions can be obtained substituting g = 0 into Eqs. (4.45)–(4.46) which yield  (4.48) ! = C3 + C4 erfc[− Prl ξ /2] where the corresponding Prandtl number is defined using the liquid viscosity. Denote !c the contact temperature at ξ = 0. The temperature distributions in the wall, !w , and in the splat, !s , which satisfy the boundary conditions (4.34) are  (4.49) !w = !c erfc[− Prw ξ /2] √ erf[ Prs ξ /2] $ !s = !c + (!∗ − !c ) #√ (4.50) erf Prs ∗ /2 where !c =

es !∗ #√ $ es + ew erf Prs ∗ /2

(4.51)

√ √ es = ks ρs cs and ew = kw ρw cw are the thermal effusivities of the splat and wall, respectively. At this stage only the value of ∗ is unknown. It should be determined from the boundary conditions related to the phase transition, in particular, from the local thermal balance at the solidification front.

Drop solidification The boundary conditions (4.36) and (4.37) for the temperatures !l and !s in the liquid and splat regions yield   2Ste ks ∗ !s ( ) − ! l (∗ ) (4.52) ∗ = Prl kl Ste =

cvl (Td0 − Tw0 ) L

(4.53)

where Ste is the Stefan number, cvl is the specific heat of liquid at constant volume and L is the latent heat of fusion per unit mass of the wall material. Finally, Eqs. (4.52) and (4.53) with the help of Eqs. (4.47) and (4.49)–(4.51) yield the following transcendental integral equation for ∗ : √ ks Prl π ∗ S!∗ − L(1 − !∗ ) = (4.54)  kl 2Ste where

# $ Prl exp − Prl ∗2 /4 L= I (Prl , ∗ , ∞) # $ √ ew Prs exp − Prs ∗2 /4 #√ $ S= . ew erf Prs ∗ /2 + es √

(4.55) (4.56)

4.2 Flow in a Spreading Viscous Drop

115

Figure 4.11 Function L(Pr, ∗ ), defined in Eq. (4.55), as a function of ∗ for various values of

the Prandtl number (Roisman 2010). Reproduced with permission.

This equation can be solved numerically when all the thermophysical parameters are known. The function L(Prl , ∗ ) in Eq. (4.55) is not given in an explicit form, since it involves the integral function I (Prl , ∗ , ∞), which depends on the scaled velocity g(ξ ). However, it can be evaluated for any given Pr and ∗ . The results of the calculations of L(Prl , ∗ ) are shown in Fig. 4.11 for various values of the Prandtl number. For each value of the Prandtl number the function L(Prl , ∗ ) is a monotonously increasing function of ∗ . Negative values of ∗ correspond to the case of remelting of the substrate if it consists of the same material as the impacting liquid drop. The dimensional height Z ∗ of the solidification front increases in time and can be expressed as √ (4.57) Z ∗ = ∗ νt.

Drop impact without phase transition Using the boundary conditions (4.27) and (4.28) as well as Eq. (4.34) in Eqs. (4.45) and (4.46), and accounting for the fact that g = 0 at ξ < 0, the following expressions for the temperature fields !l and !w in the liquid and solid regions are obtained !l =

el + es I (Prl , 0, ξ ) , el + es I (Prl , 0, ∞)

at

ξ >0

(4.58)

 √  Prw el erfc − ξ , at ξ < 0, (4.59) !w = el + ew I (Prl , 0, ∞) 2 √ where el = kl ρl cvl is the thermal effusivity of liquid, with kl being the thermal conductivity of liquid and ρl being the liquid density.

116

Drop Impact onto a Dry Solid Wall

Figure 4.12 Function I(Pr, 0, ∞) for the liquid phase as a function of the Prandtl number,

calculated using Eq. (4.46), in comparison with the approximate solution (4.63) for small Prandtl numbers (Roisman 2010). Reproduced with permission.

The corresponding dimensional contact temperature at z = 0 can now be determined as Tc =

el Td0 + ew I (Prl , 0, ∞)Tw0 el + ew I (Prl , 0, ∞)

(4.60)

and the heat flux at the interface z = 0 is φq =

el ew (Tw0 − Td0 ) √ √ . [el + ew I (Prl , 0, ∞)] π t

(4.61)

The dimensionless function I (Pr, 0, ∞) defined in Eq. (4.46) can be calculated numerically for various values of the Prandtl number. The results of the calculations are shown in Fig. 4.12. In the entire range of the Prandtl number the value of I (Pr, 0, ∞) is smaller than unity. In the limiting case Pr → ∞ (an instantaneous contact of two solid semi-infinite bodies) the value of I (Pr, 0, ∞) approaches unity. These results explain the enhanced cooling effect of drop or spray impact. For very small Prandtl numbers, typical for liquid metals, the flow viscosity can be neglected, the scaled velocity g(ξ ) can thus be approximated by its outer asymptotic solution ξ − ∗ − γ (∗ ), and the function I (Prl , ∗ , ∞) can be estimated using Eq. (4.46) as √     χ Prl ∞ Prl 2 χ − 2Prl exp − (ζ − ∗ − γ ) dζ dχ I (Prl , ∗ , ∞) = √ 4 π ∗ ∗ +γ √   √  5 Prl ∗ Prl ∗ 2 (4.62) = exp − ( + γ ) erfc √ ( − 4γ ) . 5 5 2 5

4.2 Flow in a Spreading Viscous Drop

Linearization of Eq. (4.62) in the limit of small Prandtl numbers yields √ 1 4γ (0) Prl I (Pr, 0, ∞) ≈ √ + , γ (0) ≈ 0.6. √ 5 π 5

117

(4.63)

As shown in Fig. 4.12, the approximate solution (4.63) agrees fairly well with the numerical solution for the Prandtl numbers Pr < 0.1.

4.2.4

Isothermal Case When the liquid properties do not depend on temperature or the temperature field is uniform, Eqs. (4.31)–(4.33) are reduced to f − g = 0

(4.64)

1 g + 2gg + ξ g + g − g 2 = 0. 2

(4.65)

The ordinary differential equation (4.65) for g can be solved numerically subject to the boundary conditions g = 0,

g = 0,

at

ξ =0

(4.66)

g = 1,

at

ξ → ∞.

(4.67)

Equations (4.65)–(4.67) form a nonlinear boundary value problem. The ordinary differential equation (4.65) was solved as the initial value problem for different values of the second derivative g (0) in order to find the solution which approaches to the boundary condition (4.67) as ξ → ∞, i.e. using the “shooting” method. Finally, two solutions of Eq. (4.65) are determined corresponding to the adjusted initial conditions g (0) = −0.618925 and g (0) = 1.0354. The solution satisfying g (0) = −0.618925 is associated with the separated boundary layer with the negative wall shear stress. This solution predicts the flow in the positive z-direction at ξ < 4. This unphysical solution is not considered further. The solution corresponding to g (0) = 1.0354 is shown in Fig. 4.13. The typical boundary layer thickness corresponding to g (ξ ) = 0.99 is uniform (i.e. it does not depend on the radial coordinate r) and can be estimated from the numerical solution as √ (4.68) δ0.99 = 1.88 νt. This value is much smaller than the corresponding thickness of the boundary layer in the two-dimensional first Stokes’ problem. The reason is in the presence of the negative component of the velocity in the normal-to-the-wall direction and in the positive velocity gradient in the radial direction. An expanding boundary layer generates an additional liquid flow also in the vertical direction. As shown in Fig. 4.13, the value of the function ξ − g(ξ ) asymptotically approaches some constant γ = 0 as ξ → ∞. The value of the constant γ = 0.60 is determined from the numerical solution of Eq. (4.65).

118

Drop Impact onto a Dry Solid Wall

g(ξ), g′(ξ)

g′(ξ) ξ – g(ξ)

γ

ξ Figure 4.13 Scaling functions g(ξ ) and g (ξ ) (Roisman 2009). Reprinted with the permission of AIP Publishing.

The velocity field in the boundary layer can now be finally expressed in the following form   √   r z ν z (4.69) vr = g √ , vz = −2g √ √ νt t νt t while the expression for the wall shear stress is obtained as √ τw = 1.0354 νρrt −3/2 .

4.2.5

(4.70)

Evolution of the Lamella Thickness At the distances from the substrate significantly exceeding the typical thickness of the boundary layer, z > δ, the function g(ξ ) approaches its asymptote g(ξ ) → ξ − γ . The corresponding axial velocity in the spreading lamella outside the boundary layer is, therefore, √ ν 2z (4.71) vz = − + 2γ √ , at z > δ. t t If one assumes an initial drop shape in the form (4.9), the additional axial velocity component related to viscosity results in the following corrected lamella thickness distribution, which is obtained using the velocity field outside the boundary layer    2.78r2 −2 0.01 + 0.37 exp − + hν (t ) (4.72) hL = (t + τ ) (t + τ )2 √ t 4γ hν = (4.73) √ . 5 Re

4.2 Flow in a Spreading Viscous Drop

119

The viscous term hν is negligibly small at the initial stages of drop spreading if the Reynolds number is very high. However, at long times and small lamella thicknesses this additional term becomes significant.

4.2.6

Lamella Flow Governed by Viscosity The thickness of the boundary layer can be expressed using Eq. (4.68). The dimensionless thickness can, therefore, be roughly estimated as  (4.74) hBL = 1.88 t/Re where the drop initial diameter is used as the length scale and the impact velocity as the velocity scale. The time tBL , at which the boundary layer reaches the free surface of the lamella at the axis, can be estimated from the condition hBL = h. At times t > tBL the flow in the lamella is governed mainly by the balance between liquid inertia and viscosity. The flow on a spherical substrate in this regime has been estimated in Bakshi et al. (2007). In the case of a flat substrate their analysis can be simplified. The evolution equation for the approximate lamella thickness on a planar surface can be obtained in the dimensionless form 3 h˙ 9 h˙ 2 =0 (4.75) + h¨ − 5 h Re h2 where the first two terms are associated with the inertial effects and the third one is associated with the viscosity. The solution for the film thickness can be expressed as −1  9/5  h0  h 15 15 V0 + − dh. (4.76) t = tBL + 14Reh0 h9/5 14Reh h 0 At some time instant the velocity in the lamella almost vanishes while the thickness approaches to some asymptotic value determined in Bakshi et al. (2007). The residual lamella thickness can be estimated from the solution (4.76) as h9/14 0 hres =  5/14 h−1 + 14ReV 0 /15 0

(4.77)

and also evaluated using the matching conditions as hres ≈ 0.96η1/5 Re−2/5 ,

(4.78)

where η is defined in Eq. (4.7). The lamella formed by an axisymmetric drop impact onto a rigid spherical target of the dimensionless diameter Ds is a more general case of the impact onto a flat target (which corresponds to Ds → ∞). The comparison between the theory and the results of the numerical simulations (Fukai et al. 1995) of drop impact onto a flat solid substrate at Re = 6020, and We = 117 is shown in Fig. 4.14. Let us approximate the evolution of the lamella thickness in the case of drop impact onto a flat substrate at large times by Eqs. (4.72) and (4.73) and neglect the small constant τ . This rough estimation yields tBL ≈ 0.87η2/5 Re1/5 . The thickness of the lamella

120

Drop Impact onto a Dry Solid Wall

Figure 4.14 Drop impact onto a flat surface at Re = 6020. Numerical simulations (Fukai et al. 1995) of the drop height at the axis as a function of time in comparison with the thickness of the boundary layer (4.74) and with the theoretical predictions (4.76) for the viscous lamella regime (Roisman 2009). Reprinted with the permission of AIP Publishing.

at the time instant t = tBL and the rate of the decrease in the thickness at this time instant are obtained in the following form h0 ≈ 1.77η1/5 Re−2/5 ,

h˙ 0 = V0 ≈ 2.78η−1/5 Re−3/5 .

(4.79)

Finally, the expression for the residual film thickness resulting from the normal impact of a drop onto a smooth flat substrate estimated using η = 0.39 [see Eq. (4.8)] and Eq. (4.77) can be written in the following form hres = 0.79Re−2/5

(4.80)

which is valid for high values of the impact Reynolds number.

4.3

Initial Phase of Drop Impact

4.3.1

Compressibility Effects at Early Stages At the short initial stage of drop contact with a rigid substrate the compressibility effects are dominant. These effects are associated with generation and propagation of a shock wave in the drop (Lesser and Field 1983, Haller et al. 2003). In most of  the considered cases the ratio of the viscous length to the initial drop size is rather small, ν/(Dc)  1, where c is the speed of sound in the drop. The compressible initial stage of high-speed impact of drops of diameters larger than 1 µm can be thus treated using the inviscid approach.

4.3 Initial Phase of Drop Impact

4.3.2

121

Incompressible Flow At early times of drop impact and during the initial deformation stage the remote asymptotic flow field expressed by Eqs. (4.2) and (4.3) is not valid. Description of the kinetics of drop spreading at this early stage remains mainly empirical (Mongruel et al. 2009). These data can be considered as an extension of the earlier measurements of drop spreading diameter for normal drop impact (Rioboo et al. 2002). In Mongruel et al. (2009) scaling analysis for the evolution of the lamella radius and thickness are proposed, based on the assumed characteristic time of the lamella ejection, ν/V02 . However, the effect of the capillary forces and the formation of the rim are neglected in this analysis. Understanding of the flow in the drop during the initial stage of its impact and deformation is rather important, especially in the case of impact onto structured, rough, eroding or soft/elastic targets, since the pressure and the shear stresses at this phase are especially high. This problem is extremely difficult, since the flow in the spreading drop is singular at t = 0, due to an instantaneous deceleration of the drop front in the direction normal to the surface, from the impact speed, V0 , to zero. In several papers (Courbin et al. 2008, Guiot et al. 2007) the dimensionless pressure at the impact point (scaled by ρV02 ) is assumed in the form p ∼ t −1/2 at t  1. On the other hand, if the radius of the wetted spot a is much smaller than the drop radius (which corresponds to t < 1/2) the velocity field in the drop can be accurately approximated by the irrotational flow past a disk of radius a. The velocity potential of such a flow is well known (Batchelor 2000). At the impact point this velocity potential is expressed as φ0 = −2a/π . At the initial stages (t < 12 ) the shape of the deforming drop can be approximated by a truncated sphere. The dimensionless radius of the wetted spot rendered dimensionless by D is, therefore, a ≈ 1/4 − (1/2 − t )2 . Therefore, the stagnation pressure p at this point can be obtained from the Bernoulli equation (1.18) in Section 1.4 in Chapter 1 as p=

dφ0 1 1 − 2t 1 − = + √ , 2 ∂t 2 π t − t2

at

t
1/2 the pressure at the impact point decays nearly exponentially and is expressed by Eq. (4.10). It should be emphasized that the expressions (4.81) and (4.10) are valid only if compressibility effects in the liquid are negligibly small, namely at times larger than tc = V0 /c, where c is the speed of sound in the fluid. The important data for the height of the rim bounding a spreading lamella at the initial stages can be found in de Ruiter et al. (2010). These experiments clearly demonstrate that the radius of the rim is not determined solely by the drop viscosity, but depends also on the surface tension. Nevertheless, at high Reynolds numbers the dimensionless lamella thickness is accurately scaled by the characteristic viscous length h ∼ (t/Re)1/2 . Therefore, the drop spreading diameter D1 at the time t = 1 (rendered dimensionless

Drop Impact onto a Dry Solid Wall

10 Numerical results: Roisman et al. (2009) Eggers et al. (2010) Theory

p

1

0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 t Figure 4.15 Dimensionless pressure as a function of the dimensionless time at the initial phase of

drop impact and deformation. The results of the numerical simulations of Roisman et al. (2009) and Eggers et al. (2010) in comparison with the expression (4.81) (Marengo et al. 2011). Reprinted with permission from Elsevier.

by D/V0 ) is scaled as D1 ∼ h−1/2 ∼ Re1/4 . The experiments show that this scaling is correct only for high Reynolds numbers. The experimental data for D1 /Re1/4 is shown in Fig. 4.16 as a function of the capillary number Ca = We/Re. The best fit for the experimental data (Roisman et al. 2002, Pasandideh-Fard et al. 1996, Chen 1977) for D1 yields D1 ≈ Re1/4 [0.65 − 0.34 exp(−3.24Ca)].

(4.82)

0.8 0.7

1/4

0.6

D1/Re

122

0.5 0.4 0.3 0.2

-1

0

1

2

3

4

5

6

7

8

9

Ca Figure 4.16 Experimental data (Roisman et al. 2002, Pasandideh-Fard et al. 1996, Chen 1977) for

the scaled diameter D1 /Re1/4 of spreading drops at the time instant t = 1 as a function of the capillary number Ca = We/Re.

4.4 Maximum Spreading Diameter

123

This empirical correlation can also be used for the drop spreading modeling at dimensionless times larger than 1.

4.4

Maximum Spreading Diameter The value of the maximum spreading diameter is only slightly dependent on the wettability of the substrate if the impact Reynolds and Weber numbers are high (Rioboo et al. 2002). The dimensionless maximum spreading diameter is, therefore, usually expressed using only the Reynolds and Weber numbers. Several approaches to predict the maximum spreading diameter exist in the literature. The approach based on energy balance considerations, the approach based on the estimation of the typical length scales for spreading, and the approach employing consideration of the momentum balance of the rim bounding a spreading lamella. The energy balance considers the initial kinetic and surface energy of the impacting and spreading drop and accounts for the energy lost due to viscous dissipation during drop spreading. One example can be found in Ukiwe and Kwok (2005) where the maximum spreading diameter is expressed as a root of the equation (We + 12)Dmax = 8 + D3max [3(1 − cos θ ) + 4WeRe−1/2 ]

(4.83)

where Dmax is the dimensionless maximum spreading diameter, rendered dimensionless by the initial drop diameter D, and θ is the dynamic contact angle. Further models for drop spreading and maximum spreading diameter, also based on the energy balance approach, can be found in Attané et al. (2007). It should be emphasized that the energy balance approach in its usual form is incorrect, since it does not account for the edge effects in the lamella. In Roisman et al. (2009) it was shown that the energy losses in the viscous boundary layer near the rim, which are never considered in such models, are significant; see Section 1.11 in Chapter 1. On the other hand, the energy dissipated in this boundary layer can be accounted for only if the flow in the spreading drop is known. Several attempts have been made to evaluate correct scales for the maximum spreading diameter for various impact regimes. Such semi-empirical models can predict the dimensionless maximum spreading diameter Dmax for a rather wide range of impact parameters Dmax = 0.61M 0.332 ,

from Scheller and Bousfield (1995)

Dmax = −0.7 + 0.8M 0.3 ,

from Bennett and Poulikakos (1993)

(4.84) (4.85)

where M = We1/4 Re1/2 . Some further models based on the length scale estimates are considered in Eggers et al. (2010). The main assumption in such models is that in a viscous regime Dmax depends solely on the Reynolds number, while in the capillary regime the maximum spreading diameter depends only on the Weber number. Following the analysis in Clanet et al. (2004) these two regimes are switched at P = WeRe−4/5 = 1. The scaling in

124

Drop Impact onto a Dry Solid Wall

Clanet et al. (2004) shows that the maximum spreading diameter, Dmax Re−1/5 , of a drop impacting onto a dry superhydrophobic substrate is a function of the parameter P = WeRe−4/5 . For the impact of a low-viscosity liquid (P < 1) the maximum drop diameter is determined exclusively by the capillary number and can be scaled as Dmax ∼ We1/4 . In the viscous regime (P > 1), the maximum drop diameter is determined mainly by the impact Reynolds number, Dmax ∼ Re1/5 . Note also that a recent comparison of the predictions of the empirical Eq. (4.84) and the scaling model of Eggers et al. (2010) revealed that the empirical expression and the latter scaling model are in good agreement with each other and the data in the experimentally covered range (Sahu et al. 2015). Examining the correlations and models available in the literature, one may note significant differences between them, even in relation to the phenomena that have been widely investigated, such as the maximum spreading. For example, the existing correlations attempt to describe the maximum spreading diameter as a power law of the impact velocity, Dmax ∼ V0α . The differences arise in the identification of the exponent α: for example Scheller and Bousfield (1995) employ the exponent α = 0.332, PasandidehFard et al. (1996) estimate it to be in the range 0.23 < α < 0.38, while Bennett and Poulikakos (1993) propose α = 0.5. This underlines the difficulty inherent in many correlations, in which authors use dimensionless quantities, but may vary only one of several parameters. A single, universal and reliable expression for Dmax cannot be established as a combination of the Reynolds and Weber numbers alone, since a model based on the physical understanding of the phenomenon should account for several modes of spreading. The first mode corresponds to the relatively small values of the dimensionless groups, We < 10 or Re < 102 , when the flow in the entire deforming drop is governed by the viscous and capillary forces (Roisman et al. 2009). If both dimensionless groups are large enough, We 10 or Re 102 , the drop shape can be presented as a radially spreading lamella bounded by a rim. The flow in the lamella is inertia dominated, it does not depend on the impact parameters, whereas the rim motion is governed by the surface tension and viscosity. The second mode corresponds to the case when the capillary forces are so significant that the rim diameter reaches the maximum at the time instant tmax < tBL , i.e. before the moment tBL when the boundary layer reaches the upper free surface of the lamella. The third mode corresponds to the case when tmax > tBL , and the maximum spreading diameter is determined mainly by the viscous effects. Consider the third mode of spreading when the maximum spreading diameter corresponds to the instant when the flow in the thin lamella is damped by viscosity. In the limit of an infinite Weber number the volume accumulated in the rim can be neglected. The residual film thickness is approximately uniform and the residual drop shape can be roughly approximated by a disc of thickness hres . The maximum drop diameter can be roughly estimated from the mass balance as  Dmax ∞ ∼

2 = 0.78Re1/5 . 3hres

(4.86)

4.4 Maximum Spreading Diameter

125

Figure 4.17 Single drop impact onto a dry substrate. Comparison of the correlation (4.88) with

β = 0.4 for the dimensionless maximum spreading diameter with the experimental data (Roisman et al. 2002, Pasandideh-Fard et al. 1996, Chen 1977) (Roisman 2009). Reprinted with the permission of AIP Publishing.

The value of the maximum spreading diameter is influenced by the appearance of the rim since the Weber number in reality is finite (Roisman et al. 2002). The typical length scale LR associated with the rim motion can be obtained with the help of the rim relative velocity (Taylor 1959), Eq. (2.159) in Section 2.7 in Chapter 2, in the following form = 0.61Re2/5We−1/2 . LR ∼ tBL We−1/2 h−1/2 res

(4.87)

The maximum drop diameter should decrease if the length scale LR increases. Fitting the experimental data (Roisman et al. 2002, Pasandideh-Fard et al. 1996, Chen 1977) for the maximum spreading diameter using the linear combination of the terms defined in Eqs. (4.86) and (4.87), one obtains the following correlations Dmax = 0.87Re1/5 − βRe2/5 We−1/2 β = 0.4, β = 0.48,

(4.88)

for partially wettable substrates

(4.89)

for superhydrophobic surfaces

(4.90)

(Roisman 2009, Butt et al. 2014). It should be emphasized that the terms on the right-hand side of Eq. (4.88) are of the same order of magnitude as the terms determined in Eqs. (4.86) and (4.87). The results of the comparison between the predictions of Eq. (4.88) and the experimental data for the maximum spreading diameter are shown in Fig. 4.17. The agreement is rather good in the entire range of the impact parameters. The effect of the viscous stresses on the rim velocity is neglected in the above scaling analysis, since in all the experiments considered the ratio of the viscous force to the

126

Drop Impact onto a Dry Solid Wall

Figure 4.18 Sketch of a drop shape after a high velocity impact (Roisman et al. 2002).

surface tension, Wehres /Re ≈ WeRe−7/5 , is negligibly small. For very viscous liquids or very small drops this term can become significant and the proposed simple scaling relation (4.88) will no longer be valid.

4.5

Time Evolution of the Drop Diameter: Rim Dynamics on a Wall In the case of the relatively high impact velocity of a drop, the lamella is bounded by a rim (Roisman et al. 2002). It is formed due to the interplay between the surface tension forces and wettability. The dynamics of the rim motion determines the expansion of the drop diameter. Consider a polar coordinate system with the radial coordinate r and with the origin at the point of impact. The rim, with the centerline at the radius r = Rr (Fig. 4.18), expands with a radial velocity Vr . Assuming that the characteristic size of the rim is much smaller than the radius Rr , the mass and the momentum balance in the radial direction in the axisymmetric rim, respectively, take the following form 1 dWr = Rr hl (Vl − Vr ) 2π dt

for r = Rr

ρWr dVr = ρRr hl (Vl − Vr )2 − Rr σ + Rr Fw − Rr Fμ for r = Rr 2π dt

(4.91) (4.92)

where Wr is the total volume of the rim, Vr is its radial velocity, Vl is the velocity of the liquid in the lamella, hl is the thickness of the lamella, ρ is the liquid density, σ is the surface tension, t is time. The first term on the right-hand side of the momentum balance in Eq. (4.92) expresses the inertia of the liquid entering into the rim, while the second term expresses the surface tension at the free surface of the lamella, the third term is associated with the wettability of the surface and the fourth term is the viscous drag force applied to the rim. Considering the force balance at the contact line, the radial force associated with the wettability is Fw = σ cos θ ,

(4.93)

4.5 Rim Dynamics on a Wall

127

where θ is the dynamic contact angle, which depends on the capillary number Ca = μVr /σ based on the velocity Vr of the contact line. This dependence is described by Hoffman’s law (Kistler 1993). The drag force applied to the rim by the wall, in the case when the width ar of the rim is much smaller than the radius Rr (see Fig. 4.18), can be defined as  ar Fμ = τw dx (4.94) −ar

where x is the coordinate in the radial direction with the origin at the rim centreline, τw is the shear stress which can be estimated using the lubrication theory as τw (x) =

3μVr hr (x)

(4.95)

with hr (x) being the local thickness of the rim. The exact shape of the rim is not determined in the present analysis. The total value of the drag force is estimated as per Fμ ≈

6μVr ar hr

(4.96)

where hr is the height of the rim. Assuming hl  hr , ar  Rr , and the circular cross-section of the rim (which is the case when the value of the drag force is small relative to the surface tension), one obtains the following geometrical relations sin θ ar = hr 1 − cos θ

(4.97)

Wr . 2π Rr (θ − sin θ cos θ )

(4.98)

 ar = sin θ

The value defined in Eq. (4.96) using the maximum height of the rim underestimates the value of the drag force. However, the velocity Vr of the rim is of the order of V0 . Thus the ratio Fμ /σ is of the order of Ca = We/Re, where We = ρDV02 /σ is the dimensionless Weber number and Re = ρDV0 /μ is the Reynolds number. Therefore, the viscous drag force is negligibly small relative to the surface tension if Re We. This case will be considered in more detail below. Note however, that at the concluding stages of drop spreading, the thickness of the lamella becomes so thin that the inertial effects of the liquid in the lamella are no longer the dominating factor. The motion of the rim is determined by the balance of the surface tension, wettability and viscous forces. The viscous term Fμ defined in Eq. (4.96) cannot be neglected at this concluding stage of drop spreading and the following receding phase. At large times, when the flow in the lamella takes the form V¯l =

r¯ ¯t + τ¯

h¯ l =

η¯ (t¯ + τ¯ )2

(4.99) (4.100)

128

Drop Impact onto a Dry Solid Wall

Figure 4.19 Axisymmetric impact of a water drop. The image (a) is taken at 0.84 ms after the

impact and the subsequent images are taken at an interval of 1.08 ms from the previous one. The impact parameters are: the initial drop diameter D = 2.6 mm, the impact velocity V0 = 2 m/s and the target diameter Ds = 6 mm.

following from Eqs. (4.1) and (4.7) in subsection 4.1.1 in the present chapter, and the thickness of the lamella almost does not depend on the radius, Eq. (4.91) can be replaced by the total mass balance of the drop Wr = W0 − π R2r hl ,

(4.101)

where W0 =

π D3 6

(4.102)

is the initial volume of the drop. Using Eqs. (4.91), (4.92) and (4.101), the momentum balance equation yields the following equation of motion of the rim   ¯ dV¯r 12R¯ r ¯ l (V¯l − V¯r )2 − 1 − cos θ − 6Vr sin θ , for r¯ = R¯ r . (4.103) h = dt We Re(1 − cos θ ) 1 − 6R¯ 2r h¯ l

4.6

Drop Impact onto Spherical Targets and Encapsulation In many practical applications (for example in spray encapsulation, or spray agglomeration of chemical materials in a fluidized bed, spray impact onto rough or microstructured surfaces, spray coating, process of icing of the elements of planes, etc.) drop impacts are random events. In the case of impact of two spherical bodies the probability density of the impingement with a definite off-axis distance b linearly depends on b. The probability of the exactly axisymmetric impact is practically zero. Understanding the hydrodynamics of non-axisymmetric, inclined drop impact and spreading is, therefore, an important issue, which is to be elucidated for better design and optimization of such technological processes. The experimental results are acquired as the distribution of liquid on the surface of a fixed spherical target at several time instants. Successive images acquired at different time instants are shown in Figs. 4.19 and 4.20 for the axisymmetric and nonaxisymmetric impact, respectively. The impact parameters in the experiments shown in these figures correspond to the Reynolds number Re = 5200 and Weber number We = 142.

4.6 Drop Impact onto Spherical Targets and Encapsulation

129

Figure 4.20 Non-axisymmetric impact of a water drop. The image (a) is taken at 0.25 ms after the

impact and the subsequent images are taken at an interval of 1.08 ms from the previous one. The impact parameters are: the initial drop diameter D = 2.6 mm, the impact velocity V0 = 2 m/s, the off-axis distance b = 1.12 mm and the target diameter Ds = 6 mm.

In Fig. 4.21 the experimental data of Bakshi et al. (2007) for the film thickness hexp at the impact center is compared with the “inviscid residue” hexp − hν , where hν is defined in Eq. (4.73). The difference is rather significant at large times. As predicted by the theory, the value of hexp − hν does not depend on the Reynolds number even for the relatively large times after impact. The target-to-drop diameters ratio in the experiments shown in Fig. 4.21 is 2.7. It should be emphasized that during most of the time of drop spreading the values of the drop thickness hexp for both values of the Reynolds number are very similar. The inviscid part hexp − hν follows closely the inverse-square-law of time in the form predicted in Yarin and Weiss (1995).

Figure 4.21 Drop impact onto a fixed spherical target. Evolution of the film thickness h at the

impact axis as a function of time in comparison with the “inviscid” thickness estimated as hexp − hν (t ). The experimental data for hepx are from Bakshi et al. (2007). The target-to-drop diameters ratio is 2.7.

130

Drop Impact onto a Dry Solid Wall

1.2

hres/D Re2/5

1.0 0.8 0.6 0.4 0.2 0.0 0

2

4

6

8

10

Ds /D Figure 4.22 Scaled residual film thickness after drop impact onto a fixed spherical target. The

curve corresponds to Eq. (4.104).

An empirical correlation for the dependence of the residual film thickness on the impact Reynolds number (based on the normal component of the impact velocity) and on the target diameter Ds reads    0.66Ds . (4.104) hres = DRe−2/5 0.79 − 0.52 exp 0.29 − D Correlation (4.104) was obtained by the least mean square fit of the experimental data, shown in Fig. 4.22. It should be emphasized that this correlation approaches asymptotically in the limit Ds /D → ∞ to the analytical solution (4.80) obtained for the case of a flat target. The critical impact velocity (or the Reynolds number based on it) corresponding to complete encapsulation of a fixed particle can be estimated from the mass balance π D3 /6 = π D2s hres , leading to the following result    √ D5 0.66Ds 5/2 Reencaps ≈ 36 6 s5 0.79 − 0.52 exp 0.29 − . (4.105) D D

4.7

Outcomes of Drop Impact onto a Dry Wall Various outcomes of drop impact have been observed after drop impact onto a dry solid substrate. Rioboo et al. (2001) have subdivided them into several typical groups: deposition, corona splash and prompt splash (see Fig. 4.23). On non-wettable or partially wettable substrates (Fig. 4.24) receding breakup, partial and full rebound can also take place under specific conditions. The splashing threshold is usually defined as a minimum drop velocity leading to drop breakup and generation of secondary droplets. An overview on the existing models can

4.7 Outcomes of Drop Impact onto a Dry Wall

131

Figure 4.23 Outcomes of drop impact: deposition, prompt splash and corona splash (Roisman

et al. 2015). Reprinted with permission from Elsevier.

be found in Moreira et al. (2010). Stow and Stainer (1977) demonstrated that the surface roughness enhances drop splashing. Since then many authors have attempted to develop a relation between the impact parameters associated with the splashing threshold (Stow and Hadfield 1981, Vander Wal et al. 2006, Mundo et al. 1995, Range and Feuillebois 1998, Walzel 1980). Among these parameters is the critical Weber number, or capillary number, or a combination of the Weber and Reynolds numbers. One of the frequently used parameters related to splashing threshold is We1/2 Re1/4 , the well-known Kd -parameter, introduced in Mundo et al. (1995); according to Eq. (1.2) in Section 1.2 in Chapter 1, Kd is related to the dimensionless group K as Kd = K 5/8 . The experiments of Range and Feuillebois (1998) showed that the critical value of We1/2 Re1/4 depends on the substrate roughness, specifically on the arithmetic mean value of the roughness amplitude, Ra . They found that the critical value of We1/2 Re1/4 , in fact, does not depend uniquely on the value of Ra /D. The experiments of de Ruiter et al. (2010) confirm the assumption that the lamella √ thickness at the splash initiation is scaled by the viscous length μt/ρ. Based on these observations, Bird et al. (2009) proposed a splashing threshold for the inclined drop impact in the following form   Vt (4.106) We1/2 Re1/4 1 − 3.5 Re−1/2 > 127.0, Vn where Vt and Vn are the tangential and normal components of the impact velocity, respectively. The splashing condition in the form (4.106) predicts that the threshold velocity for splash monotonically increases for higher liquid viscosity. Vander Wal et al. (2006) used liquid drops of various viscosities in their impact experiments onto a very smooth substrate. They predict a constant critical capillary number, Ca = We/Re, as a splashing

Figure 4.24 Drop impact onto hydrophobic substrates. Outcomes illustrating drop receding

breakup, partial and full rebound (Roisman et al. 2015). Reprinted with permission from Elsevier.

Drop Impact onto a Dry Solid Wall

Oh

Ca

Oh0.35

We

132

Re Figure 4.25 Map depicting the outcomes of drop impact at normal ambient pressure. Boundary I

separates the corona splash and deposition domains, while Boundary II separates the corona splash and deposition/prompt splash domains (Roisman et al. 2015). Reprinted with permission from Elsevier.

threshold. Therefore, in these experiments the threshold velocity decreases for higher viscosity, which is in contradiction with the prediction (4.106). Palacios et al. (2013) demonstrated that the splashing threshold is accurately described as a constant value of We/Re1/2 . The existence of different, and sometimes contradictory expressions based on various scales, indicates that currently a complete picture of the splash conditions on a dry wall is unavailable and a reliable and universal correlation for splashing threshold does not exist. The map of the main outcomes of normal drop impact onto a dry substrate under normal ambient pressure conditions, comprised of the experimental data from Palacios et al. (2013), Vander Wal et al. (2006), Bird et al. (2009) and Pan et al. (2010) is shown in Fig. 4.25. In this map only single impacts leading to the formation of the secondary drops are considered as a splash. The region corresponding to the corona splash is bounded by the two thresholds Ca = 0.067 + 0.60Oh0.35 , Oh = 0.0044,

Boundary I,

Boundary II,

(4.107) (4.108)

where Boundary I separates the corona splash and the deposition/lamella lifting domains, while Boundary II separates the corona splash and deposition/prompt splash domains. As one can see, the most popular and widely used Kd number, Kd = We1/2 Re1/4 , does not describe sufficiently accurately any of the splashing thresholds for drop impact onto a dry substrate. It should be emphasized that the same range of impact parameters corresponds to prompt splash (Palacios et al. 2013) and deposition (Vander Wal et al. 2006), i.e. these two regimes can overlap. This means that the prompt or jetting splash is determined not

4.8 The Effect of Reduced Pressure of the Surrounding Gas

133

only by the combination of the Weber and Reynolds numbers but also by the conditions at the substrate surface, namely, by the target roughness.

4.8

The Effect of Reduced Pressure of the Surrounding Gas Studies of the splash produced by drop impact are focused mainly on the description of the splashing threshold and explanation of the splash mechanism. An extensive list of the existing correlations for the splashing threshold can be found in Rein and Delplanque (2008). Relatively recently it was discovered that drop splashing on a dry surface is determined not only by the material properties and the impact parameters but also by the parameters of the surrounding gas. The effect of the ambient pressure on the splashing threshold was observed in Xu et al. (2005). The splash was explained by the appearance of a weak shock in the gas in the neighborhood of the fast advancing contact line during the initial phase of drop spreading. According to Xu et al. (2005) the splashing threshold is the ratio of the aerodynamic stresses to the surface tension stresses: =

1 cG ρG WeRe−1/2 , 2 V0 ρ

(4.109)

where cG and ρG are the speed of sound and density of gas, respectively. Drops splash if > 0.45, which means that drop splash on a dry surface is suppressed at reduced surrounding gas pressure. This initial study has been recently further developed in Driscoll et al. (2010) where the influence of the surrounding pressure on the evolution of the drop shape, ejection of a thin liquid sheet and on the bubble entrapment has been investigated. On the other hand, Stevens et al. (2014) have shown that the threshold ambient pressure changes nonmonotonously if the liquid viscosity increases, which contradicts the correlation (4.109). Note, that the explanation of the mechanism of splash in Xu et al. (2005) has one weak point. The dimensionless characteristic time of the weak shock propagation, which moves with velocity comparable with the speed of sound, is V0 /cG (note that this is simply the Mach number Ma). In most of the experimental cases this time is of the order of 10−2 , while the corona becomes apparent at the later dimensionless times of order of unity. It is difficult to expect compressibility effects to be significant for the velocities corresponding to Ma = 10−2 , since these effects are as important as ∼ 0.1Ma2 ∼ 10−5 in comparison to 1 according to the isentropic formulas of gas dynamics (Loitsyanskii 1966). Mani et al. (2010) attempted to explain the effect observed in Xu et al. (2005) by analyzing an incompressible gas flow. They predicted that the effect of a thin air layer, which could possibly trigger a splash, is only important in the air layers thinner than the substrate roughness in most of the experiments. Also, if one considers by analogy an axisymmetric collision of two equal drops, following the logics of Mani et al. (2010) and Xu et al. (2005) the collision could lead to the appearance of two symmetric crowns. This phenomenon has never been observed in experiments. Therefore, the mechanism

134

Drop Impact onto a Dry Solid Wall

Figure 4.26 Leidenfrost effect: water droplet levitating on a vapor pillow above a very hot surface.

CFD computations of the shape of a water drop, of initial diameter 1 mm, 0.45 ms after an impact onto a hot, 300 ◦ C, solid target with the impact velocity 1.11 m/s. We = 20 (Dawi et al. 2013).

of the effect of reduced air pressure on drop splashing still remains uncovered, which has triggered additional experimental efforts (Thoroddsen et al. 2011, Zhang et al. 2012, Visser et al. 2015).

4.9

Drop Impact onto Hot Rigid Surfaces The Leidenfrost effect described in 1756 is familiar to anyone who has ever sprinkled drops of water onto a very hot skillet or a pan (Leidenfrost 1756, Tyndall 1863). At temperatures about 150 ◦ C, instead of an almost instantaneous flash evaporation, the Leidenfrost effect surprisingly allows water drops to survive for several minutes and skid and roll over the hot pan surface. Due to the initial intense evaporation at the bottom of the drop in contact with the hot skillet, a vapor layer between the drop and the hot surface is generated with a pressure sufficient to levitate the drop (Fig. 4.26). The Leidenfrost effect is not only a classical demonstration of kitchen physics but also a technically and industrially important phenomenon which severely restricts heat removal from high-heat-flux surfaces, since thermal conductivity through the vapor layer is negligibly small compared to the latent heat of water which might be exploited otherwise to remove heat. The behavior of individual cold drops impinging onto hot surfaces determines directly the efficiency of spray cooling systems, which are presently one of the most effective methods for cooling of high-heat-flux surfaces. Spray cooling is a valuable alternative in extreme cases for microelectronics, optoelectronics or radiological devices, for example for cooling in Unmanned Aerial Vehicles (UAVs) (Child 2009, Kinney 2009). The tremendous cooling potential of this technology is associated with liquid evaporation at the hot surface. Its efficiency is strongly affected by the

4.9 Drop Impact onto Hot Rigid Surfaces

135

Figure 4.27 Drop impact onto a hot surface at different temperatures: Thermodynamic

phenomena: evaporation (a), nucleate boiling (b), foaming (c), transition (d) and film boiling, the Leidenfrost effect (e).

hydrodynamics and heat transfer associated with drop impact onto hot surfaces (see Sections 5.8 and 5.9 in Chapter 5). While the basic hydrodynamics of drop impact on dry surfaces at atmospheric pressure in the isothermal case is mainly understood (see Sections 4.1–4.7 in this chapter), an accurate description of drop impact onto hot surfaces remains a challenging problem. Various aspects of this phenomenon have been investigated experimentally, namely, heat transfer associated with drop impact (Labeish and Pimenov 1984), breakup probability (Ko and Chung 1996) and the limiting temperature resulting in a drop rebound (Karl and Frohn 2000). The phenomena of drop impact onto hot surfaces are influenced significantly by the contact temperature (Cossali et al. 2005), which is a function of the initial temperatures of the wall and drop, their thermal diffusivities, the wall thickness and the Prandtl number of the liquid. If the wall temperature is high enough, the contact temperature, expressed by Eq. (4.60), exceeds the liquid saturation temperature. In this case the drop spreading is accompanied by intensive evaporation and boiling. In the nucleate boiling regime the heat transfer between the wall and drop is very high. On the other hand, if the contact temperature is significantly higher than the liquid saturation temperature, the Leidenfrost effect sets in (Leidenfrost 1756, Gottfried et al. 1966) (see Fig. 4.26), droplets levitate and heat transfer from the wall is significantly hindered by the intermediate vapor layer. Moreover, the high pressure in the vapor layer leads to the instability of spreading liquid drops accompanied by their shattering and formation of a cloud of small secondary droplets. A comprehensive review of Moreira et al. (2010) revealed that the dynamic Leidenfrost temperature corresponding to the transition to the Leidenfrost regime is a function of drop impact parameters. The abundant literature devoted to the Leidenfrost effect continues to expand at a steady pace (Biance et al. 2003, Biance et al. 2006), even though today a comprehensive theory (for example a reliable prediction of the skittering speed) is still absent. However, the driving mechanism of the Leidenfrost effect depicted in Fig. 4.26 is already well understood. Typical thermodynamic phenomena observed during drop impact onto a hot surface at different initial wall temperatures are shown in Fig. 4.27. Another important effect has been observed in Mehdizadeh and Chandra (2006). If the drop impact velocity is high enough, many very small bubbles appear in the lamella. The size of the bubbles is much smaller than the estimated length of the Taylor wave. Moreover, it is not yet clear why drop impacts with higher velocity (corresponding to

136

Drop Impact onto a Dry Solid Wall

higher pressure at the wall) lead to the formation of small bubbles. Small bubbles appear during drop impact onto extremely smooth single-crystal surfaces (Nagai and Nishio 1996) even if the wall temperature is lower than the temperature required to activate the pre-existing nuclei of the size comparable with the surface roughness. The latter authors claimed that these results should lead to the revisiting of the existing models of heat transfer in transition boiling. This result is not yet explained, and the mechanism of formation of the very small bubbles is not yet described in detail. Various hydrodynamic impact modes on hot surfaces can be identified, as shown in Fig. 4.28. At the deposition mode (A) there is no rebound observed, while at the partial rebound (B) only a part of the drop rebounds. The total rebound (C) can be subdivided into three different categories C1–C3. During the receding of the drop in Mode C1, secondary droplets will be formed at the contact line, while in Mode C3 secondary droplets will appear during the drop advancing motion in the case of very fast rising bubbles through the thin lamella of the drop. This is very close to the atomization stage. Mode C2 does not result in any secondary droplets at the rebound event. At the atomization stage (D) the drop breaks up in the case of lamella instabilities and multiple secondary droplets are formed. Fast rising and growing bubbles in the lamella produce holes and hence, instabilities. The lamella breaks up and the drop disintegrates. This phenomenon can be very weak as the primary drop breaks up into a few parts (D1), or very strong (D2) as the drop disintegrates into multiple secondary droplets. The comparison between a total rebound and atomization is clearly seen on the righthand side in Fig. 4.29. At the atomization regime the upcoming holes in the lamella can be seen in the top view, whereas at the rebound stage no holes can be observed. Generally, when the initial wall temperature increases, the breakup of the impacting drop into a myriad of very small secondary droplets happens earlier (Manzello and Yang 2002). The increased temperature leads to higher values of the pressure in the vapor layer under the liquid drop, and thus to a stronger upward acceleration of the drop, promoting the Rayleigh–Taylor instability of the drop (cf. subsection 1.10.4 in Section 1.10 in Chapter 1). The phenomena of drop impact onto a hot substrate are often described or correlated using the contact temperature (4.51). In Tarozzi et al. (2007) the contact temperature has been accurately measured using a thermographic camera. It was shown that the temperature initially reduces well below the theoretically predicted value (4.51), which is based on the analysis of the heat conduction in the thermal boundary layers in the liquid and solid. The reason is in the thinning of the thermal boundary layer in the liquid due to the flow in the spreading drop. In the analysis of this phenomenon a correct expression for the contact temperature, which accounts for the gradient of the normal velocity component in the drop, should be used. A universally valid approximation for drop impact hydrodynamics and disintegration under such conditions has not yet been developed. A simplified quasi-stationary model of Buevich and Mankevich (1982) neglects the dissipative effects accompanying drop impact. Publications using direct numerical simulations of drop impact in the Leidenfrost regime are also available (Ge and Fan 2006).

4.9 Drop Impact onto Hot Rigid Surfaces

137

A

B

C1

C2

C3

D1

D2

Figure 4.28 Drop impact onto a hot surface at different temperatures: Hydrodynamic impact

modes: deposition (A), partial rebound (B), total rebound (C) and atomization (D).

The drop behavior on a heated wall can be greatly influenced by several additional factors, for example, by surface inhomogeneity (Linke et al. 2006). Moreover, the secondary atomization can be influenced significantly by dissolving even a small amount of polymer additives in the liquid (Bertola and Sefiane 2005). The mechanics of

138

Drop Impact onto a Dry Solid Wall

breakup is more complicated for compound drops impacting onto a hot surface (Chiu and Lin 2005). The phenomena can be subdivided into several main parts: hydrodynamics of drop impact, heat transfer in the drop and in a substrate, evaporation and instability. At very high substrate temperatures significantly exceeding the saturation temperature, the liquid cannot be in contact with the solid. The flow in the vapor layer dividing the liquid from the hot substrate leads to a pressure distribution which allows the impacting drop to levitate above the surface (Leidenfrost 1756) [see the English translation in Leidenfrost (1966)]. One of the important parameters characterizing the film boiling is the minimum film boiling temperature (or the Leidenfrost temperature). The model of Berenson (1961) for the Leidenfrost temperature is based on the Rayleigh–Taylor instability of the vapor/liquid interface. In this theory the spacing between bubbles and their size are determined by the most unstable wavelength of the Rayleigh–Taylor instability [see Eq. (1.88) in Section 1.10 in Chapter 1]. The Leidenfrost temperature is then evaluated from the condition of the minimal heat flux. If the vapor density is much smaller than the liquid density, the minimum film boiling temperature predicted by Berenson (1961) can be expressed in the form Tmin = Tsat

ρv hiv + 0.127 kv



σ 3 μ2v gρ 5

1/6 (4.110)

where Tsat is the saturation temperature, ρv , μv and kv are the density, viscosity and thermal conductivity of the vapor, hiv is the latent heat of evaporation. Olek et al. (1991) proposed an empirical formula for the minimum temperature of substrate in the film boiling   27 Tcrit − Tsat (4.111) Tmin = Tsat + C 32 where Tcrit is the critical temperature, C is a dimensionless fitting coefficient depending on the liquid (C = 0.64 for water). Yao and Henry (1978) have shown that Eq. (4.111) provides the best approximation for the minimum contact temperature [defined in Eq. (4.51)] associated with film boiling. In Ohtake and Koizumi (2004) the collapse of the vapor film in the film boiling regime is explained by the non-uniformity of the surface heating. The presence of local cold spots on the surface is associated with conditions of the minimum-heat-flux temperature. The study of Bernardin and Mudawar (2004) can serve as an example of a model for the prediction of the Leidenfrost temperature corresponding to drop impact onto a rigid substrate. In this study the bubble nucleation and growth and the interaction with the thermal boundary layer are analyzed. Since the flow in the drop is not known, some empirical correlations for the energy loss are introduced in the model, which is finally fitted to the experimental data for the Leidenfrost temperature. The Leidenfrost temperature always increases when the drop impact velocity increases. This observation is confirmed by the experiments of Celata et al. (2006), Testa and Nicotra (1986), etc.

139

4.9 Drop Impact onto Hot Rigid Surfaces

–0.67 ms

0.40 ms

1.10 ms

1.60 ms

2.14 ms

3.00 ms

4.13 ms

7.13 ms

10.87 ms V0 = 1.1 m/s

V0 = 2.4 m/s

Figure 4.29 Drop impact onto a hot surface at 300 ◦ C with different impact velocities:

V0 = 1.1 m/s, leading to the rebound (left) in comparison with the impact with V0 = 2.4 m/s, leading to the atomization (right).

Yao and Cai (1988) and Bertola and Sefiane (2005) obtained the following empirical correlations for the the Leidenfrost temperature Tmin = Tsat + 135.6We0.09 Tmin = (165 + 30We

0.38 ◦

for liquids other than water

) C for water.

(4.112) (4.113)

In Fig. 4.29 the two modes of drop impact on hot surfaces are compared, namely, the rebound and atomization. The switch between the modes is caused only by the drop impact velocity V0 . The mechanism of atomization can be explained by the breakup of the lamella. These two phenomena appear always simultaneously. It should be emphasized that the thickness of the lamella only slightly depends on the liquid surface tension.

140

Drop Impact onto a Dry Solid Wall

T ,°C Figure 4.30 Regime map for different drop impact velocities and surface temperatures. The

experimental data are courtesy of Jan Breitenbach, Technische Universität Darmstadt. The additional data for atomization regime from Bertola (2015) are shown as hollow symbols.

It is mainly determined by the liquid viscosity (as shown in Section 4.2 in the present chapter). Figure 4.30 depicts the map, which delineates different regimes of drop impact onto a hot wall.

4.10

Drop Impact with Solidification and Icing Drop impacts onto a surface accompanied by deposition of the impinging liquid are frequent in several fields of technology and in nature. During ink-jet printing depositing drops reproduce a digital image or text from a computer system onto a sheet of paper, plastic foil or other materials. In case of near net shape manufacturing, drop impacts are employed to form objects layer-by-layer, e.g. for medical applications or for prototyping purposes (Rengier et al. 2010, Noguera et al. 2005). During thermal spray coating a layer of one material is placed on top of another base material. In contrast to printing, it is not used to only change the optical appearance of the base material, but rather to change the surface properties of the coated part to improve the overall performance of the compounded object (Fauchais et al. 2014). All the aforementioned processes are of technological relevance and are in focus in the present section. A process which is most frequently met both in nature and in the engineering applications is the icing of surfaces. It occurs due to natural environmental conditions and may pose a severe hazard when it occurs on technical surfaces, like those of the aircraft or power lines. Icing may occur due to the impingement of supercooled water drops, but is also possible when warmer drops impact onto cold surfaces below 0 ◦ C, as in the case of freezing rain. The outcome of such deposition events depends on the fluid dynamical peculiarities of the impacting drops, which determine the liquid spreading and thereby the coverage of the surface. Furthermore, the outcome can also be influenced by the thermodynamics

4.10 Drop Impact with Solidification and Icing

Surface at +17 ◦C

141

Surface at –17 ◦C

t = 0.00 ms

t = 0.00 ms

10 mm

10 mm

t = 2.10 ms

t = 2.10 ms

10 mm

10 mm

t = 4.20 ms

t = 4.20 ms

10 mm

10 mm

t = 12.61 ms

t = 12.61 ms

10 mm

10 mm

t = 33.62 ms

t = 33.62 ms

10 mm

10 mm

Figure 4.31 Comparison of the fluid behavior during a drop impact onto surface at +17 ◦ C (left)

and −17 ◦ C (right). In the case of the cooled surface, an increasing viscosity causes a much slower receding. The lines added to the images on the right side indicate the first point of the oblique impact.

of liquid drying during printing, and the time delay of solidification in the case of such hazardous phenomenon as icing. When drop impact onto a surface directly triggers the freezing process, the resulting iced area would have almost the shape of the area which is covered by a drop without phase change being involved. Since a freezing delay may occur during which the drop still moves, the outcome, i.e. the finally iced area of such an impact event, strongly depends on the moment at which freezing occurs.

4.10.1

Ice Nucleation and Dendrite Growth on a Substrate Figure 4.31 shows a qualitative comparison of drop spreading behavior on a substrate at +17 ◦ C and −17 ◦ C. Drops are shown at different instants during the first 34 ms of the impact process. Note that the time intervals between the first three and the last three frames are constant but are chosen to be longer for the last frames to account for the

Drop Impact onto a Dry Solid Wall

0.02 0.015 deq (m)

142

0.01 0.005 0

Tsub = –17 ◦ C Tsub = +17 ◦ C 0

0.01

0.02

0.03

t (s) Figure 4.32 Temporal evolution of the spreading diameter deq =



4 Ac /π . Ac is the contact area

between the liquid and the surface.

decreasing speed of the process. The drop motion in the images in Fig. 4.31 is from the right to the left, resulting in a thin liquid lamella on the right, while the rest of the drop continues its spreading in the direction of the initial motion. The comparison shows that there is no obvious difference in the spreading behavior in the first two frames, since inertia dominates this stage and heat transfer between the drop and the surface is still insignificant. A first change in the behavior can be recognized in the third frame: the thin liquid film on the right-hand side of the spreaded drop starts to recede on the warm surface, while no motion of this kind happens in the case of the cold substrate due to an increased viscosity which is caused by the fast cooling down of the thin liquid lamella in this area. This difference becomes more obvious in the fourth frame, where a thick rim has already formed during the ongoing receding on the warm surface, whereas the splat remains almost in its initial spreaded shape in the case of the cold surface. Thereby, a further cooling down due to a larger contact area between the liquid and the surface is facilitated. A quantitative comparison of the fluid behavior during the first 32 ms after drop impact is illustrated in Fig. 4.32. It shows the temporal evolution of the area-equivalent √ diameter deq = 4 Ac /π , where Ac is the contact area between the liquid and the surface. The drop impacting onto the warm surface begins to recede on the right-hand side, while it is still spreading on the left-hand side. On the cold surface, the receding on the right-hand side is strongly damped by an increased viscosity, whereas the spreading on the left-hand side is not yet affected and still dominated by inertia. Thereby, a drop impact onto a cooled surface results in a slightly larger area covered with liquid than onto a warm surface, as Fig. 4.32 illustrates. The typical time of spreading is scaled as tspread ∼ D/vimp , where D is the initial drop diameter, and vimp is the normal component of the impact velocity. The typical time of drop cooling by conduction without phase change is tcool ∼ h2 /α, where h stands for the lamella thickness and α is the thermal diffusivity of the liquid. The residual drop

4.10 Drop Impact with Solidification and Icing

143

thickness is expressed in the form h = 0.79 D Re−2/5 , in which the Reynolds number Re is defined using the drop diameter as the length scale, Re = ρ vimp D/μ (Roisman 2009). The ratio of the characteristic cooling time to the characteristic hydrodynamic time is, therefore, tcool tspread

∼ Pr Re1/5 , with Pr ≡

μ . ρα

(4.114)

The Prandtl number of water ranges from Pr = 7 at 20 ◦ C to approximately Pr = 37 at −20 ◦ C, and the Reynolds number in the present experiments from Re = 5980 to Re = 1370. Therefore, the characteristic time ratio in Eq. (4.114) is between 40 at room temperature and 157 at −20 ◦ C, resulting in tspread  tcool for the entire range of possible liquid temperatures. Since the time scale of drop spreading is much smaller than that of the cooling time of the liquid, freezing before the spreading has been completed has never occured in these experiments. When spreading is complete and the contact area has reached its maximum, the heat transfer to the surface is the largest as well. After that moment freezing is observed for the first time. In the case of a freezing sessile drop, a freezing delay of varying duration was reported in Boinovich et al. (2014) and Tourkine et al. (2009), meaning that the drop temperature is well below the liquid melting point but no solidification takes place, i.e. the liquid is supercooled. Since the drop is sessile on top of a surface, fluid motion inside it can be only induced by the free convection. However, the same phenomenon, i.e. a freezing delay, has been observed in the present experiments with drop impact, in which fluid motion is involved. Freezing begins at times after impact varying from microseconds up to several minutes. Before freezing has started, the liquid is still mobile and receding permanently alters its shape and hence, the contact area with the underlying surface. Therefore, the shape of the frozen ice and the area of the surface which is covered with ice depends on the freezing delay. The outcomes at the end of the captured videos are categorized according to the respective shape into five groups, from which four are shown in Fig. 4.33. Freezing immediately after the moment of maximum spreading leads to an oval shape, as depicted in the top left column. In this case, the area of the surface covered by ice is the largest. If the freezing delay is longer, the liquid shape continues to alter, tending to that of a sessile droplet. During the continuous receding beginning from the splat shape, at one point the liquid splits into several smaller droplets, from which a part may freeze, while the others stay liquid, as depicted in the bottom right column. The two additional intermediate morphologies are depicted in the top right and bottom left frames in Fig. 4.33. The fifth of the observed outcomes, which is not shown in this figure, is the one where the drop has split into several smaller droplets which stay liquid during the entire period of observation. Figure 4.34 shows the ratio of the number of liquid drops Nliq to the total number of performed impact experiments N0 as a function of the freezing delay after drop impact. In case of an observed freezing event, Nliq /N0 reduces and ultimately reaches 0.34 which corresponds to the percentage of drops remaining liquid at the end of the observation period. For a completely random occurrence of freezing events within a number of liquid

Drop Impact onto a Dry Solid Wall

Splat tdelay = 0...40 ms Occurrence = 22%

Splat/rivulet transition 40...125 ms Occurrence = 15%

Frozen rivulet tdelay = 125...400 ms Occurrence = 7%

Liquid & frozen drops > 400 ms Occurrence = 22%

Figure 4.33 Outcomes of a drop impact onto a cooled surface. Freezing delay tdelay determines final shape of frozen liquid. Percentages represent the occurrence of the respective outcome in these experiments; a residual 34 % of the outcomes contain liquid droplets.

drops, the freezing rate is proportional to the number of liquid drops as d(Nliq ) ∼ Nliq . d(tdelay )

(4.115)

According to this, there would be a linear relation between ln(Nliq /N0 ) and time. In case of the present experiments, where the liquid was probably further cooled down during the freezing delay, the increasing supercooling would be accompanied by an increase in the freezing probability (Hobbs 2010). Following this, the magnitude of the freezing rate |d(Nliq /N0 )/d(tdelay )| should increase. Figure 4.34, however, shows

(-)

1

Nliq N0

144

0.5

0

0.2

0.4

0.6

0.8

1

tdelay (s) Figure 4.34 Relative number of liquid drops, scaled by the total number of experiments, Nliq /N0 , as a function of freezing delay time. Linear fit for the last ten points which are slightly influenced by liquid cool down and fluid motion.

4.10 Drop Impact with Solidification and Icing

145

Figure 4.35 Propagation of a freezing front of dendrites within a receding drop.

that |d(Nliq /N0 )/d(tdelay )|, in fact, decreases, with the largest decrease being observed during the first phase after drop impact. Fluid motion, as well as heat transfer, are the highest during this phase, obviously resulting in the highest freezing rates observed in these experiments. The temperature within the liquid tends to approach an equilibrium corresponding to the initial surface temperature, and fluid motion calms down during the time accompanied by a decrease in the contact area between the liquid and the surface. Hence, the freezing rate decreases over time and the dependence between ln(Nliq /N0 ) and time is accurately approximated by a linear relation, as shown for the last 10 points in Fig. 4.34. A set of images showing the propagation of the freezing front within a drop is shown in Fig. 4.35. Since the liquid still cools down during the receding phase, finally approaching the surface temperature, it solidifies in a supercooled state in all the observed cases. Hence, the observed freezing front is rather an envelope of a growing dendritic structure than a real solid/liquid interface like in the case of planar solidification at the melting point in the Stefan problem (Tikhonov and Samarskii 2011). As seen in the figure, nucleation begins at one single point, followed by a radial motion of the front without major deviations from a circular shape. The front motion has been analyzed for all cases except those in which a sessile droplet froze. In the latter, the threedimensional character of the front propagation is non-negligible; hence, such cases were not used for further examination. The freezing velocity in a supercooled liquid increases

146

Drop Impact onto a Dry Solid Wall

Figure 4.36 Spreading, receding and total rebound of a water drop with an initial temperature of 0 ◦

C on an undercooled superhydrophobic surface at −10 ◦ C. The drop diameter is 1.59 mm, and the impact velocity is 3.30 m/s. The horizontal straight line indicates the level of the substrate surface (Li et al. 2015). With permission of Springer.

with increasing supercooling, as shown in Shibkov et al. (2003) for a single dendrite tip, or in Pasieka et al. (2014) for an aluminum dish filled with supercooled water. When nucleation begins at a single point, the remaining liquid still cools down after freezing has started, thereby potentially leading to an increase in the freezing velocity.

4.10.2

Drop Impacts onto Superhydrophobic Surfaces The influence of solidification during drop impact is more apparent for superhydrophobic surfaces. With either the drop temperature or the substrate temperature above 0 ◦ C, total rebound was observed as shown in Fig. 4.36. When both the drop and the substrate temperatures were below 0 ◦ C, the drop exhibited an elongated contour (at 7.25 ms), a sharp edge (at 9.25 ms) or a cubic shape (at 64.75 ms), as shown in Fig. 4.37. Such shapes are impossible with a fully liquid drop, indicating that the internal ice structures had already formed before the onset of the receding phase. Formation of such ice/water mixtures can only be caused by the rapid growth of ice dendrites. The occurrence of the irregular shapes was observed to be random in time and location. Occasionally, zigzag drop contours appeared immediately upon impact. It can be argued that the transition from total rebound to partial rebound was not due to the impalement by dendrites. Since total rebound was observed for impacts of warm drops above 0 ◦ C onto substrates at all tested temperatures from 5 to −20 ◦ C, the possibility of impalement is excluded. It is therefore inferred that partial solidification prevented the total rebound.

4.10 Drop Impact with Solidification and Icing

147

Figure 4.37 Partial rebound of a supercooled drop at −4 ◦ C on a superhydrophobic surface at

0 ◦ C. The drop diameter is 1.57 mm, and the impact velocity is 3.38 m/s. The horizontal straight line indicates the level of the substrate (Li et al. 2015). With permission of Springer.

This assumption is confirmed by the experiments on drop impact onto a wall of variable temperature. The impact of supercooled drops was conducted on superhydrophobic substrates of decreasing temperature, from 0 to −20 ◦ C. The drop rebound remained unaffected until the substrate temperature decreased to −2 ◦ C. At lower substrate temperatures, the central part of the drop appeared to be frozen on the solid surface and consequently prevented total rebound. The residual frozen area became larger as the substrate temperature dropped, as shown in Fig. 4.38. The images are selected at the instants when the drop/wall interface reached a minimum diameter during the receding phase, which is designated as the minimum receding diameter dmin .

–4°C

–10°C

–20°C

Figure 4.38 Effect of the substrate temperature on the receding of supercooled water drops on

superhydrophobic surfaces. The images have been taken at the instant of the minimum receding diameter. The diameters of the impinging drops were approximately 1.6 mm, and the impact velocities were around 3.4 m/s (Li et al. 2015). With permission of Springer.

Drop Impact onto a Dry Solid Wall

Minimum receding diameter Speed of ice accretion on SHSs Growth speeds of ice dendrites

0.9

0.7

0.7

0.5

0.5

0.3

0.3

dmin

0.1 −0.1

vicing m/s

0.9

d*min

148

0.1

−20

−15

−10 Ts °C

−5

0

−0.1

Figure 4.39 Dimensionless minimum receding diameter and the speed of ice accretion resulting

from impact of supercooled water drops at −4 ◦ C on superhydrophobic substrates from 0 to −20 C. The growth speeds of ice dendrites are taken from Shibkov et al. (2003). The diameters of the impinging drops are approximately 1.6 mm, and the impact velocities are around 3.4 m/s (Li et al. 2015). With permission of Springer. ◦

Figure 4.39 quantifies the relationship between the minimum receding diameter, dmin , and the substrate initial temperature, TS . The dimensionless minimum receding diame∗ = dmin /dmax . The value of dmax ter is scaled by the maximum spreading diameter as dmin was measured before the periphery of the flattened drop separated in the form of secondary droplets. The speed of ice accretion on superhydrophobic substrates is defined as vicing = dmin /(2tcontact ), where tcontact is the icing time measured from the instant of impact to the instant when dmin was reached. The value of dmin increased monotonically with the reduction of the temperature TS until −12 ◦ C and leveled off when TS dropped below −15 ◦ C. The data scatter could be attributed to the randomness in the formation of the initial nucleation sites and the irregular influence of ice dendrites on the drop deformation. The final deposited drops on the substrate had irregular shapes. The minimum receding diameter indicates the coverage of ice on the substrate. The speeds of ice accretion on substrates are also plotted in Fig. 4.39 in comparison with the growth speeds of free ice dendrites at the corresponding temperatures (Shibkov et al. 2003). It is seen that the speed of ice accretion on the substrate is faster than the growth rate of ice dendrites in sessile drops. This is an indication that the front of the dendrites expansion propagates under a certain angle to the substrate. This angle is usually approximately equal to α = 30◦ . Only the line of intersection of the surface of the dendrite front with the substrate could be measured. The speed of ice propagation on the substrate is, therefore, vicing = vdendrite / sin α, which is twice the velocity of a single dendrite propagation, vdendrite . This conclusion is confirmed by the results shown in Fig. 4.39.

4.11 References

4.11

149

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5

Drop Impact onto Dry Surfaces with Complex Morphology

Surface texture, e.g. roughness, porosity, wettability and chemical composition can significantly affect the outcome of drop impact. Section 5.1 deals with the splashing threshold on rough, textured and also porous solid surfaces. In Section 5.2 an impact of a single Newtonian drop near a hole in a flat substrate is considered as a simplified model of drop spreading on a porous substrate. The experiments described in Section 5.3 deal with drop impacts of such different liquids as water and oily Fluorinerts onto suspended thin membranes with microscopic pores of different wettability. They reveal that liquid penetration is possible even through a non-wettable porous medium if the impact velocity is high enough. A similar conclusion stems from the experiments with water drop impacts onto membranes coated with much less permeable nanofiber layers discussed in Section 5.4. In the case of nanofiber mats deposited onto impermeable surfaces, drop splashing and bouncing after impact can be fully suppressed, as the experiments of Section 5.5 show. The reason for the phenomena observed in Sections 5.3–5.5 is the hydrodynamic focusing of liquid brought by a millimeter-sized drop into micron-sized pores. The theory of the hydrodynamic focusing phenomenon is given in Section 5.6, and the results are illustrated experimentally by the amazing fact that liquid velocity in the jets which penetrated through the entire porous medium thickness is higher than that in the impacting drop, even though the viscous dissipation in flow through porous medium is extremely high. Liquid penetration following drop impact onto a nonwettable porous medium is also visualized in the experiments with the entrained seeding particles in Section 5.7, which also contains the evaluation of the critical filter thickness which can be fully penetrated in spite of the viscous dissipation in the pores. Drop impacts onto hot surfaces covered with nanofiber mats also reveal significant enhancement of surface cooling due to the hydrodynamic focusing. The latter sustains the contact of liquid coolant with the hot surface underneath and thus facilitates complete liquid vaporization and significant heat removal in the form of latent heat of evaporation (Section 5.8). Even the Leidenfrost effect can be suppressed at the high-temperature surfaces covered by nanofiber mats (Section 5.9) due to the hydrodynamic focusing which forces liquid brought by an impacting drop to adhere to the surface. Wettability of a dielectric surface by impacting liquid drop can be dramatically altered by an electric field applied to the surface. As shown in Section 5.10 the dynamic electrowetting-ondielectric (DEWOD) phenomenon can prevent bouncing of water drops even from a Teflon surface.

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5.1

Drop Splashing on Rough and Textured Surfaces Surface morphology in many cases significantly influences the drop spreading and the splashing threshold. The surface patternation, i.e. the topology of the surfaces, introduces additional parameters into the physical problem, such as the height of the features (pillars, bumps and so on), the wavelength, the size ratio, etc. Roughness gradients of the surface, like the wettability gradients, lead to the emergence of the drop motion parallel to the surface. In such cases the direction of the drop rebound can be controlled (Malouin Jr et al. 2010). The effect of the surface microstructure on the deformation of the spreading alumina micro-drops impacting with velocity of 90 m/s (the conditions corresponding to plasma spaying) has been studied in Shinoda et al. (2007). It has been shown that the surface structure influences the diameter of the splat and the stability of its shape. Asymmetric drop spreading on structured substrates has been observed in Sivakumar et al. (2005). A drop splash on rough and textured surfaces was studied in Xu (2007). In many cases the surface morphology enhances the prompt splash occurring during the very first phase of drop impact and its initial deformation and spreading. In the case of textured surfaces (square micro-pillars arranged in square lattice) the dimensions of pillars, in particular their height, affect the splashing very strongly. Moreover, at some limiting pillar height the corona splash can be suppressed. In Tsai et al. (2010) the effect of the micro-pillar arrangement on splash was investigated. They have demonstrated that the threshold Weber number for splash is determined by the ratio between the inter-pillar distance and the pillar thickness. The increase of the inter-pillar distance leads to an increase in the critical Weber number, or in other words to splash suppression. Interesting experiments on drop impact onto hydrophobic microgrids have been reported in Brunet et al. (2009). This case can be considered as a combination of the structured surface and porous target. Drop collision with such surfaces leads to the formation of arrays of tiny droplets behind the grid. Such studies can help to better understand the phenomena of drop collision with porous surfaces. The threshold Weber number corresponding to the formation of droplets is less than unity and depends on the relative holes area in the grid. In order to determine the effect of the surface roughness, wettability and porosity, various materials have been used as targets. Examples of the scanning electron microscope images of the surfaces of the targets, used in the experiments, are shown in Fig. 5.1. Roughness of the porous targets is irregular in all cases. This is also true for a number of impermeable targets, denoted as rough targets in the following. Several bronze and stainless steel targets were manufactured by face turning to achieve larger roughness values. These targets have a regular groove structure. The main properties characterizing the surface roughness can be measured using a commercial tactile roughness measurement instrument. The following are the main length scales characterizing the targets: r Amplitude average roughness – Ra r Mean of five individual values of the distances between the highest peak and the deepest valley) for five consecutive sample lengths – Rz

157

5.1 Drop Splashing on Rough and Textured Surfaces

50 μm

50 μm

200 μm

50 μm

1 mm

1 mm

Figure 5.1 Scanning electron microscope images of porous targets used in the experiments with

drop impact (left to right): bronze, ceramics, borosilicate glass, polyethylene, polytetrafluoroethylene, stainless steel (Roisman et al. 2015). Reprinted with permission from Elsevier.

r Average height of protruding peaks above roughness core profile – R pk r Mean width of a profile element – Rsm The value of R pk can be determined also for the porous targets for which the depth of the “valley” is apparently infinite. Range and Feuillebois (1998) pointed out that the average roughness Ra is insufficient to describe the splashing threshold on a rough surface. It is clear that the ratio Ra /D (with D being the drop diameter) cannot be used as a relevant scale characterizing the influence of roughness on the splashing threshold. Various combinations of the Reynolds and Weber numbers and roughness properties of the substrates should be tested to evaluate their effect on the experimental data. The two main parameters describing best the deposition–prompt splash limit are the Weber number and the ratio R pk /Rsm . The dimensionless parameter R pk /Rsm expresses the characteristic slope of the substrate topography. The experimentally obtained map for the prompt splash on rough substrates at the atmospheric pressure is shown in Fig. 5.2. The effect of viscosity on the critical Weber number is small. Since the target morphologies are irregular, there is no sharp boundary between the regions corresponding to splash or deposition. Near the splashing threshold a domain exists where the splashing probability varies from zero to one. Accordingly, two thresholds can be determined based on the experimental data: r the lower splashing threshold, below which the probability of prompt splash is negligibly small, r the upper deposition threshold, above which the drop deposition has negligibly small probability.

Drop Impact onto Dry Surfaces with Complex Morphology

We

158

Figure 5.2 Map of prompt splash/deposition on rough substrates at the atmospheric pressure,

p = 1 bar (Roisman et al. 2015). Reprinted with permission from Elsevier.

The lower splashing threshold is estimated from the experimental data on rough substrates in the form Wesplash = 10.5(R pk /Rsm )−0.7 .

(5.1)

The average value of the deposition/splashing threshold in Fig. 5.2 corresponds to Weaverage = 10.2(R pk /Rsm )−0.83 .

(5.2)

The interpretation of the experimental results on drop impacts onto porous substrates is even more complicated, since several additional scales are added in the problem: the substrate porosity and the characteristic pore diameter. Surprisingly, the limiting Weber number for splash, defined in Eq. (5.1) describes rather accurately not only impacts on rough substrates but also on porous surfaces. This means that the surface porosity does not influence the lower splashing threshold. However, the substrate porosity influences the upper deposition limit. In other words, drop deposition without splash is more probable on porous substrates. This phenomenon is caused by the possibility of a fast partial drop penetration into the target already during the impact stage, as in Fig. 5.3. The main dimensionless parameter which influences

Figure 5.3 Prompt splash/deposition on rough substrates at the atmospheric pressure, p = 1 bar (Roisman et al. 2015). Reprinted with permission from Elsevier.

159

5.2 Drop Impact Close to a Pore

We/(1 + 0.24X 0.5)

1000

deposition data: porous substrates rough substrates

800 600 400 200 0 0.00

0.02

0.04

0.06

0.08

0.10

Rpk/Rsm Figure 5.4 Map of prompt splash/deposition on rough substrates at the atmospheric pressure,

p = 1 bar (Roisman et al. 2015). Reprinted with permission from Elsevier.

the upper deposition limit on porous targets is the modified Reynolds number χ=

ρR pk U0  μ

(5.3)

where  is the porosity. It should be emphasized that in the experiments no correlation between the pore size and the splashing threshold could be identified. With the help of the parameter χ the upper deposition threshold on porous and rough substrates is estimated in the form We = 29.0(R pk /Rsm )−0.68 . (5.4) 1 + 0.24χ 0.5 This equation is obtained by fitting of the experimental data, as shown in Fig. 5.4. In addition to surface roughness and porosity, surface wettability can significantly affect the outcomes of drop impact at sufficiently low velocity. Deng et al. (2012) formed porous fractal-like deposits of candle soot on a substrate and coated the deposit with a 25-nm layer of silica. The coating was calcinated and silanized, after which it became superamphiphobic, i.e. repelled both water and hexadecane drops being simultaneously superhydrophobic and superoleophobic.

5.2

Drop Impact Close to a Pore Drop impact onto a dry or wetted interface is one of the elements of various industrial applications and one of the favorite topics of many research groups (Yarin 2006). The phenomena of drop impact onto a smooth dry substrate are already sufficiently elucidated (see Chapter 4). The evolution of drop diameter is determined by the Reynolds number, the Weber number and the wettability (Roisman et al. 2002); see Sections 4.3 and 4.4 in Chapter 4. The evolution of the dimensionless height of the lamella at the

160

Drop Impact onto Dry Surfaces with Complex Morphology

early initial stage is universal. It almost does not depend on the impact parameters (Roisman et al. 2009); see subsection 4.3.2 in Chapter 4. At the later stages the lamella height is influenced by the flow in the near-wall viscous boundary layer (Roisman 2009); see subsection 4.2.2 in Chapter 4. Much less is known about the mechanisms of drop impact onto porous substrates, which is relevant for ink-jet printing, needleless injection, rain–soil interaction, etc. The effect of roughness on the splashing threshold of an impacting drop has been demonstrated in Range and Feuillebois (1998). It was shown, however, that not only the mean roughness amplitude influences the magnitude of the threshold velocity (see Section 5.1 in this chapter) but also the target material, probably through its wettability properties. This effect is not fully understood and no reliable model has been proposed for such a phenomenon. Among the first studies devoted to drop penetration into porous plates is the one of Marmur (1988). Kellay (2005) focused on the investigation of the impact of a small water drop onto a granular material initially covered by a thin water film. Such drop impacts lead to substrate erosion, water entrainment and pattern formation. Moreover, at some threshold of drop impact frequency the geometry loses its axial symmetry and three-dimensional structures appear on the substrate. Target porosity plays an important role in the studies of drop impact onto paper (Kannangara et al. 2006) – the topic associated with ink-jet printing. An example of the direct numerical simulations of such phenomena can be found in Reis et al. (2008), including the description of drop spreading and penetration into a porous target. The higher is the porosity value and the more open are the pores, the less spreading is observed over the substrate surface, which is in agreement with the experimental data shown in Fig. 5.5. Recently the research of these phenomena was significantly extended by numerous experimental and numerical studies. The principal interest is associated with the surface morphology of the porous structure. In most of the studies random porous surfaces are used. Because of the small dimensions involved, the mechanisms of liquid penetration into pores cannot be easily investigated in detail. A model experimental situation is called for, which is the focus of the present section. The main aim of the experimental and numerical study described below is the investigation of a single drop impact onto a porous substrate. In order to achieve a better understanding of the physics behind the drop impact onto porous media the process is decomposed into simple submodels. One of the simplifying approaches is to model a porous medium as a group of unconnected cylindrical vertical pores. The model experiments with the setup shown in Fig. 5.6 revealed various interesting phenomena, e.g. the development of a liquid jet behind the pores. By further simplifying the model and reducing it to a single drop impact close to one pore it is possible to investigate these phenomena in detail. The experimental setup for the investigation of a single drop impact and its spreading on porous plates was the same as shown in Fig. 5.6. A plexiglas pad with an array of unconnected pores of 1 mm in diameter was used as a target. The images were recorded by a high-speed camera filming from below the plexiglas pad. Water drops of 2.5 mm in diameter were formed using a drop generator installed at a variable height above the

5.2 Drop Impact Close to a Pore

161

Figure 5.5 Drop impact on C1730III, CRS601II (porosity  = 59%, average pore diameter

3–5 µm), CMS606II (porosity  = 63%, average pore diameter 15 µm) and CPS6011II (porosity  = 65%, average pore diameter 30 µm) (from top to bottom). Time between two subsequent images is 1.85 ms. Drop diameter D = 2.35 mm; the impact velocity V0 = 2.35 m/s.

target. In the experiments the impact velocity, the distance between the impact axis and the holes, the size and the number of the holes have been varied. An example of the image sequence in Fig. 5.7 shows the moments when the drop flows over a pore. At the bottom of each image the time t after the drop impact is recorded. In the left image the drop just spreads straight over the pore. The shape of

Figure 5.6 Experimental setup.

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Drop Impact onto Dry Surfaces with Complex Morphology

t = 0.000025 sec

1

t = 0.00005 sec

2

t = 0.000075 sec

3

Figure 5.7 Time sequence of drop spreading over three holes. View from the bottom.

the lamella is almost axially symmetric, its flow is almost unperturbated. At the instants shown in the images in the middle and on the right-hand side the hole starts to decelerate the lamella spreading and behaves as an obstacle. The lamella front strongly deforms and is no longer circular. With time this effect becomes more pronounced, as is seen in the right-hand side image in Fig. 5.7. This phenomenon can be explained as follows. Probably the liquid of the spreading lamella enters the hole and does not overflow it. The hole serves as a local sink. The shape of the front near the hole is defined by the local velocity in the lamella and by the relative velocity of the rim. The phenomenon is thus similar to the breakup of a free liquid sheet by an obstacle (Clanet and Villermaux 2002). In Fig. 5.8 the case with a higher impact velocity is shown. At the outside edge of the pores narrow liquid jets start to form. They are ejected in the radial direction centered at the impact point. This case is rather interesting. Such a phenomenon has never been observed before, as to our knowledge. The appearance of the jets cannot be immediately explained. On the other hand, it could be rather important to understand this phenomenon, since it probably influences the splashing threshold and the process of drop penetration into a porous target. It should be emphasized that the appearance of the jets is not occasional. The jets emerged systematically starting from a definite threshold velocity. In Fig. 5.9 the maximum spreading for three different impact velocities is shown. The impact velocity in the left image is V0 = 1.4 m/s. The lamella is disturbed in its propagation over several holes, as has already been visible in Fig. 5.7. The jet formation

t = 0.0015 sec

2

t = 0.002 sec

3

t = 0.0025 sec

Figure 5.8 Drop spreading over three holes leading to the jet formation. View from the bottom.

4

5.2 Drop Impact Close to a Pore

t = 0.0027 sec

1

t = 0.0027 sec

2

163

t = 0.0037 sec

3

Figure 5.9 Maximum spreading state. The impact velocities are the following: in the left image, V0 = 1.4 m/s, in the middle, V0 = 3.1 m/s, in the right-hand side image, V0 = 4.4 m/s. View from the bottom.

does not take place. In the middle image the impact with the velocity of V0 = 3.1 m/s is shown. The drop spreads to a larger diameter than in the case on the left. The liquid flow sticks behind the holes more intensely and in addition the jet formation is triggered. The recognizable dependence of these effects on the impact velocity is elucidated in the right-hand side image in Fig. 5.9, where the impact velocity was V0 = 4.4 m/s. Not only do the jets appear behind the holes but even the lamella flow between the holes is strongly deformed, breaks up and leads to the appearance of finger-like jets. The experiments show that the behavior of a drop spreading over a hole, and generally on a porous material, is a more complex phenomenon than it might be initially assumed. To better understand the observed phenomena and, in particular, the mechanisms leading to the jets formation, drop impact near a hole has been simulated numerically. These simulations are described below. Drop impacts have been numerically simulated using the interFoam solver of OpenFOAM v1.6. InterFoam models a free surface flow with a volume-of-fluid method based on a finite volume discretization. Only one hole has been considered in the simulations in this study. The drop spreading was assumed symmetric relative to the plane which includes the impact and the hole axes. In order to reduce the computation cost only a half of the spreading drop was computed. The results of the comparison of the numerical predictions of drop spreading over a single hole with the experimental data are shown in Fig. 5.10. The numerical simulations

Figure 5.10 Comparison of the numerical simulations with experiments. Top view. The simulation

results are shown in the top row, the experimental images at the same time moments are in the bottom row.

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Drop Impact onto Dry Surfaces with Complex Morphology

Figure 5.11 Splash of a drop at a hole. Qualitative comparison of the numerical simulations (on

the left) with the experiments (on the right). Side view.

predict the emergence of the jets when the lamella flows over the hole. However, the upper view does not allow one to explain the mechanism of their formation. Much more information is obtained by plotting the numerically predicted shape of the lamella in the cross-section through a hole. An example of such an image is shown in Fig. 5.11. The flow in the lamella partially enters the hole and impacts the frontal part of the hole surface. This impact generates two main high-speed jets: one free uprising jet and the jet penetrating the hole, which reminds one of the jet partition described in Section 3.1 in Chapter 3. The deviation of the spreading lamella and its acceleration towards the hole is probably caused by the very high shear stresses at the contact line released when the contact line passes over the hole edge. The uprising jet can be clearly seen in the experiments (see the right-hand side image in Fig. 5.11). It should be emphasized that despite a relatively high velocity of the lamella spreading (and accordingly, the high Weber number) the wettability plays a significant role in the process of the lamella deviation over the hole. In Fig. 5.12 two cases are considered with different substrate wettabilities expressed by the apparent dynamic contact angles:

Figure 5.12 Effect of the substrate wettability on the lamella spreading over a hole. Side view with the hole cross-section. The contact angle is 30◦ on the left and 110◦ on the right.

5.3 Drop Impact onto Porous Surfaces

t* = 0.07 ms

t* = 0.27 ms

t* = 0.52 ms

t* = 0.77 ms

t* = 1.02 ms

t* = 1.32 ms

165

Figure 5.13 Liquid penetration into a hole. Impact velocity V0 = 11 m/s, drop diameter is D = 1.5 mm, distance to the pore 2.5 mm. Side view.

θ = 30◦ and θ = 110◦ . All the other impact parameters are the same as before. The lamella deviation is much more pronounced in the case of a wettable substrate (θ = 30◦ ). This deviation is enhanced by the capillary forces appearing at the hole walls. It should be emphasized that the flow over a hole is rather complicated and threedimensional; see the view in the symmetry plane in Fig. 5.13. Parts of the flow interact with the hole walls and generate two streams along the walls in the azimuthal direction. They collide at the opposite side of the wall. Such a collision also contributes to the emergence of the jets (see Section 3.1 in Chapter 3). It is clear that inertia plays a major role governing the flow in the lamella and generation of the jets. In Fig. 5.14 several cases of numerical simulations of drop impacts with different impact velocities are shown. While at relatively small impact velocities the lamella front is simply disturbed by the presence of the hole, at higher velocity it leads to the jet formation behind the hole. In the case considered in Fig. 5.14 the threshold velocity is 7 m/s.

5.3

Drop Impact onto Porous Surfaces Questions on penetration of liquid drops into porous media arise in relation to filtration through porous non-woven membranes, coalescence filters and ordered fibrous media with fiber diameters from several tens of microns down to electrospun nanofibers (Tien 1989, Brown 1993, Yarin et al. 2005). Single collector capture mechanisms intercept a single drop per event, at one point on a fiber or at the pore surface. Multi-fiber capture mechanisms such as straining, pore bridging and pore blocking are also present in practice. Various physical mechanisms of drop interception and drainage are discussed in the above-mentioned references. Some approaches to the development of novel

166

Drop Impact onto Dry Surfaces with Complex Morphology

V0 = 3 m/s

V0 = 4 m/s

V0 = 5 m/s

V0 = 6 m/s

V0 = 7 m/s

V0 = 8 m/s

V0 = 9 m/s

V0 = 10 m/s

V0 = 11 m/s

Figure 5.14 Maximum drop spreading at various impact velocities. Drop diameter is

D = 1.5 mm, distance from the impact point to the pore center is 2.5 mm, pore diameter 1 mm.

coalescence filters were triggered by the development of a number of the so-called superhydrophobic materials with high static contact angles of more than 150–160◦ , in particular, those mimicking the Lotus effect (Jiang et al. 2004, Gao and McCarthy 2006, Gao and McCarthy 2009, Marmur 2007). Drop impacts onto porous non-wovens are characteristic not only of coalescence filtration but also of protective clothes used as a barrier for warfare liquid aerosols, e.g. such nerve agents as VX (Reneker et al. 2007). Observations with the help of high-speed cameras revealed many intriguing features of drop impact onto solid and liquid surfaces and significantly facilitated our understanding of this fascinating phenomenon (Yarin 2006, Thoroddsen et al. 2008). A recent ramification of such investigations encompassed drop impact onto microand nano-textured porous surfaces (Tsai et al. 2009, Tsai et al. 2010, Nguyen et al. 2010, Lee and Lee 2011). Such impacts are accompanied by non-trivial physical effects, which require understanding. In addition to the interesting physical aspects, the attention to drop impacts onto nano-textured porous surfaces was fueled by the interest in developing novel water-repelling surfaces, with electrospun Teflon nanofiber mats being a natural candidate (Han and Steckl 2009). On the other hand, drop impact onto nanofiber mats (both electrospun or not) revealed that under certain conditions they promote dynamic wettability rather than repelling water (Srikar et al. 2009, Lembach

5.3 Drop Impact onto Porous Surfaces

167

et al. 2010, Sinha-Ray et al. 2011, Weickgenannt et al. 2011a, Weickgenannt et al. 2011b, Tsai et al. 2009, Lee and Lee 2011). Moreover, it was recognized that due to the large disparity between the drop and the inter-fiber pore sizes (the drop diameter D of the order of 1 mm and the pore size d of the order of 1–10 µm), liquid accumulates the kinetic energy at the pore entrances and protrudes into them with a speed of the order of U = (D/d )V0 , where V0 is the impact velocity. This hydrodynamic focusing effect (see Section 5.6 in this chapter) is reminiscent of the mechanism of formation of shaped-charge (Munroe) jets, considered in Sections 13.1 and 13.2 in Chapter 13, and is kindred to the widely known way of opening wine bottles by a sufficiently strong palm hit onto their bottom. A possible wettability-driven liquid spreading on non-uniform electrospun mats is a slow phenomenon (on the scale of 10 s, see Section 5.5 in this chapter), which is immaterial in the domain of the dynamic wettability (on the scale of a few milliseconds, see Section 5.5 in this chapter); Lembach et al. (2010). The idea that it might be possible to achieve an ultimate fluid-repellent coalescence filter made of electrospun Teflon or other superhydrophobic nanofibers is of significant interest, sounds attractive and is widely discussed. However, as the above-mentioned results on drop impact on electrospun nanofiber mats show, static hydrophobicity of such mats does not characterize dynamic behavior under the conditions of drop impact, and cannot prevent water penetration into these nanofiber mats. The latter means that dynamic transition from the Cassie–Baxter to Wenzel state (see Section 1.8 in Chapter 1) is possible for such systems, even though the static one is not. The present section as well as the following Sections 5.5–5.7 in this chapter consider drop impact on microscopic pores and pores in nano-textured surfaces and reveal in detail the effect of hydrodynamic focusing which arises in such cases. In the present section the drop-impact experiments were conducted using the following microscopically porous materials and liquids: (i) hydrophilic bare nylon grids (see Fig. 5.15a) were used as microscopically porous membranes, while water drops impacted onto them; (ii) hydrophobic Teflon-coated grids obtained by dip coating of nylon grids into a Teflon solution (see Fig. 5.15b) were used as microscopically porous membranes, while water drops impacted onto them (note, that water is a polar, high surface tension fluid); (iii) drops of a non-polar, low surface tension Fluorinert fluid FC 7500 impacted onto bare nylon grids and Teflon-coated grids; (iv) drops of another nonpolar, relatively low surface tension fluid, hexane, were impacted onto bare nylon grids and Teflon-coated grids. Nylon grids were installed in the experimental setup schematically shown in Fig. 5.16. It consisted of an adjustable platform, a syringe pump, a high-speed CCD camera, an external light source, a computer and a stand assembly. Nylon grids were placed centrally on the adjustable platform which can be moved in the X –Y direction (in the horizontal plane). To ensure reproducibility of drop sizes, liquids were delivered by syringe pump with a constant flow rate of 5 ml/h. A 25G needle was used to form water drops by gravity-driven dripping, whereas an 18G needle was used to form FC 7500 and hexane drops. In the experiments with drop impacts onto bare nylon grids it was found that at low impact velocities (1–2 m/s) there was no visible penetration of water drops to the other

168

Drop Impact onto Dry Surfaces with Complex Morphology

(a)

(b)

Figure 5.15 SEM image of: (a) bare nylon grid, (b) Teflon-coated nylon grid. The image shows

some blocked pores (the white areas), which were very infrequent. The blockage is due to a thin Teflon film. The majority of the pores are always open (the dark areas) (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

side of the grid (Fig. 5.17). However, it can be seen from Figs. 5.17b–5.17f that although there was no penetration to the other side, the water drop did not retract, but rather was pinned. When the drop impact velocity was increased above 2 m/s, visible penetration of water through bare nylon grids was observed (Fig. 5.18). However, it was also observed

Nozzle

CCD Camera

Adjustable Stand Assembly Syringe Pump

Sample

Adjustable Platform External Light Source Figure 5.16 Schematic of the experimental setup (Sahu et al. 2012). Reproduced with permission

from The Royal Society of Chemistry.

5.3 Drop Impact onto Porous Surfaces

(b)

(a)

t = 0 ms

(c)

t = 6 ms (e)

(d)

t = 18 ms

169

t = 12 ms (f)

t = 24 ms

t = 30 ms

Figure 5.17 Impact of water drop onto bare nylon grid with low impact velocity of 1 m/s. The

panels correspond to: (a) t = 0 ms, (b) t = 6 ms, (c) t = 12 ms, (d) t = 18 ms, (e) t = 24 ms and (f ) t = 30 ms. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

that at lower impact velocities (2–3 m/s), after the water drop has penetrated through the nylon grid, it mostly retracted back presumably due to the effect of the surface tension (Figs. 5.19a and 5.19b). At a higher impact velocity (3.46 m/s), a part of the water drop that penetrated through the nylon grid did not retract back and dripped down as smaller drops (Figs. 5.19c and 5.19d). It is clear that in the latter case water still had enough kinetic energy to overbear surface tension. The scanning electron microscope (SEM) image of a Teflon-coated nylon grid is shown in Fig. 5.15b. Even though the grid has some blocked pores (the white areas) due

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 5.18 Water drop images on bare nylon grids in 2 ms after the impacts. The impact

velocities: (a) 2.23 m/s, (b) 2.44 m/s, (c) 2.64 m/s, (d) 2.82 m/s, (e) 3.0 m/s, (f ) 3.16 m/s, (g) 3.31 m/s and (h) 3.46 m/s. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

170

Drop Impact onto Dry Surfaces with Complex Morphology

(a)

(b)

(c)

(d)

Figure 5.19 Water drop impact with the impact velocity of 2.23 m/s at the time moments:

(a) t = 2 ms and (b) t = 4 ms. Water drop impact with the impact velocity of 3.46 m/s at the time moments: (c) t = 2 ms and (d) t = 4 ms. The comparison of panels (a) and (c) shows how the amount of water penetrating through nylon grid increases with an increase in the impact velocity. Panel (b) show that at a lower impact velocity, the surface tension is capable of retracting almost all penetrated water, whereas panel (d) shows that at a higher impact velocity, the surface tension is incapable of preventing full penetration of a significant part of the impacting drop. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

to a thin clogging film of Teflon, the occurrence of such blockage was very rare. The majority of the pores were always open (the dark areas). Generally, the dip coating of Teflon over nylon grids was successfully conducted without pore blockage. The difference in wettability of bare nylon grids and Teflon-coated nylon grids was evaluated as follows. Water drops were gently put on each of these two types of grids at different places, and their contact angles observed. Two representative images are depicted in Fig. 5.20. It is seen that the bare nylon grids were partially wettable with water, with the static contact angle varying in the range of 55◦ –65◦ . On the other hand, the Teflon-coated nylon grids were rather non-wettable, with the static contact angle varying in the range of 125◦ –140◦ . The low-velocity (1 m/s) impact of a water drop dripped onto a Teflon-coated nylon grid is shown in Fig. 5.21. It is seen that after an initial spreading, the water drop recedes under the action of the surface tension and even tends to bounce back from the grid, albeit the lower part of it remains in contact. No visible water penetration through such grids was found at such low impact velocities (up to 2.44 m/s). This is different from

5.3 Drop Impact onto Porous Surfaces

(a)

171

(b)

Figure 5.20 Static contact angle of water drops on (a) a bare nylon grid and (b) on a Teflon-coated

nylon grid. The images show that Teflon coating changed the partially wettable nylon grids into rather hydrophobic ones under static conditions. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

the observations described above for bare nylon grids. However, for drop impact with impact velocities of 2.64 m/s and above, there is visible penetration to the other side of the Teflon-coated nylon grids resembling that for bare nylon grids (Fig. 5.22). Sometimes, it was found that the upper part of the penetrating water drop splashes and breaks up into smaller drops, as is characteristic of receding splashes on hydrophobic surfaces (Yarin 2006). Still, a part of the penetrated drop (with impact velocity of 2.64 m/s) can be lifted back to the rear side of the Teflon-coated nylon grid due to the surface tension effect (Figs. 5.23a and 5.23b). For drop impacts with impact velocity of 3.46 m/s onto (a)

(c)

(b)

t = 0 ms

t = 4 ms (f)

(e)

(d)

t = 12 ms

t = 8 ms

t = 16 ms

t = 20 ms

Figure 5.21 Water drop impact onto Teflon-coated nylon grid with a low impact velocity of 1 m/s

at (a) t = 0 ms, (b) t = 4 ms, (c) t = 8 ms, (d) t = 12 ms, (e) t = 16 ms and (f ) t = 20 ms. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5.22 Water drop impact onto Teflon-coated nylon grid 2 ms after the first contact. The

impact velocities of (a) 2.64 m/s, (b) 2.82 m/s, (c) 3.0 m/s, (d) 3.16 m/s, (e) 3.31 m/s and (f ) 3.46 m/s. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

(a)

(b)

(c)

(d)

Figure 5.23 Water drop impact onto Teflon-coated nylon grid with the impact velocity of 2.64 m/s

at (a) t = 2 ms and (b) t = 8 ms. An impact with the impact velocity of 3.46 m/s at (c) t = 2 ms and (d) t = 4 ms. The comparison of panels (a) and (c) shows how the amount of water penetrating through the grid increases with the increase in the impact velocity. Panel (b) shows that at a lower velocity, surface tension is able to stop and uplift almost all the penetrated water behind the rear side of the grid. On the other hand, panel (d) shows that at a higher impact velocity no “lift-up” is possible anymore and a significant part of water fully detaches from the rear side of the grid. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

5.3 Drop Impact onto Porous Surfaces

12

Bare Nylon Teflon-coated Nylon

10

10

8

8

6

6

4

4

2

2

0 2.2

2.4

2.6 2.8 3 Impact velocity (m/s)

3.2

3.4

Volume fraction (%)

Volume fraction (%)

12

173

0

Figure 5.24 Fraction of water drops which penetrated through micropores after drop impact onto

bare nylon or Teflon-coated nylon grids. The data was obtained using the images recorded 2 ms after drop impact. In the case of nylon grids, there is no water penetration at impact velocities below 2.23 m/s, in the case of Teflon-coated nylon grids there is no water penetration at impact velocities below 2.64 m/s (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

the Teflon-coated nylon grid, no “lift-up” was found and some part of the penetrated drop detached from the parent drop (Fig. 5.23d) similar to the higher-speed impacts onto bare nylon grids (Fig. 5.19d). The inspection of the images in Figs. 5.17–5.19 and 5.22–5.23 shows that penetrating water jets have diameters of about 0.01 to 0.05 cm. Given the distance of about 30 µm between the pores in the grid, these values recast into simultaneous penetration of water through 10–250 pores. Volume fractions of the impacting drops which penetrate through the grid were evaluated using images taken 2 ms after the impact. It was assumed that water visible below the rear surface of the grid forms a body of revolution, which allows the evaluation of volume fraction based on two-dimensional images. Figure 5.24 shows the penetrated fraction of water drop versus the impact velocity of the initial drop for the two cases: bare nylon grids and Teflon-coated nylon grids. It is instructive to see that at lower impact velocities the Teflon coating diminishes the penetrated volume fraction. However, as the impact velocity increases up to 3.31 m/s, the penetrated volume fractions in both cases (bare nylon grid and Teflon-coated grid) become the same, and the grid wettability has no effect on the penetration process. At higher impact velocities penetration becomes fully dynamic. Figure 5.25 shows the sequence of images for the drop of FC 7500 (its surface tension and kinematic viscosity are 16 mN/m and 0.77 cSt, respectively) for the impact velocity of 1 m/s onto a bare nylon grid. It can be seen that the liquid penetrates through the grid

174

Drop Impact onto Dry Surfaces with Complex Morphology

(a)

(b)

(c)

t = 2 ms

t = 0 ms

(d)

t = 4 ms

t = 6 ms

Figure 5.25 Drop of FC 7500 impacting onto a bare nylon grid with the impact velocity of 1 m/s

at (a) t = 0 ms, (b) t = 2 ms, (c) t = 4 ms, (d) t = 6 ms after the impact. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

even at such a low impact velocity (compared with the threshold velocity of 2.23 m/s for water penetration through such grids). This can be attributed to the fact that for penetration through a membrane of thickness h, the impact velocity needed for the full penetration (through the entire membrane thickness) is of the order of V0 = νh/d 2 ; see Section 5.7 in the present chapter, where ν is the kinematic viscosity and d is the pore size. An impact with such velocity can overbear viscous dissipation in the pores and deliver the liquid entering the pores to the rear side of the grid. The kinematic viscosity of FC 7500 is 0.77 cSt, which is less than that of water, which explains the reason for easier penetration of FC 7500 (compared to that of water) through the entire membrane at lower impacting velocities. An important observation can be made for impacts of FC 7500 drops onto bare nylon grids. At a certain threshold velocity of about 2 m/s, FC 7500 had started to emerge after penetration as separate tiny jets originating from the pores of the nylon grid, as seen in Fig. 5.26a. The leading parts of these tiny jets then break up into tiny droplets (presumably due to the Rayleigh capillary instability, see subsection 1.10.1 in (a)

(b)

(c)

(d)

(e)

Figure 5.26 Drop of FC 7500 impacting onto a bare nylon grid with impact velocity of 2.64 m/s.

Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

5.3 Drop Impact onto Porous Surfaces

175

(a)

(b)

(c)

(d)

(e)

Figure 5.27 Impact of FC 7500 drop onto a bare nylon grid with the impact velocity of 3.46 m/s.

Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

Chapter 1), whereas the residual parts then coalesce to form a single jet, similar to the one accompanying water drop impact, as can be seen in Figs. 5.26b–5.26c. The merger jet, in its turn, breaks up into bigger secondary droplets, again due to the action of the surface tension (the capillary instability). Figure 5.27 shows the sequence of images for the FC 7500 drop impact with impact velocity of 3.46 m/s. The appearance of tiny jets is also apparent in this case: see Fig. 5.27 which depicts the processes of penetration, coalescence and jet breakup similar to those of Fig. 5.26. The tiny jets are rather blurred in Fig. 5.27a, which is due to the high impact speed of the drop. The effect of the impact velocity on the outcome of the FC 7500 drop impact is illustrated in Fig. 5.28. The penetration patterns for this liquid corresponding to various impact velocities can be clearly distinguished. Similar experiments were conducted with the FC 7500 drop impacts onto Tefloncoated nylon grids. The experiments showed no visible difference in the penetration pattern of drops of FC 7500 through Teflon-coated grids compared to those for the bare nylon grids. Hexane has a kinematic viscosity of 0.45 cSt, which is even lower than that of FC 7500. The results for the hexane drop impacts onto bare nylon grid are shown in Fig. 5.29 for three impact velocity values: V0 = 1 m/s, V0 = 1.4 m/s and V0 = 3.46 m/s. It is seen that hexane drops fully penetrate (through the entire thickness) through bare nylon grids at the very low velocity of 1 m/s compared to that of water drops for which the first penetration was observed at 2.23 m/s. This can be attributed to a much lower kinematic viscosity of hexane, and thus the reduced dissipation of kinetic energy inside pores. After penetrating at the impact velocity of 1 m/s, the hexane blob retracts back to the rear surface of the nylon grid under the action of surface tension. Hexane drop

176

Drop Impact onto Dry Surfaces with Complex Morphology

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 5.28 Effect of the impact velocity on the penetration pattern of drops of FC 7500 impacting onto bare nylon grid. The impact velocities: (a) 1.0 m/s, (b) 1.41 m/s, (c) 1.73 m/s, (d) 2.0 m/s, (e) 2.23 m/s, (f ) 2.44 m/s, (g) 2.64 m/s, (h) 2.82 m/s, (i) 3.0 m/s, ( j) 3.16 m/s, (k) 3.31 m/s and (l) 3.46 m/s. All the images correspond to 2 ms after drop impact. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

impact at the high speed of 3.46 m/s results in separate jets coming out of several micropores of the nylon grid as is seen in Fig. 5.29. The relatively low surface tension of hexane (18.43 mN/m) compared to that of water is insufficient to merge the multiple jets after penetration (cf. Figs. 5.29 and 5.18). The impacts of hexane drops onto a Teflon-coated nylon grid at low and high impact velocity (V0 = 1 m/s and V0 = 3.46 m/s, respectively) are depicted in Fig. 5.30. The outcomes look similar to those for a bare nylon grid. At low velocity a blob which penetrated through the grid retracts back to the rear surface, whereas at high impact velocity separate jets are visible behind the nylon grid. The results for the FC 7500 and hexane drops show that tiny jets behind the grid break up into tiny secondary droplets presumably due to the capillary instability (see

177

5.4 Drop Impact onto Suspended Nanofiber Membranes

V0 = 1 m/s

V0 = 1.4 m/s

V0 = 3.46 m/s

t = 0 ms

t = 2 ms

t = 4 ms

t = 6 ms

t = 8 ms

Figure 5.29 Hexane drop impact onto a bare nylon grid at different impact velocities. The

sequence of the images for each velocity value corresponds to the time span from 0 ms (the moment just before the impact) to 8 ms. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

subsection 1.10.1 in Chapter 1). That makes it extremely difficult to evaluate the amount of liquid penetrated through the grid. Therefore, the results similar to those for water in Fig. 5.24 are unavailable for FC 7500 and hexane. It should be emphasized that for both these low surface tension and low-viscosity liquids, FC 7500 and hexane, penetration through pores was much easier (at a lower impact velocity) than that for water. Therefore, the experiments described in the following Section 5.4 were conducted for water drops alone.

5.4

Nano-textured Surfaces: Drop Impact onto Suspended Nanofiber Membranes Electrospun nanofiber mats are porous permeable materials composed of individual non-woven polymer nanofibers (with diameter of about several hundred nanometers) which are randomly orientated in the mat plane. The size of the inter-fiber pores is of

V0 = 1 m/s

V0 = 3.46 m/s

t = 0 ms

t = 2 ms

t = 4 ms

t = 6 ms

t = 8 ms

Figure 5.30 Hexane drop impact onto a Teflon-coated nylon grid at the impact velocities of V0 = 1 m/s and V0 = 3.46 m/s. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

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Drop Impact onto Dry Surfaces with Complex Morphology

2 mm

Figure 5.31 Top view of PAN nanofiber mat at different magnifications. On the right-hand side a

water drop softly deposited on the nanofiber mat is also seen (Lembach et al. 2010). Reprinted with permission. Copyright (2010) American Chemical Society.

the order of several micrometers (Reneker et al. 2007, Lembach et al. 2010, Yarin et al. 2014); see Fig. 5.31 where polyacrylonitrile (PAN) nanofiber mat is shown (PAN is a partially wettable polymer with water contact angle on a cast sample of about 30–40◦ ). The electrospun nanofiber mats are usually produced from polymers which are either partially wettable or non-wettable. The pores in the mats are filled with air. This makes them typically poorly wettable by water. Moreover, due to the presence of texture on several scales (roughness or nanopores of the order of 1–10 nm, or sometimes even beads of the order of several hundred nanometers, on individual nanofibers, nanofiber diameters of several hundred nanometers, and pore sizes of the order of several micrometers) electrospun nanofiber mats are nano-textured materials and typically behave similarly to hydrophobic materials. Thus, water drops deposited softly onto nanofiber mats remain practically spherical at their surface (see Fig. 5.31). To study drop behavior when impacting onto nanofiber mats, Sahu et al. (2012) conducted experiments with suspended electrospun nanofiber membranes. The membranes were partially supported by nylon grids (see Fig. 5.15), which possessed a 20 µm average pore size, with a pitch of 40 µm and 60 µm in orthogonal directions and a 34 µm thickness. These supporting grids prevented any visible vibrations following drop impact onto the electrospun mats. On the other hand, the grids could not affect liquid penetration, since the hydraulic resistance of the grids was much lower than that of the mats. A representative scanning electron microscopy (SEM) image of PAN nanofiber mat on a grid is shown in Fig. 5.32. It should be emphasized that the duration of nanofiber deposition onto nylon grids mostly decreased pore sizes in the mat, while the increase of the mat thickness was non-monotonous. Drop impact with impact velocity of 3.46 m/s onto nylon grids coated with PAN nanofibers electrospun for 5–60 s is shown in the images in Fig. 5.33. It was found that

179

5.4 Drop Impact onto Suspended Nanofiber Membranes

Figure 5.32 SEM image of electrospun PAN nanofibers deposited onto a bare nylon grid for 60 s.

(a) The overall view, (b) a zoomed-in view over an opening in the grid (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

the threshold velocity for water penetration through the nanofiber-covered nylon grid (with the mat deposited for 5 s) was 2.43 m/s. This value is higher than that of the corresponding threshold value of 2.23 m/s for bare nylon grids. Therefore, nanofiber mats deposited on nylon grids significantly reduce permeability. The presence of nanofibers results in a significant increase in viscous dissipation during water flow inside the pores, as discussed in Section 5.7 in the present chapter. The latter factor determines the thickness of nanofiber mat which can be fully penetrated at a given impact velocity. Alternatively, viscous dissipation determines a higher critical impact velocity for a full penetration of a given mat thickness in comparison with that for the bare nylon grid underneath. Note that full penetration of water through the mat means that some part of the water drop appears on the other side of the mat (thus crossing the full mat thickness), rather than the entire liquid delivered as a drop passed through the membrane. It should

(a)

(b)

t=5s (g)

t = 35 s

(c)

t = 10 s (h)

(d)

t = 15 s (i)

t = 40 s

(e)

t = 20 s (j)

t = 45 s

(f)

t = 25 s (k)

t = 50 s

t = 30 s (l)

t = 55 s

t = 60 s

Figure 5.33 Impacts of the identical water drops onto PAN nanofiber mats electrospun onto nylon

grids for different time t. The impact velocity was 3.46 m/s, the initial drop diameter was 2 mm. The electrospinning time t values were: (a) t = 5 s, (b) t = 10 s, (c) t = 15 s, (d) t = 20 s, (e) t = 25 s, (f ) t = 30 s, (g) t = 35 s, (h) t = 40 s, (i) t = 45 s, ( j) t = 50 s, (k) t = 55 s and (l) t = 60 s. The images correspond to 2 ms after the moment when the drops touched the target. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

Drop Impact onto Dry Surfaces with Complex Morphology

3.6

3.4

3.2 Impact velocity (m/s)

180

3

2.8

2.6

2.4

2.2

2 0

10

20

30 40 50 Electrospinning time (s)

60

70

Figure 5.34 The critical velocity for water penetration after drop impact versus the

electrospinning time of PAN nanofiber mats onto nylon grids. The square symbols correspond to the experiments where water penetration was observed, whereas the triangular ones correspond to the experiments without water penetration after drop impact. The initial drop diameter was 2 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

be emphasized that water penetration after drop impact onto a nanofiber mat deposited on a grid is mostly determined by the mat permeability, whereas the grid permeability effect is negligible since the grid is much more permeable than the mat. Figure 5.34 depicts the critical threshold velocity of drop impact needed for full water penetration through nanofiber mats electrospun for different times onto nylon grid. The critical velocity linearly increases with the deposition time in the range studied. The nanofiber mats deposited over nylon grids were inspected before and after drop impacts to corroborate that no big holes in the mats were created by drops, and water penetration was indeed associated with the initial mat porosity. It was found that drop impact does not create any holes in the nanofiber mat as the pore size and the overall mat morphology still remain approximately the same. In addition, the effect of multiple drop impacts at the same place of nanofiber mats was studied. It was observed from the images that no big holes appear even after the eighth impact, albeit some local nanofiber rearrangement could happen. Several other materials were used to form nanofiber mats on nylon grid. For example, nylon 6/6 nanofibers were electrospun over bare nylon grids for a few seconds. The rate of nylon 6/6 nanofiber deposition was very rapid compared to that of PAN

5.4 Drop Impact onto Suspended Nanofiber Membranes

181

(a) V0 = 1 m/s

t = 0 ms (b) V0 = 2.64 m/s

t = 4 ms

t = 8 ms

t = 12 ms

t = 16 ms

t = 20 ms

t = 0 ms (c) V0 = 3.46 m/s

t = 2 ms

t = 4 ms

t = 6 ms

t = 8 ms

t = 10 ms

t = 0 ms

t = 2 ms

t = 4 ms

t = 6 ms

t = 8 ms

t = 10 ms

Figure 5.35 Water drop impact onto electrospun nylon 6/6 nanofiber mats on a bare nylon grid at different velocities: (a) V0 = 1 m/s, (b) V0 = 2.64 m/s and (c) V0 = 3.46 m/s. The images correspond to different time instants from the moment of impact (approximately at t = 0 ms). Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

nanofibers. The accumulation of nylon 6/6 nanofibers beyond 10 s would result in very thick nanofiber mats, which exceed the critical nanofiber mat thickness, estimated theoretically (see Section 5.7 of the present chapter) and also significantly reduce the pore size to 2–4 µm. Therefore, nylon 6/6 nanofibers were collected for 5 s. The pore size in these mats was of the order of 1–4 mm. Figure 5.35 shows a sequence of images illustrating water drop impact onto nylon 6/6 nanofiber mats electrospun over a bare nylon grid. It is seen that at a low impact velocity a water drop spreads after impact and its contact line becomes pinned with no visible penetration through the mat. An increase in the impact velocity to V0 = 2.64 m/s results in a water drop corona splash after the impact. A full water penetration in the case of nylon 6/6 begins from the impact velocity of 3 m/s, compared to the impact velocity of 2.43 m/s in the case of PAN nanofibers. This is attributed to the reduced pore size of the nylon 6/6 nanofiber mat compared to that of PAN, which results in a higher viscous dissipation of kinetic energy inside the pores in the case of nylon 6/6. At the impact velocity of 3.46 m/s a water drop fully penetrates through the nylon 6/6 nanofiber mat on a bare nylon grid. The penetrated water drop can either retract back to the rear side of the grid under the action of surface tension, or break off as a merged jet and smaller secondary droplets resulting from the capillary instability (see subsection 1.10.1 in Chapter 1). Polycaprolactone (PCL) nanofibers were electrospun onto bare nylon grids for different times in the range of 10 to 120 s. These supported nanofiber mats were subjected to water drop impacts with velocities in the range 1 m/s to 3.46 m/s. The penetration charts for both PAN and PCL nanofiber mats obtained in the water drop impact experiments

Drop Impact onto Dry Surfaces with Complex Morphology

3.6 3.4 3.2

Impact velocity (m/s)

182

3 2.8 2.6 2.4 2.2 2 1.8

0

10

20

30

40

50

60 70 80 90 100 110 120 130 140 150 Electrospinning time (s)

Figure 5.36 Penetration chart for PCL and PAN nanofiber mats on bare nylon grids. The solid line

corresponds to PCL and the dashed line to PAN (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

were superimposed and presented in Fig. 5.36. Each point in the penetration chart corresponds to a certain combination of the impact velocity and the electrospinning time. The square symbols correspond to the penetration events, whereas the triangular symbols correspond to the non-penetration ones (for PCL nanofibers). The threshold velocity corresponding to the same time of deposition during electrospinning, appears to be different for PAN and PCL. Namely it is higher for PCL mats compared to that for PAN. This can be attributed to the interplay of the difference in the thickness and different pore sizes in the mats, and different wettability of PCL and PAN (much lower for the former). However, the penetration chart in Fig. 5.36 shows that for any wettability, after a threshold velocity is surpassed, water penetration inevitably happens. The static contact angle of water drops on cast Teflon was measured as 115–120◦ , whereas the static contact angle of water drops on the electrospun Teflon membrane on nylon grid was in the range 140–150◦ . Figure 5.37 shows the SEM micrograph of the electrospun 5 wt% Teflon AF 1600 nanofibers collected on a nylon grid. The fiber cross-sectional diameter ranges from 400 nm to 1 µm. It can also be seen from the SEM images in Figs. 5.37a and 5.37b that at some places the fibers flattened and ribbonlike structures appeared. The flattened structures could probably be merged nanofiber bundles which resulted from fibers landing at the counter-electrode while still wet due to an incomplete solvent evaporation.

5.4 Drop Impact onto Suspended Nanofiber Membranes

(a)

183

(b)

(c)

Figure 5.37 SEM images of electrospun Teflon nanofiber mat collected on nylon grid. The pore

size of the nylon grid is 20 µm and the collected nanofibers effectively reduce the pore size of the substrate to 3–6 µm. The thickness of the nanofiber mat is of the order of 5–8 µm. (a) The overall view. (b) Teflon fibers over a pore of the nylon grid. SEM images of electrospun Teflon nanofiber mat before (a) and after (c) drop impact at the impact location. The comparison of panels (a) and (c) does not show any visible damage to the mat caused by drop impact (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

SEM images of Teflon nanofibers collected on a nylon grid shown in Fig. 5.37a reveal that the pores in the grid were not entirely blocked by the nanofibers but rather diminished to the level of 3–6 µm when a 5–8 µm thick mat was electrospun. The mat is superhydrophobic. Superhydrophobicity of Teflon nanofiber mats is also illustrated in Fig. 5.38a where a water jet impacts obliquely at a relatively low velocity of 1–2 m/s at the mat. It is seen that the jet bounced back from the mat. Figure 5.38b shows blowing off of drops from the mat surface by low-speed air blowing. Significant water repellency of Teflon nanofiber mats visible in Fig. 5.38 does not necessarily mean that such mats are impenetrable to water under dynamic conditions of drop impact. In the experiments of Sahu et al. (2012), water drops were impacted onto Teflon nanofiber mats of thickness in the range of 5–8 µm deposited onto bare nylon grids. Drops had the impact velocity of 2.82 m/s and water penetration through the mat and grid was observed. Figure 5.39 shows the images of water drop impact onto Teflon mats at the impact velocity V0 = 3.46 m/s. It is seen that the water drop

184

Drop Impact onto Dry Surfaces with Complex Morphology

(a)

(b)

Figure 5.38 Two frames from a movie demonstrating superhydrophobicity of the Teflon nanofiber

mat. A water jet issued from an inclined syringe manually impacts onto the Teflon mat with a velocity of 1–2 m/s. (a) Water jet is repelled from the Teflon nanofiber mat on a nylon grid. (b) Drops are blown off from the Teflon nanofiber mat. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

partially penetrated through the Teflon nanofiber mat. After the impact the penetrated portion of water drop might retract back to the surface of the supporting nylon grid or break off as a jet or individual secondary droplets (as in Fig. 5.39). The penetration patterns depended on the impact velocity and thickness of the nanofiber mat. After the impact, the portion of the water drop remaining on the nanofiber mat surface retracts and completely rebounds from the surface. The inspection of the nanofiber mat after drop penetration showed that no damage was caused to the nanofibers and that the porous structure after penetration resembles the one before the penetration which is evident from Figs. 5.37a and 5.38c. The portion of the nanofiber mat shown in these figures corresponds to the impact location. It only covers an area of 0.2 mm × 0.2 mm. Some fibers are seen slightly re-oriented either by drop impact or water drying. However, it can be seen that no significant damage (big holes) was observed. Note that in the present section, it was shown that a millimeter-sized water drop softly deposited on a non-woven electrospun nanofiber mat produced from PAN stayed almost spherical for several minutes. A completely different outcome was observed when a similar water drop impacted onto the same nanofiber mat on a solid substrate

t = 0 ms

t = 2 ms

t = 6 ms

t = 10 ms

t = 14 ms

t = 18 ms

Figure 5.39 Water drop impact onto electrospun Teflon nanofiber mat on a nylon grid. The impact

velocity V0 = 3.46 m/s. Scale bars, 1 mm (Sahu et al. 2012). Reproduced with permission from The Royal Society of Chemistry.

5.4 Drop Impact onto Suspended Nanofiber Membranes

185

(a)

(b)

(c)

(d)

Figure 5.40 Different modes of drop impact onto a PAN nanofiber mat: (a) deposition,

(b) fingering without splash, (c) receding splash and (d) advancing splash. Time span is 1.5 ms, the drop diameter is 2 mm and the impact velocity is 1.7 m/s (a), 2 m/s (b), 2.3 m/s (c) and 2.7 m/s (d) (Lembach et al. 2010). Reprinted with permission. Copyright (2010) American Chemical Society.

at a speed of about 2–3 m/s. In the latter case, the drop first rapidly spreads on a nanofiber mat surface as on a dry, rigid substrate but then remains pinned in the spreadout configuration and does not recede or bounce, as on a completely wettable substrate (see Fig. 5.40). In the experiments with drop impacts onto suspended nanofiber membranes of PAN, nylon 6/6, PCL and Teflon, which practically encompass the entire wettability range, it was found that there exists a relatively low threshold velocity at which water drops fully penetrate (i.e. some water is disconnected from the rear side) through any of these porous nano-textured membranes irrespective of their wettability. It shows that after the threshold velocity has been surpassed, the static wettability plays only a secondary role

186

Drop Impact onto Dry Surfaces with Complex Morphology

in water penetration, which is mainly a dynamic process. The less wettable the membrane is, the higher is the threshold velocity; however, this velocity is still below 3.5 m/s. An especially non-trivial result is that superhydrophobicity of the porous nano-textured Teflon skeleton with interconnected pores is incapable of preventing water penetration due to drop impact, even at relatively low impact velocities close to V0 = 3.46 m/s. It should be also emphasized that electrospun nano-textured Teflon membranes, which possess very low hydraulic permeability, are shown to be incapable of fully repelling water under the dynamic conditions corresponding to drop impact.

5.5

Drop Impact onto Nanofiber Mats on Impermeable Substrates and Suppression of Splashing Drop impacts onto nanofiber mats on impermeable substrates (in distinction from drop impacts onto suspended membranes discussed in the previous section) revealed several non-trivial outcomes. In general, drop impacts onto impermeable surfaces involve such phenomena as spreading, receding, splashing and bouncing (Yarin 2006); see Chapter 4. However, drop impacts onto partially wettable PAN nanofiber mats on impermeable substrates demonstrated that receding, splashing and bouncing were practically eliminated; see Fig. 5.40. As Fig. 5.40a shows, the drop first spreads on the nanofiber mat surface as on a dry, rigid, completely wettable substrate but then remains pinned in the spread-out configuration and does not recede. Figure 5.40 demonstrates that an impacting drop, depending on its velocity and diameter, can develop different spreading modes ranging from deposition-like to splash-like ones. Several features of drop behavior shown in Fig. 5.40 (deposition, fingering, advancing and receding splash) are also familiar for drop impacts on impermeable surfaces (Yarin 2006). However, there are striking distinctions characteristic of drop impacts onto nanofiber mats on impermeable substrates. Namely, a corona splash, similar to that of drops impacting on thin films of water (see Section 2.2 in Chapter 2 as well as Sections 6.2, 6.6 and 6.7 in Chapter 6), was never observed, neither was bouncing. For drop impacts onto nanofiber mats it is impossible to determine definite thresholds for impact conditions corresponding to such specific secondary impact outcomes as fingering without splash, receding splash and advancing splash (see Fig. 5.40). A range of impact conditions can be identified in which several types of the outcomes can occur with a certain probability for the same impact conditions. It is explained by the fact that fingering and advancing and receding splashes are the results of an instability, which is initiated by the initial perturbations of drop surface and which cannot be completely controlled in the experiment, especially with such random nano-textured surfaces as those of nanofiber mats. In an attempt to generalize the results, the diagram is shown for two independent dimensionless parameters: the Reynolds number, Re = ρDV0 /μ, and the parameter Kd = [D3V05 ρ 3 /(μσ 2 )]1/4 , where ρ, μ and σ denote liquid density, viscosity and surface tension, D and V0 the drop diameter and normal impact velocity; Kd = K 5/8 [see Eq. (1.2) in Section 1.2 in Chapter 1 and Section 4.7 in Chapter 4].

187

5.5 Suppression of Splashing on Nanofiber Mats

140 120

140

Deposition, no fingers Advancing splash

120

Fingering Receding splash

100

Kd

Kd

100

80

80

60

60

(a) 3000

4000

5000 Re

6000

(b) 3000

4000

5000 Re

6000

Figure 5.41 Observed outcomes of drop impact onto PAN nanofiber mats at various parameters

near the threshold for these secondary morphologies (Kds = 87). (a) Results for the deposition and advancing splash regimes, (b) results for the fingering and receding splash (Lembach et al. 2010). Reprinted with permission. Copyright (2010) American Chemical Society.

The experimental results for all secondary morphologies are in good agreement over the complete parameter space (Fig. 5.41). The dimensionless parameter Kd allows a clear delineation of the domain with respect to drop deposition without fingers and advancing splash (see Fig. 5.41a). However, as is shown in Fig. 5.41b, in a certain parameter domain various impacts with seemingly the same parameters can produce different outcomes (fingering and receding splash). The drop splashing threshold for a flat smooth impermeable dry substrate corresponds to Kds = 57.7 (Mundo et al. 1995, Yarin 2006). In the case of drop impact onto a nanofiber mat the threshold value of Kds separating deposition without fingers and advancing splash is higher, approximately Kds = 87, which indicates that the nanofiber coating of the target surface suppresses advancing splash. The receding splashes, the regime in which secondary drops are formed in receding fingers, were mostly seen in conjunction with the advancing splash and seldom observed as the sole splash phenomenon on nanofiber mat surfaces. The rarity of the receding splash outcome can be explained by the properties of the nanofiber mat surface. Namely, a large contact angle hysteresis characteristic of nanofiber mat surfaces renders liquid on the surface practically immobile and requires a large amount of energy for drop separation from a finger. However, if a drop possesses the required critical energy, it will likely be separated during the advancing splash stage. Another interesting observation is that occasionally a drop can eject a secondary tiny droplet upon impact that is intercepted later on by a moving finger originating from the primary drop. In such cases, no secondary droplets are ultimately lost, which means that such events do not meet a standard definition of splash. The cumulative mass of the splashing tiny droplets in an advancing splash (with droplets leaving the surface) was about 1–2% of the primary drop mass, which was evaluated by measuring diameters and the number of such droplets. All the phenomena described in the present section so far happen on the scale of several milliseconds. In the case of such partially wettable (by water) nanofibers as the

188

Drop Impact onto Dry Surfaces with Complex Morphology

(a)

(b)

(c)

Figure 5.42 Water spreading inside a PAN nanofiber mat. (a) Images with wetted spot

configurations at different time moments (t = 0 s corresponds to the moment when water in the drop deposited onto nanofiber mat has come to rest). (b) Contours of the wetted spot inside the nanofiber mat at different time moments. The images are taken here for a time interval less than 6 s when the wettability-driven spreading in the mat is far from being over. The entire process lasts several minutes. (c) Close-to-circular contours of a wetted spot (Lembach et al. 2010). Reprinted with permission. Copyright (2010) American Chemical Society.

PAN nanofibers shown in Fig. 5.40, some additional phenomena are observed, albeit on a much longer time scale. The observations of these long-term developments are facilitated by the fact that the refraction indexes of the nanofibers and water are such that if a certain level of moisture concentration is reached in the nanofiber mat, it becomes transparent and the underlying darker copper surface becomes visible through the nanofiber mat. This allows observation of slow water spreading inside nanofiber mats. The captured images were used to characterize the growing dark area visible through the transparent nanofiber mat spots (Fig. 5.42). In particular, in several cases the wetted nanofiber mat areas are rather fractal-like looking. It is emphasized that Fig. 5.42b shows the worst case of the fractal-like spreading, and many images of the spreading process were quite circular (Fig. 5.42c). The darker areas have not been observed on the targets coated with very thick nanofiber mats. Water spreading inside such mats could not be detected with this method. Some areas stayed non-transparent throughout the entire spreading process, even though they were already surrounded by transparent fully wetted areas. In such cases the wet region inside nanofiber mats was more three-dimensional than twodimensional. Only in the case of two-dimensional propagation of moisture, could measurements be carried out with PAN nanofiber mats, with the 100–200 µm thickness being the best for observations. A visible front of contrast between the wetted and practically non-wetted part of nanofiber mat r f (t ) in the axially symmetric cases of wettability-driven spreading was √ found to follow the t-law, namely (Lembach et al. 2010) √ (5.5) r f = 1.492 amt + const with t being time and am being the moisture transport coefficient, a physical constant of the process (Luikov 1966). Its value was found as am = 8 × 10−4 cm2/s. √ It should be emphasized that the appearance of the t-law follows from the fact that the moisture transport equation is a parabolic equation which describes diffusion-like

5.6 Hydrodynamic Focusing on Nanofiber Mats

189

processes, which are in the present case the processes of wettability-driven imbibition of water into pores in the PAN nanofiber mats. These slow processes are exactly those processes which are responsible for the Lucas–Washburn velocity of the wettabilitydriven imbibition (Levich 1962, de Gennes 1985, de Gennes et al. 2004).

5.6

Hydrodynamic Focusing in Drop Impact onto Nanofiber Mats and Membranes Drop impact onto suspended nanofiber membranes (see Section 5.4 in this chapter), as well as drop impact and spreading on nanofiber mats deposited on an impermeable substrate (see Section 5.5 in the present chapter) revealed several fascinating and puzzling phenomena. These phenomena manifest the hydrodynamic focusing effect, as explained in the present section. It is a non-trivial fact, indeed, that drop impacts at about 3.5 m/s onto nano-textured electrospun membranes can result in water penetration even through hydrophobic Teflon nanofiber mats, i.e. the dynamic focusing effect can fully dominate the wettability effect. During hydrodynamic focusing the kinetic energy brought by a drop which impacts onto a wall with a small opening compared to the drop size is focused into the opening, which results in very high speeds of flow into it. This effect is related to the formation of shaped-charge jets (the Munroe jets or the von Neumann effect; see Sections 13.1 and 13.2 in Chapter 13). In the Munroe jets focusing results from the kinetic energy delivered by an explosion to a ductile metal (a liner on the explosive conical cavity which collapses), when the metal flows with a very high speed like an inviscid liquid (see Section 1.1 in Chapter 1) through a tiny opening. A similar hydrodynamic focusing in the form of jetting (an ejecta sheet) was predicted in the neck between an impacting drop and the target liquid surface (Weiss and Yarin 1999), which was confirmed in the experiments in the subsequent work (Thoroddsen 2002). Due to the hydrodynamic focusing, as is shown below, the initial velocity of liquid penetration into individual pores U ≈ (D/d )V0 can be several orders of magnitude higher than the drop impact velocity V0 , since the drop size D, which is of the order of 1 mm, is much larger than the pore size in the nanofiber mats d, which is of the order of several micrometers. During the hydrodynamic focusing, the initial penetration velocity U is also much larger than the Lucas–Washburn velocity corresponding to the wettabilitydriven imbibition (Levich 1962, de Gennes 1985, de Gennes et al. 2004), which makes the latter immaterial, while water drops are penetrating into Teflon electrospun nanofiber mats. Moreover, the hydrodynamic focusing resulting in dynamic penetration happens well below the static impregnation threshold, i.e. at ρV02 < 4σ /d (with ρ and σ being the liquid density and surface tension), already at ρV02 /(4σ /d ) ≈ 0.025 (Lembach et al. 2010), in distinction from the case when drops and orifices are of the same order of magnitude. The effect of surface roughness of the impermeable non-porous media on drop impact is clearly distinct from the phenomena observed in drop impacts onto permeable porous nano-textured surfaces (Yarin 2006, Weickgenannt et al. 2011b, Sivakumar et al. 2005).

190

Drop Impact onto Dry Surfaces with Complex Morphology

Consider in more detail the failure of the static impregnation threshold when it is applied to the experiments with drop impact onto nanofiber mats. Liquid brought softly to the nanofiber mat surface can impregnate pores if the nanofiber mat surface is partially wettable. The speed of wettability-driven impregnation of pores is given by the following expression related to the Lucas–Washburn formula Vi ∼ σ d cos θ /(8μH ) (Washburn 1921, Levich 1962), where H is the pore length and its permeability is taken as a2 /8 as in the Hagen–Poiseuille flow, with d being the pore diameter and a = d/2 its radius. The advancing contact angle is denoted as θ , which is less than π /2 in this case. On the other hand, non-wettable pores can be filled in a static situation (when liquid softly comes into contact with the pores) if the pressure difference between the liquid at the pore entrance and gas inside p is larger than the Laplace pressure, i.e. p > −2σ cos θ /d, with θ > π /2 in this case (Zheng et al. 2005). In the case of drop impact p = ρV02 /2. It is easy to see that with | cos θ | ∼ 1, non-wettable mats with pores d = 10−3 cm would not be filled with water at impact speeds of the order of 1– 2 m/s, since the static inequality does not hold. Indeed, the ratio of the right-hand side to the left-hand side in the inequality is of the order of 10 and the static condition predicts no filling of non-wettable nanofiber mats. However, it does not account for the dynamic nature of the impact filling process, which is responsible, for example, for the rapid filling of PAN nanofiber mats described in Section 5.5 in the present chapter, or for the filling of non-wettable polycaprolactone (PCL) nanofiber mats in Srikar et al. (2009). The dynamic nature of the pore filling process after drop impact onto porous media manifests itself as the hydrodynamic focusing according to the following scenario. In the experiments of Section 5.5 drop sizes are typically of the order of D ∼ 10−2 – 10−1 cm, whereas the pore sizes are of the order of 10−3 cm. Therefore, drop impact onto a single pore can be imagined as an abrupt impact of a solid wall with an orifice in the middle onto an upper half-space filled with water. To be able to apply the calculation technique rooted in the Cauchy formula of complex analysis (Henrici 1974, Polya and Latta 1974), consider a kindred two-dimensional problem. Namely, a plane at y = 0 with a slit in the middle at −a < x < a (where a = d/2 is analogous to the pore radius) imposes a pressure impulse on the liquid filling the upper half-plane y > 0. This problem belongs to the class of impulsive fluid motions described in Section 1.5 of

τ Chapter 1. Accordingly, the pressure impulse = lim τ →0 0 p dt (where pressure p→∞

p → ∞ and the impact duration τ → 0) is of the order of one. The pressure impulse is applied at −∞ < x ≤ −a, y = 0 and a ≤ x < ∞, y = 0 to the liquid filling the upper half-plane. Flows arising in response to the pressure impulse are known to be potential, with flow potential φ = − /ρ being a harmonic function (see Section 1.5 of Chapter 1). This means that φ satisfies the Laplace equation (for any fluid irrespective of its viscosity, and rheological constitutive equation, in general) ∂ 2φ ∂ 2φ + =0 ∂x2 ∂y2

(5.6)

with the Cartesian coordinate x being associated with the impact plane, and y being normal to it, with the coordinate origin at the pore center.

5.6 Hydrodynamic Focusing on Nanofiber Mats

191

The value of the pressure impulse corresponding to drop impact can be evaluated as follows. Assume first that liquid in the drop comes to rest after an impact on a solid obstacle as fast as the pressure waves in it related to its compressibility deliver the information from the preceding layers (as it happens at the very first moments after an impact). Then, the convective part of the normal force acting on the obstacle is of the order of ρV0 cD2 , where c is the speed of sound in the liquid (see Section 1.1 in Chapter 1). In addition, the “water hammer” part of normal force acting on the obstacle related to the deceleration is of the order of −ρD3 A, where the liquid acceleration is of the order of A = −V0 c/D. Therefore, the “water hammer” part of the normal force acting on the obstacle is also of the order of ρV0 cD2 . Hence, the pressure from the liquid experienced by the obstacle, as well as the pressure which the drop bottom experiences from the obstacle, are of the order of p ∼ ρV0 c (see Section 1.1 in Chapter 1). If one assumes that liquid comes to rest after a certain spreading, then the short-living compressibility effects can be neglected, the normal load at the wall due to the dynamic pressure is of the order of ρV02 D2 and the “water hammer” load is of the same order, since in this case A = −V0 /(D/V0 ) = −V02 /D. Thus, in the latter case the pressure which the drop bottom experiences from the obstacle is of the order of p ∼ ρV02 , much less than in the compressible case. However, due to the fact that in the compressible case τ ∼ D/c, whereas for the latter incompressible stage τ ∼ D/V0 , the value of the pressure impulse ∼ ρV0 D is the same irrespective of the deceleration mechanism. Therefore, the potential value in liquid in contact with the wall is φ(X ) = φ0 = −V0 D over −∞ ≤ X ≤ −a, y = 0 and, a ≤ X ≤ ∞, y = 0 φ(X ) = 0 over

−a < X < a, y = 0

(5.7) (5.8)

with x = X , with the notation X being used as a dummy variable in Eq. (5.9) The distribution of the potential (a harmonic function) in the liquid filling the upper half-plane is found using the Cauchy formula, which in the present case inevitably reduces to Poisson’s integral formula for the upper half-plane (Polya and Latta 1974)  1 ∞ φ(X , 0)y dX . (5.9) φ(x, y) = π −∞ (x − X )2 + y2 The integral in Eq. (5.9) is evaluated using Eqs. (5.7) and (5.8) and the resulting flow potential needed to calculate the flow through the opening is given by   2ay φ0 . (5.10) φ(x, y) = − arctan 2 π x + y2 − a2 The corresponding velocity vector is v = ∇φ. Therefore, the y-component of v over the opening is found as  ∂φ  V0 D 2a vopening (x) = = . (5.11) ∂y y=0 π x2 − a2 Note that vopening < 0. Therefore, as expected, after drop impact, liquid begins to flow into the opening, i.e. in the negative y-direction. At the opening edges, at x = ±a, vopening = −∞, since there the pressure impulse is discontinuous. This is typical for

192

Drop Impact onto Dry Surfaces with Complex Morphology

y

−D 2

−a

Liquid drop at the “moment of impact”

a

D 2

x

Pore Figure 5.43 Sketch of drop impact onto a single pore (Sahu et al. 2015). Reprinted with

permission from Elsevier.

problems of penetration of solid plates into an incompressible liquid (Batchelor 2002). In reality, the velocity at the orifice edges will be diminished by viscosity due to the no-slip condition, which is excluded from the consideration in the present analysis. However, it can still be expected that liquid penetration into the orifice will take the shape of an upside down corona. The velocity minimum is expected at the opening center, where |(vopening )min | = U , with U given according to Eq. (5.11) by U =

4D V0 . π d

(5.12)

The theory described above and the resulting expression (5.12) for the initial velocity of penetration U dealt with impact of a solid wall with a single gap onto an upper half-space filled with water. The next approximation of interest is related to drop impact onto a single pore considered as an abrupt impact of a solid wall with a gap (orifice) in the middle onto a semi-circular liquid domain (Fig. 5.43, where the planar case is still assumed for simplicity), as in Sahu et al. (2015). The assumption of a semi-circular liquid domain at the beginning of water penetration is an approximation, albeit a realistic one, according to the experimental observations of Brunet et al. (2009), and the numerical modeling of Reis et al. (2008), since sheet jetting below the impacting drop is extremely fast (on the scale of 10 µs). Moreover, it should be emphasized that as Eq. (5.28) obtained below in the present section shows, a detailed shape of liquid over a pore (semi-circular or a half-plane) makes practically no difference in the initial rate of water penetration when the drop size D is much larger than the pore size d, as in the present case. Note also, that the expected dynamic effects in the present case are so dominant according to Lembach et al. (2010), Sahu et al. (2012), Sinha-Ray et al. (2011), Weickgenannt et al. (2011a) and Weickgenannt et al. (2011b) (see Sections 5.4–5.9 in the present chapter), that surface tension at the liquid front penetrating into a pore would be negligible, and the situation under consideration is radically different from the one in Lorenceau and Quéré (2003) where the dynamic effects were relatively weak, while surface tension determined the penetration threshold.

193

5.6 Hydrodynamic Focusing on Nanofiber Mats

(a)

z ω=

B φ=0

i

–a –1 A φ = –1 E

0 a 1 C F φ = –1 D φ=0

φ=0

ω

η

(b) y

1+z 2

( ( 1–z

x

D

–1 B φ=0

0 A

ωE 1 ωF

∞ E C F φ = –1 D ξ φ = –1 φ = 0

Figure 5.44 Conformal mapping onto the upper half-plane (Sahu et al. 2015). Reprinted with

permission from Elsevier.

In the case depicted in Fig. 5.43, the Laplace equation (5.6) is solved not with the boundary conditions (5.7) and (5.8), but rather with the ones corresponding to the pressure impulse applied to the drop at −D/2 < x ≤ −a, y = 0 and a ≤ x < D/2, y = 0. In the Laplace equation (5.6) and hereinafter in this case the coordinates x and y and the pore half-width are rendered dimensionless by the drop radius D/2 (with the dimensionless a = 2a/D), and the potential φ – by V0 D. Below the bars over the dimensionless parameters are omitted for brevity. Therefore, the dimensionless boundary conditions imposed on the solution of Eq. (5.6) in the present case read φ(x, y = 0) = φ0 = −1 over −1 ≤ x ≤ −a and over a ≤ x ≤ 1

(5.13)

φ(x, y = 0) = 0 over −a < x < a

(5.14)

φ(x, y) = 0 at x2 + y2 = 1, y > 0

(5.15)

In particular, the boundary condition (5.14) implies that at the opening there is no impulse transmitted to the impacted liquid, i.e. the overpressure there is equal to zero. It is easy to see that the interior of the semi-circle corresponding to the liquid domain in the complex plane z = x + iy in Fig. 5.44a is conformally mapped onto the upper half-plane ω = ξ + iη (with i being the imaginary unit) in Fig. 5.44b by the following mapping function [which is seen from the boundary correspondence; Mathews and Howell (2006)]   1+z 2 . (5.16) ω= 1−z Accordingly, the boundary conditions (5.13)–(5.15) take the following form in the ω-plane φ = 0 at −∞ ≤ ξ < 0

(5.17)

φ = −1 at 0 ≤ ξ < ξE

(5.18)

φ = 0 at ξE ≤ ξ < ξF

(5.19)

φ = −1 at ξF ≤ ξ < ∞

(5.20)

194

Drop Impact onto Dry Surfaces with Complex Morphology

where the mapped ξ -coordinates of points E and F are given by     1−a 2 1+a 2 ξE = , ξF = . 1+a 1−a

(5.21)

The field of the harmonic potential φ(ξ , η) in the upper half-plane with the boundary conditions imposed at the real axis (5.17)–(5.20) is given by Poisson’s integral formula for the upper half-plane of the type of Eq. (5.9), which is now written in the following form  φ(κ, 0) η ∞ dκ (5.22) φ(ξ , η) = π −∞ (κ − ξ )2 + η2 where κ is a dummy (real) variable. The integral in Eq. (5.22) is evaluated using Eqs. (5.17)–(5.20), and the resulting flow potential needed to calculate the initial flow through the pore is found as        ξE − ξ ξ ξF − ξ 1 π φ(ξ , η) = − arctan + arctan + − arctan . (5.23) π η η 2 η The expression for the potential φ(ξ , η) (5.23) is recast into φ(x, y) using the fact that the conforming mapping (5.16) is identical to the following coordinate transformation ξ=

(1 − x2 − y2 )2 − 4y2 , [(1 − x)2 + y2 ]2

η=

4y(1 − x2 − y2 ) . [(1 − x)2 + y2 ]2

(5.24)

The corresponding velocity vector is v = ∇φ. Therefore, the dimensionless ycomponent of v over the pore −a < x < a is found as  ∂φ  vopening (x) = ∂y y=0   1 1 1 4 (1 − x2 ) − − (5.25) =− π (1 − x)4 (ξF − Q) (ξE − Q) Q (1 − x2 )2 Q= . (5.26) (1 − x)4 The latter shows that at the center of the pore entrance the dimensionless y-component of the velocity is vopening |x=0;y=0 = −

2(1 − a)2 . πa

(5.27)

The corresponding magnitude of the dimensional velocity component toward the pore at its center |vopening |x=0;y=0 | = |vopening |min | = U is   d 2 4D V0 1 − U = . π d D

(5.28)

It should be emphasized that the magnitude of this penetration velocity U is much larger than the drop impact velocity V0 , since the ratio D/d 1 (d = 2a).

195

5.6 Hydrodynamic Focusing on Nanofiber Mats

5 4 3 2

V

1 0 −1 −2 −3 −4 −5 −1

−0.8 −0.6 −0.4 −0.2

0 X

0.2

0.4

0.4

0.8

1

Figure 5.45 Velocity distribution at the drop bottom for a pore with a = 0.5. The central part

corresponds to the pore, the left and right parts to the wall (Sahu et al. 2015). Reprinted with permission from Elsevier.

Equation (5.28) is also in agreement with Eq. (5.12) where the impacting drop was considered as a complete upper half-plane rather than the semi-circle of Fig. 5.43a. In particular, Eq. (5.28) recovers the result of Eq. (5.12) in the limit D/d → ∞. Also, in both cases the y-component of velocity diverges at the sharp edges of the pore, i.e. according to Eqs. (5.25) and (5.26) v|x=±a = −∞.

(5.29)

On the other hand, at the contact line of the drop it acquires a high positive vertical velocity, since according to Eq. (5.25) v|x=±1 = ∞.

(5.30)

When a high pressure is applied to a diminishing mass of liquid near the contact line, there should be a propensity for a prompt splash in the form of liquid rising from the wall near the contact line (Yarin 2006), which is clearly expressed by the infinite vertical velocity there as per Eq. (5.30) (in reality restricted by viscosity). This phenomenon is similar to a finite impulse propagating over a whip near its edge: without energy losses the whip edge velocity would be infinite (since the mass of the affected edge tends to zero); in reality there are energy losses, but the velocity is still very high and expresses itself as a whistle of the whip in air. The entire velocity distribution predicted by Eq. (5.25) at the drop bottom is depicted in Fig. 5.45. It shows that over almost the entire pore, the velocity is practically uniform

196

Drop Impact onto Dry Surfaces with Complex Morphology

Figure 5.46 Potential distribution at the moment of impact. (a) a = 0.01, (b) a = 0.1, (c) a = 0.5

(remember that here a stands for the dimensionless a) (Sahu et al. 2015). Reprinted with permission from Elsevier.

and close to the central value given by Eqs. (5.27) and (5.28). The infinite velocity values at the pore edges, inevitable in the inviscid formulation, will be smoothened in reality by viscous shear stresses. An additional illustration of the potential distribution (5.23) in the drop after the impact onto a wall with a pore at the center is shown in Figs. 5.46a–c. Since the pressure values are related to –φ, the regions with high pressure correspond to the darker domains near the wall surrounding the pore in these figures. The sharp pressure loss at the orifice at the moment of impact revealed by the results in Fig. 5.46 should produce a narrow liquid jet entering the pore at high speed. Such jets, indeed, are observed in the experiments with drop impacts onto suspended grids (see Figs. 5.26a, 5.27a and 5.28d–h), where Fluorinert fluid FC 7500 with very low surface tension is used to prevent jet merging at the rear side of the membrane. Note also, that according to Darcy’s law, air in the pores is easily displaced by penetrating liquids, since its viscosity is about one hundredth of that of a liquid. Velocity of penetration of liquid into pores of nanofiber mats can be studied not only as a plane problem as in the two cases considered above, but also as a more realistic axisymmetric problem. Then, a drop impact onto a single pore can be considered as an abrupt impact of a solid wall with a cylindrical orifice in the middle onto an upper halfspace filled with liquid. A plane at z = 0 with the cylindrical orifice (pore) in the middle at 0 < r < a (where r and z are the radial and axial cylindrical coordinates centered at the orifice center) applies a pressure impulse at r ≥ a, z = 0 to the liquid filling the upper half-plane (z > 0), where a is the pore radius. The axisymmetric Laplace equation for the potential of the impact flow reads   ∂φ ∂ 2φ 1 ∂ (5.31) r + 2 = 0. r ∂r ∂r ∂z Its solution for liquid is to be found for 0 ≤ r ≤ ∞, and z ≥ 0, and the potential distribution in the liquid in contact with the wall is posed in the form  0, 0≤r 6.77 m/s. According to Eq. (5.41), in the present case U = 1.3 × 103 m/s through a single pore (roughly speaking, about 8 m/s in the case of 160 pores). The density and surface tension of FC 7500 are ρ = 1.614 g/cm3 and σ = 16.2 g/s2 , respectively. Therefore, in the present case the ratio (ρV02 )/(4σ /d ) = 0.66, and penetration of liquid in the pores happened below the static penetration threshold due to the hydrodynamic focusing. In the dynamic penetration process the hydrodynamic focusing plays the dominant role, while wettability and surface tension are secondary. Note that water penetration into porous medium after drop impact under a significant overpressure is obviously possible if ρV02 4σ /d; however, due to the hydrodynamic focusing (when drop diameter D is much larger than the pore diameter d) water penetration is also possible when ρV02 < 4σ /d, which is much less obvious.

200

Drop Impact onto Dry Surfaces with Complex Morphology

5.7

Impact of Aqueous Suspension Drops onto Non-Wettable Porous Membranes: Hydrodynamic Focusing and Penetration of Nanoparticles Coalescence filters are an example of porous media collecting drops from an oncoming gas or liquid flow (Contal et al. 2004, Frising et al. 2005, Filatov et al. 2007). Drops penetrate the filter membrane and accumulate inside. As a result, filter permeability decreases, whereas the pressure drop in gas or liquid, which is required to sustain the flow, increases. Some groups expressed expectations that a hydrophobic filter medium will prevent water drops from penetrating inside, thus facilitating water collection and removal at the front surface (see Section 5.4 in this chapter). The hydrodynamic focusing discussed in the preceding Section 5.6, however, is expected to overcome hydrophobicity and let the impacting water drops penetrate into hydrophobic media. In the present section this is directly demonstrated using the entrainment of seeding nanoparticles. In a broader context, the entrainment of seeding particles is not merely an observation tool but is of interest by itself in such applications as ink-jet printing on smart textiles, Park et al. (2012). The aim of the present section is in the experimental and theoretical investigation of dynamic liquid penetration due to the hydrodynamic focusing, and the entrainment and deposition of nanoparticles suspended in drops impinging onto a porous filter membrane. Glass fiber filter (1 mm thick and 47 mm in diameter) with 2.7 µm pores and polytetrafluoroethylene (PTFE) depth filter (1 mm thick, 47 mm in diameter) with 10 µm pores were used in the experiments of Sahu et al. (2015) discussed in the present section. Glass fiber membrane is wettable with water, whereas PTFE is non-wettable. The surfaces of the membranes observed using an optical microscope Olympus BX-51 are shown in Fig. 5.48. They are non-woven fibrous materials. The glass fiber filter reveals distinct pores and fibers (Figs. 5.48a–c), whereas the PTFE filter reveals some film-like structures with open pores attached to fibers (Figs. 5.48d–f). The pore size is determined by the particle size that will be retained with 100% efficiency under specified conditions. The pore size of the membranes used in the present section was 2.7 µm for the glass fiber filter (wettable) and ≤10 µm for the PTFE membrane (non-wettable). The pore size at the surfaces of both membranes seems to be larger because the lower lying fibers which block pores are out of focus and thus are practically invisible. The surface roughness is approximately equal to the fiber diameter in the membrane. The roughness of the glass fiber filter Ra = 2–4 µm, and that of the PTFE membrane is Ra = 5–8 µm. The contact angle of water on the PTFE membrane was measured as 160◦ (in the superhydrophobic range), as is seen from the image of a softly deposited water drop (Fig. 5.49). On the other hand, it was impossible to measure the contact angle of water drop on the glass fiber filter as it spreads immediately as it comes in contact with the wettable filter. Black titanium nanoparticles (60–80 nm) were suspended in water. A nanoparticle suspension was prepared by adding 0.1 g of nanoparticles to 20 g of water (0.5 wt %). The surface tension and viscosity of such aqueous suspensions are practically indistinguishable from those of water. The suspension was then sonicated using probe sonicator

5.7 Impact of Aqueous Suspension Drops

201

Figure 5.48 Optical microscope images of glass fiber filter (a)–(c) and PTFE membrane

(d)–(f ) showing the fibers and the pore sizes (Sahu et al. 2015). Reprinted with permission from Elsevier.

(QSonica Q500) for 1 min to make it homogeneous. Sonication was applied to prevent nanoparticle cluster formation and sedimentation. Fresh suspension drops were used in drop impact experiments right after the sonication. Drops of nanoparticle suspension were dripped onto porous membranes using the experimental setup sketched in Fig. 5.16. Suspension drops were released from 18G and 14G needles. Their impact velocity V0 was measured at the impact moment using high-speed imaging. The drops (17 µl, formed from a 18G needle with the inner and outer diameters of 0.838 mm and 1.27 mm, respectively; and 26 µl, formed from a 14G needle with the inner and outer diameters of 1.6 mm and 2.108 mm, respectively) were released onto glass fiber membranes from the heights of 0 cm (soft deposition) to 70 cm

Figure 5.49 Contact angle of water drop on the PTFE membrane (Sahu et al. 2015). Reprinted

with permission from Elsevier.

202

Drop Impact onto Dry Surfaces with Complex Morphology

tage

ar S

Line

y x

Sta Su ge pp or t

A Blade ple am A S tage ar S Line

BasPelate

Figure 5.50 Schematic of the experimental setup for cutting sample membranes to detect

nanoparticles embedded inside (if any) (Sahu et al. 2015). Reprinted with permission from Elsevier.

(the gravity-driven impact). The measured impact velocities are listed in Table 5.1. In addition, similar drops from a 14G needle were released from the impact heights of 180 cm and higher onto glass fiber and PTFE membranes, with the measured impact velocity of 5.32 m/s and higher. After an impact of suspension drop, the porous membranes were left to dry for a few hours and then were mounted on the setup shown in Fig. 5.50. This setup included Table 5.1 Spread factor measured for drop impact onto the glass fiber membrane at different impact velocities. The experimental results listed in the table are the averages of two duplicate experiments. Here ζE is the experimental spread factor (on porous medium), while ζT is the spread factor predicted by Eqs. (5.42) and (5.43) corresponding to an impermeable surface. Needle gauge

Impact velocity, V0 (m/s)

Weber number, We

ζE

ζT

δ

Drop volume (µl)

14G

0 1.45 2.64 3.5

0 107.3 355.5 624.9

2.21 2.37 2.78 2.91

– 3.53 4.31 4.73

1.49 1.55 1.63

26

18G

0 1.47 2.95 3.92

0 95.4 384.0 678.1

1.87 2.25 2.61 2.70

– 3.42 4.31 4.74

1.52 1.65 1.76

17

5.7 Impact of Aqueous Suspension Drops

203

two high-performance low-profile ball-bearing linear stages placed orthogonally to each other. A sharp razor blade was attached to the stage translating in the vertical direction (the Y -axis). The sample was placed horizontally (the horizontal plane includes the X -axis). The cross-section A–A corresponded to the viewing direction of the DinoLite optical microscope with respect to the sample. Namely, after a nanoparticle-seeded drop dripped onto a sample of a porous membrane and the latter was completely dried, the nanoparticles would be embedded inside if water penetrates the pores. Then, the porous membrane was sliced by the razor blade in several locations along the X -axis (Fig. 5.50), and the camera was used to observe the nanoparticle distribution in the cut cross-sections of the sample. The images of nanoparticles in the cut cross-sections of the membranes were taken for the slices produced after every translation of 0.5 mm along the X -axis. After that, the images were processed using ImageJ, Matlab and Adobe Photoshop CS2. The top view of the dried glass fiber membrane and the side view of its cut crosssection at the location where the penetration depth was maximal (Fig. 5.51) reveal the nanoparticle deposition domain. After a drop impacts onto the front surface of the membrane, it spreads over it, as well as some part of the suspension penetrates into the pores. Nanoparticles are entrained by the solvent flow until they are intercepted by the glass fibers in the membrane. The entire domains with the deposited nanoparticles and their boundaries are clearly visible at the front surface of the membrane and in its depth. It should be emphasized that the characteristic time of nanoparticle entrainment by flow evaluated as τ = 2ρ part a2part /9μ (with ρ part being the particle density, a part its radius and μ being liquid viscosity) is of the order of 10−9 s. Compared to the penetration time of the order of 10–100 µs, it shows that these nanoparticles were fully entrained by liquid flow inside the membrane until they ultimately touched the pore wall and were intercepted there. Note also, that ultimately water will evaporate from a wetted membrane. However, water filtration inside the membrane resulting from evaporation at the free surface cannot re-entrain nanoparticles adhering to the pore walls. Indeed, the adhesion force acting on a nanoparticle is of the order of Fa = Aa part /(6ε2 ), where A is the Hamaker constant, and ε is the least possible spacing between the nanoparticle and the pore surfaces (Derjaguin et al. 1975). The evaporation-driven flow of velocity ve would impose the drag force acting on the nanoparticle of the order of Fd = 6π μa part ve . To detach the nanoparticle from the pore wall, it should be Fd ≥ Fa , which yields ve ≥ A/(36π με2 ). Taking for the estimate A = 10−12 erg, ε = 3 × 10−8 cm, and μ = 1 cP, we find ve ≥ 103 cm/s, which is much higher than any evaporation-driven filtration velocity. Therefore, once a nanoparticle adhered to the pore, it could hardly be removed by any further water flow still occurring in the membrane. A partial bouncing and splashing after a drop impact is characteristic of non-wettable membranes. Figure 5.52 compares the outcome of the suspension drop impact onto a non-wettable PTFE membrane with that of the impact onto a wettable glass fiber membrane. The splashing patterns seen from the top are drastically different in these cases, with splashing on the non-wettable PTFE membrane being much more pronounced. However, it has been shown in Sections 5.4–5.6 in the present chapter, that above a certain threshold impact velocity, the static wettability of the surface does not play any

204

Drop Impact onto Dry Surfaces with Complex Morphology

Figure 5.51 Top views of the glass fiber membrane after drop impact at 1.4 m/s [panel (a)] and

3.49 m/s [panel (b)]: the final domains of the nanoparticle deposition at the membrane surface. The impact velocities were measured at the impact moment. Drops were formed using a 14G needle. Scale bars: 1 cm. (c) The side view of the cut cross-section of the membrane corresponding to the maximum penetration depth of a drop which impacted at 1.4 m/s: the nanoparticle distribution in depth. The lower boundary of the deposition domain is traced by a line. Scale bar: 1 mm. In all the images the membrane domains with no nanoparticles deposited look white, while those with the deposited nanoparticles look black. In panel (c) the dark strip under the membrane is the supporting aluminum plate (Sahu et al. 2015). Reprinted with permission from Elsevier.

significant role in penetration of water drops into porous media due to the hydrodynamic focusing. In the present case, there was no visible penetration of water into the non-wettable PTFE membrane in the 0–100 cm range of the impact heights, in distinction from the glass fiber membrane. However, when the impact height was 180 cm, a central fraction of the primary drop indicated in Fig. 5.52a by the arrow at the frame corresponding to t = 8 ms penetrated through the non-wettable PTFE membrane. Drop penetration visualized by the nanoparticle entrainment into non-wettable PTFE membranes is the main aim of the present section. Five different impact velocities of 5.32 m/s, 5.64 m/s, 6.05 m/s, 6.47 m/s and 6.8 m/s where drop penetration happens were employed, and at each of them six trials were done. After the experiments the samples were cut in two different directions: (i) beginning from the drop impact surface, and (ii) in the opposite direction (see Figs. 5.53a and 5.53b, respectively). It was observed that the cutting direction affects the depth of the area of nanoparticle deposition inside

5.7 Impact of Aqueous Suspension Drops

205

(a)

(b)

t = 0 ms

t = 0 ms

t = 2 ms

t = 1 ms

t = 5 ms

t = 2 ms

t = 8 ms

t = 5 ms

Figure 5.52 Sequence of images of drops of aqueous suspensions of nanoparticles impacting

onto: (a) PTFE membrane, and (b) glass fiber membrane. The drops were issued from a 14G needle at the height of 180 cm, and had the impact velocity of 5.32 m/s (measured at the impact moment). Scale bar is 1 cm (Sahu et al. 2015). Reprinted with permission from Elsevier.

the membrane, which means that the membrane material with the attached nanoparticles can be shifted to some extent by the razor blade. Overall, the experiments demonstrated that the seeding nanoparticles suspended in liquid drops can penetrate to a significant depth into non-wettable PTFE membranes being entrained by water flux. To minimize the effect of the cutting direction on the measured ultimate penetration depth, the average profiles found in the cuts beginning from the drop impact surface, and in the opposite direction are used further on for comparison with the theoretical predictions (see Figs. 5.54 and 5.55). It should be emphasized that the error bars in

Figure 5.53 The side view of the cut cross-section of PTFE depth membranes at the locations

corresponding to the maximal penetration depth. Panel (a) corresponds to the case when the membrane was cut beginning from the surface where the drop impact took place, whereas panel (b) shows the cut done in the opposite direction. The drops were issued from a 14G needle from the height of 180 cm, and had the impact velocity of 5.32 m/s (measured at the impact moment). The boundaries of the nanoparticle domains are traced by lines. Scale bars are 1 mm (Sahu et al. 2015). Reprinted with permission from Elsevier.

Drop Impact onto Dry Surfaces with Complex Morphology

(b)

0.6 0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0

0.0

−0.1

−0.1 −1.5

−1.0

−0.5

0.0

X

0.5

1.0

1.5

average t = 0.6

0.6

Y

(a)

Y

206

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

X

Figure 5.54 (a) The experimentally measured ultimate penetration front into PTFE depth

membrane is shown by symbols with the error bars corresponding to variance within different samples. (b) The predicted penetration front corresponding to the dimensionless time t = 0.6. The drops were issued from a 14G needle. The impact height was 180 cm and the corresponding impact velocity was 5.32 m/s. The coordinates are rendered dimensionless using the membrane thickness h. As discussed in the theoretical part of this section, time is rendered dimensionless by h/U (U is the initial water penetration velocity); b = 0.8 (this value of b defined below corresponds to a 26 µl drop impacting onto a membrane of thickness h = 0.1 cm, with liquid penetrating with a = 0.435, where a = 2a/D is the pore size rendered dimensionless by the drop size) (Sahu et al. 2015). Reprinted with permission from Elsevier.

Fig. 5.54a span the utmost boundaries of the particle domain resulting from cutting in the opposite directions, and the average profile shown by the solid line corresponds to the domain boundary inside the membrane before cutting with high confidence. The three-dimensional reconstruction of the domain where nanoparticles were deposited inside the non-wettable membrane by penetrating water is shown in Fig. 5.56.

Figure 5.55 The ultimate point of nanoparticle deposition corresponding to the results of Fig. 5.54

is shown by the symbol (PTFE depth membrane). The predicted locations of the tip of the propagating front of water are shown by the line, which is extended beyond the visible ultimate deposition of nanoparticles. Time is rendered dimensionless by h/U (Sahu et al. 2015). Reprinted with permission from Elsevier.

207

5.7 Impact of Aqueous Suspension Drops

1 2 3 4 5 6 7 0

Y

−0.2 −0.4 1 2 3 4 5 6 7

−0.6 −0.8 2 1 0 X

−1 −2

0

1

2

3

4

5

6

7

Z

Figure 5.56 Three-dimensional reconstruction of the penetration domain in the PTFE membrane

seeded by nanoparticles using data acquired from different sections of the drop impact surface. The drops were issued from a 14G needle, and cuts were done from the drop impact surface. The impact height was 180 cm and the corresponding impact velocity was 5.32 m/s. The inset shows a sketch of the cuts used in the reconstruction. The degree of grayness signifies the depth at various points and corresponds to the y-axis (Sahu et al. 2015). Reprinted with permission from Elsevier.

The data was acquired using seven different cuts across the drop stain at the surface, as shown in the inset in this figure. It should be emphasized that in the present case the dynamic pressure in liquid at the moment of impact ρV02 (the ambient pressure is obviously unimportant) is less than the capillary pressure 4σ /d. Indeed, in the present case ρV02 = 2.83 × 105 g/(cm s2 ), and 4σ /d ≥ 2.88 × 105 g/(cm s2 ), whereas the static penetration condition requires ρV02 to be sufficiently larger than 4σ /d (Bazilevsky et al. 2008). Therefore, water penetration into the PTFE membrane illustrated in Figs. 5.52–5.56 is clearly due to the hydrodynamic focusing, demonstrated in Sections 5.4–5.6 in this chapter, where the velocities and pore sizes were even smaller, and ρV02 was much less than 4σ /d. This fact is also corroborated by the remarkable experimental results in Brunet et al. (2009) where the threshold values corresponding to water penetration after drop impact onto a grid revealed that penetration is possible when ρV02 /(4σ /d ) ≈ 0.05. The part of the primary drop which does not penetrate into the membrane forms secondary droplets which roll off the upper surface of the PTFE membrane. In the case of the glass fiber membrane, splashing at the upper surface is diminished, as well as a wider puddle of water penetrates through the membrane, as is seen in Fig. 5.51b. The latter, however, cannot be completely attributed to the dynamic penetration (the hydrodynamic focusing), since the slower wettability-driven flow of nanoparticle suspension inside the membrane is sustained for a long time after the impact (see Fig. 5.42 in Section 5.5 in the present chapter). The dynamic spreading of a drop over porous media is different from that over the impermeable substrates. The impact velocity, surface tension, solvent viscosity, surface roughness, pore size, wettability and permeability of the porous medium affect the flow resulting from drop impact. Drop spreading over the impermeable surfaces is usually

208

Drop Impact onto Dry Surfaces with Complex Morphology

characterized by the spread factor ζ , with the following empirical relation widely used [Scheller and Bousfield (1995); see Eq. (4.84) in Section 4.4 in Chapter 4] 

ζ = 0.61(Re Oh) 2

0.166

We = 0.61 Oh

0.166 = 0.61(ReWe1/2 )0.166

(5.42)

where ζ = 2Rmax /D, with Rmax being the ultimate spreading radius, D the drop volumeequivalent diameter, and the Reynolds, Ohnesorge and Weber numbers given by (see Section 1.2 in Chapter 1) Re =

ρDV0 , μ

Oh =

μ , (ρσ D)1/2

We =

ρDV02 . σ

(5.43)

In Eq. (5.43) the impact velocity is denoted as V0 , and the liquid density, viscosity and surface tension are ρ, μ and σ , respectively. It should be emphasized that the empirical Eq. (5.42) is fully confirmed by the numerical and theoretical predictions of the more recent work Eggers et al. (2010). Indeed, the maximal drop spreading which determines the spread factor ζ available according to the numerical and modeling results in Fig. 9 in Eggers et al. (2010) reveals the value of ζ ≈ 3.2 for We = Re = 800. This is exactly the value which is found from Eq. (5.42) with We = Re = 800. Note, however, that even though Eq. (5.42) agrees with the predictions in Eggers et al. (2010) at the intermediate values of Re and We, its asymptotic behavior differs from the scaling laws considered in Eggers et al. (2010). Similarly, the recent experiments in Fig. 3 in Lagubeau et al. (2012) revealed for We = 214 and Re = 2690 the value of ζ ≈ 3.5–3.6, whereas Eq. (5.42) yields ζ ≈ 3.53. The values of the spread factor ζ in the present experiments with drop impacts onto the porous glass fiber membrane where splashing was minimal could be easily established. They are contrasted in Table 5.1 with those predicted by Eqs. (5.42) and (5.43) for an impermeable medium. It is clearly seen that Eqs. (5.42) and (5.43) significantly overestimate the measured values of the spread factor in the case of porous membranes. In the present experiments the spread factor is diminished due to liquid penetration into the pores; see Starov et al. (2003). It should be emphasized that the experimental values of the spread factor at V0 = 0 m/s correspond to the purely wettability-driven spreading, for which no comparable value can be obtained using Eqs. (5.42) and (5.43). Table 5.1 shows that for 26 µl drops the ratio δ of the calculated spread factor for the impermeable surface [Eqs. (5.42) and (5.43)] to the one measured experimentally increases from 1.49 to 1.63 as the impact velocity increases from 1.45 m/s to 3.5 m/s, and for 17 µl drops from 1.52 to 1.76 in the 1.47 m/s to 3.92 m/s velocity range. This clearly is associated with liquid penetration into the membrane. The data in Table 5.1 show that the spread factor measured at the same impact velocity is higher for the larger drops compared to the smaller ones. Note also, that the spread factor on the porous medium in the range 2.21 < ζ < 2.7 measured in the experiments of the present section (Table 5.1) is close to the prediction ζ ≈ 2.6 in Fig. 4 in the numerical simulations in Reis et al. (2008).

5.7 Impact of Aqueous Suspension Drops

209

y z 1

−b

0

x

b

Figure 5.57 Sketch of the membrane which is impacted by a drop from below. The bottom of the

spread-out drop corresponds to −b ≤ x ≤ b, y = 0. The coordinates x and y, as well as b, are rendered dimensionless by the membrane thickness h. This means, in particular, that the dimensionless effective penetration width of liquid after drop impact 2a is now replaced by 2b = a(D/h) (Sahu et al. 2015). Reprinted with permission from Elsevier.

For the non-wettable PTFE membrane, splashing is so significant (Fig. 5.52a) that reliable measurements of the spread factor ζ are impossible. After drop impact with velocity V0 , liquid penetrates into the inter-fiber pores in the membrane with the initial velocity U determined by the hydrodynamic focusing (Section 5.6 of the present chapter). Then, liquid from the impacting drop is spreading in the non-wettable membrane according to Darcy’s law, and its velocity v possesses a potential φ related to pressure p as (Loitsyanskii 1966, Barenblatt et al. 1989) v = ∇φ,

φ=−

k p μ

(5.44)

where k is the permeability. The effective average filtration velocity v through tortuous pores in the porous medium involved in Darcy’s law is defined as the volumetric flow rate divided by the total (pores and solid fibers) cross-sectional area. It would be much less than the initial flow velocity along a single straight pore U , approximately of the order of v = 1 m/s (see Section 5.6 in this chapter). Then, the filtration Reynolds number Refiltration = ρvd/μ ≈ 10. Note, that on the depth scale of about 10–100 µm it will rapidly diminish to the level of about Refiltration = 1 due to the dissipation of the kinetic energy inside the pores [Sahu et al. (2012), see below in the present section]. Darcy’s law is known to be applicable in the Refiltration ≤ 10 range (Charbeneau 2006). Consider a membrane sketched in Fig. 5.57. In the present case it is convenient not to use the complex potential χ (z) = φ(x, y) + iψ (x, y), with ψ being the stream function related to the hydrodynamic potential φ by the Cauchy–Riemann conditions, but rather to employ another function of a complex variable z = x + iy associated with the conjugate velocity V = dχ / dz, namely iV (z) = v + iu where u and v are the x- and y-velocity components, respectively.

(5.45)

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Drop Impact onto Dry Surfaces with Complex Morphology

The real part of iV satisfies the following boundary conditions at the membrane surfaces v|y=0 = U

for |x| ≤ b

(5.46)

v|y=0 = 0

for |x| > b

(5.47)

v|y=1 = 0.

(5.48)

In the domain shown in Fig. 5.57 the function iV (z) can be found from the boundary conditions for its real part at the surfaces using the Palatini formula which follows from the Schwartz formula for a disk (Lavrentiev and Shabat 1973, Henrici 1974). The general solution thus obtained reads   i ∞ i ∞ π (κ − z) π (κ − z) dκ + dκ v(κ, y = 0) coth v(κ, y = 1) tanh i V (z) = − 2 −∞ 2 2 −∞ 2 (5.49) where κ is a dummy real variable over the entire x-axis. It should be emphasized that if z is located at any of the domain surfaces, the first integral in Eq. (5.49) is evaluated in the Sochocki–Plemelj sense. Note that the velocity field (5.49) corresponds to the situation where liquid has already spread throughout the entire domain of a membrane. However, it is tempting to use it to predict the liquid front propagation, since the membranes of interest are thin. Substituting the boundary conditions (5.46)–(5.48) into Eq. (5.49), we arrive at  iU b π (κ − z) dκ (5.50) coth iV (z) = − 2 −b 2 which yields inside the domain

  sinh[π (b − z)/2] iU iV (z) = U − ln . π sinh[π (b + z)/2]

(5.51)

Accordingly, the velocity components u and v rendered dimensionless by U are given by the following dimensionless expressions inside the domain    sinh[π (b − z)/2] 1 (5.52) u(x, y) =  − ln π sinh[π (b + z)/2]    sinh[π (b − z)/2] 1 ln (5.53) v(x, y) = 1 +  π sinh[π (b + z)/2] with (•) and (•) denoting the real and imaginary parts, respectively. Then, the liquid front propagation is tracked by the following kinematic equations dX (s) = u[X (s), Y (s)] dt dY (s) = v[X (s), Y (s)] dt

(5.54) (5.55)

where s is the Lagrangian parameter of a liquid element at the filtration front at time t = 0, and X and Y are the current x and y coordinates of this element, respectively,

5.7 Impact of Aqueous Suspension Drops

0.5

211

t = 0.6 t = 0.45 t = 0.3 t = 0.15

0.4

Y

0.3

0.2

0.1

0.0 −1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

X Figure 5.58 Filtration front configurations at the dimensionless time moments shown in the inset.

Time shown in the inset is rendered dimensionless by h/U . In the present case b = 0.8 (Sahu et al. 2015). Reprinted with permission from Elsevier.

which satisfy the following initial conditions t = 0,

X = s with −b ≤ s ≤ b,

Y → 0.

(5.56)

Time t is rendered dimensionless by h/U , where h is the membrane thickness, and X , Y and s are rendered dimensionless by h. The system of Eqs. (5.54) and (5.55) with the velocity components from Eqs. (5.52) and (5.53) and the initial conditions (5.56) was integrated numerically using the Kutta– Merson method. Some representative results are shown in Fig. 5.58. It should be emphasized that as t → ∞ the filtration front would continue to widen after it has reached the rear side of the strip. The reason for that is that the boundary condition (5.46) implies that water pumping into the membrane will continue forever. In the experiments with drop impact, however, water is supplied through the membrane boundary only on the scale of t = 0.6 (see Fig. 5.54b and the discussion below). Therefore, water spreading outside the filtration front corresponding to t = 0.6 in Fig. 5.58 is impossible in the non-wettable membranes used in the present experiments. Note also that the flow velocity in the membrane rapidly becomes very small due to viscous dissipation discussed in detail below. That implies, as the experimental data show, that the nanoparticle transport ceases at t = 0.6, and the furthest boundary marked by the nanoparticles visualizes an effective ultimate filtration front. A detailed comparison of such predictions with the experimental data is shown in Figs. 5.54 and 5.55 and is discussed below. The liquid which filtrates through the membrane after a suspension drop impact is seeded with nanoparticles which are gradually intercepted by the pore walls. Therefore, the outer boundaries of the visible nanoparticle domains in the cut cross-sections observed experimentally (see Figs. 5.51c, 5.53a, 5.53b, 5.54 and 5.56) visualize the filtration fronts where water flow was already so weak (due to viscous dissipation)

212

Drop Impact onto Dry Surfaces with Complex Morphology

that it could not pull nanoparticles more. Such a front is termed the ultimate filtration front. The predicted configurations of the filtration fronts at different time moments similar to those in Fig. 5.58 are compared to the experimentally measured configurations shown in Figs. 5.54 and 5.55 for the PTFE membranes, which are unaffected by wettabilitydriven flow. In Fig. 5.54 the measured ultimate penetration front inside the PTFE membrane is compared to the predicted front at time moment t = 0.6. The predicted penetration front is sufficiently close to the experimentally observed front. This front thus corresponds to the dimensional time moment t = 0.6h/U . Taking for the estimate h = 0.1 cm and U ≈ 102 m/s, we find that the ultimate front has been reached in 6 µs. The result is plausible, since according to Fig. 5.40 in Section 5.5 of the present chapter, liquid spreading inside thin porous membranes ceases on the 100 µs scale. The kinetic energy of the penetrating water is rapidly dissipated by viscous friction at the pore walls. At a certain critical membrane thickness the entire initial kinetic energy will be dissipated, which means that the maximum penetration depth is reached. Namely, the kinetic energy Ek of water penetrating into the pores is of the order of   D 2 3 2 3 D. (5.57) Ek ≈ ρU D ≈ ρ V0 d In the hydrodynamic focusing process the only resistance to flow in the pores is associated with viscous friction at the pore walls, since the resistance associated with the surface tension would be negligibly small. Accordingly, the viscous shear stress S can be estimated as   μ D U V0 . (5.58) S≈μ ≈ d d d The length of a tortuous pore would be of the order of the drop size D, which means that the pore surface area is of the order of dD, while the viscous friction force acting on water in a single pore is about SdD. The number of pores through which water is penetrating is of the order of (D/d )3 . Therefore, the total viscous force is of the order of SdD(D/d )3 . The estimate of the number of pores takes account of the pores in all the directions. Accordingly, the work done by viscous forces while water has penetrated a wetted membrane thickness hw is of the order of SdD(D/d )3 hw , which is the total viscous dissipation Ed . Using Eq. (5.58), one finds that    3 D D μ V0 dD hw . (5.59) Ed ≈ d d d When the entire kinetic energy of water is fully dissipated at the critical membrane thickness, Ek ≈ Ed . Then, according to Eqs. (5.57) and (5.59) the critical wetted membrane thickness is found as (Sahu et al. 2012, Sahu et al. 2015) hw,critical =

ρV0 d 2 . μ

(5.60)

5.7 Impact of Aqueous Suspension Drops

(a)

213

(b)

Figure 5.59 Normalized penetration depth  versus the Weber number. Straight lines are the linear fits according to Eq. (5.62). Panel (a) corresponds to the case where the PTFE membrane was cut beginning from the drop impact surface, whereas panel (b) corresponds to the case where the PTFE membrane was cut in the opposite direction. The value of C for panel (a) is 0.864 and for panel (b) is 0.791 (Sahu et al. 2015). Reprinted with permission from Elsevier.

It should be emphasized that the critical thickness of a non-wettable membrane is understood as the minimal thickness which can never be penetrated by water. Evaluating it by the tip of the nanoparticle-deposition domain, i.e. using the ultimate penetration fronts as is done in the present section, is a slight underestimate in the experimental data. Equation (5.60) means that We hw,critical = const × D Ca

 2 d D

(5.61)

where Ca = μV0 /σ is the capillary number. Equation (5.61) can be transformed as  = C × We

(5.62)

where  = (hw,critical /D)/[Ca−1 (d/D)2 ], which can be found from the experimental data (i.e. measured), and C is a constant to be found by fitting this equation to the experimental data. The measured values of  obtained by observing the ultimate penetration fronts are plotted in Fig. 5.59 versus the Weber number, and also fitted by a linear fit expected according to Eq. (5.62). This is done for the five different impact velocities of 5.32 m/s, 5.64 m/s, 6.05 m/s, 6.47 m/s and 6.8 m/s. In the dynamic process accompanying water drop impact onto non-wettable membranes, water can penetrate into the interconnected pores to a significant depth, in distinction from penetration due to significant overpressure. In such dynamic penetration processes the hydrodynamic focusing plays the dominant role, while wettability and the surface tension are secondary. In the experiments of the present section the seeding nanoparticles were entrained and deposited inside non-wettable porous membranes, which underlines the results associated with the hydrodynamic focusing and the dynamic wettability.

214

Drop Impact onto Dry Surfaces with Complex Morphology

5.8

Drop Impact onto Hot Surfaces Coated by Nanofiber Mats The continuously rising demand for faster central processing units, the miniaturization and breakthrough developments in the field of semiconductor, optical and radiological components and the emergence of unmanned aerial and ground vehicles (UAVs and UGVs) result in a growing need for more powerful cooling technologies. These must be capable of removing heat fluxes of up to 1 kW/cm2 , and even more. Among the different approaches to cooling, which include, for example, natural and forced convection, heat pipes or micro-channel heat sinks, spray cooling is presently one of the most promising methods (Kandlikar and Bapat 2007, Kim 2007, Manglik and Jog 2009, Panão and Moreira 2009, Visaria and Mudawar 2009, Yarin et al. 2009, Yan et al. 2010). The tremendous cooling potential of spray cooling is associated with liquid evaporation at the hot surface. Thereby the efficiency is strongly affected by the hydrodynamics and heat transfer associated with drop impact onto hot surfaces. However, the typical receding motion of the spread-out liquid lamellae on hot metal surfaces leads to complete drop bouncing and interruption of cooling in many cases (see Fig. 4.28). Moreover, due to the insulating vapor layer established between the drop and the surface the heat flux reduces significantly in the Leidenfrost regime at higher temperatures (see Figs. 4.29 and 4.30). A novel approach in drop and spray cooling of microelectronic devices employs coating of hot surfaces with electrospun non-woven polymer nanofiber mats (Srikar et al. 2009, Sinha-Ray et al. 2011, Weickgenannt et al. 2011a, Weickgenannt et al. 2011b). Electrospun nanofiber mats are nano-textured permeable materials comprised of individual polymer nanofibers (with diameter of about several hundred nanometers) which are randomly orientated in the mat plane (see Fig. 5.31 in Section 5.4 in the present chapter). The size of the inter-fiber pores is of the order of several micrometers, and mat thickness can be several hundred microns. The electrospun nanofiber mats are usually produced from polymers which are either partially wettable or non-wettable (Reneker et al. 2007, Reneker and Yarin 2008). The benefits of using nanofiber mats in drop and spray cooling applications are mainly based on their influence on the hydrodynamic behavior of the impacting drops, as well as an increased heat transfer area. The experimental and theoretical results discussed in Sections 5.4–5.6 of the present chapter showed essentially two significant features of drop impact onto nanofiber mats. First of all, receding, splashing and bouncing during the impact on polymer nanofiber mats seem to be practically eliminated. Drop spreading after impact is similar to that on an impermeable surface, but the drop contact line is pinned as the maximum spread diameter is reached. Second, drop spreading is accompanied by filling of the pores, which are almost instantaneously impregnated underneath the area encircled by the pinned contact line of a spread-out drop. In the case of partially wettable nanofibers, liquid spreads inside the nanofiber mat over an area significantly larger than the one encircled by the pinned contact line (see Fig. 5.42 in Section 5.5 in this chapter). These features are beneficial for enhancement of heat transfer, since they increase contact area between water and the underlying hot surface.

5.8 Drop Impact onto Hot Surfaces Coated by Nanofiber Mats

(a)

(b)

215

(c)

Figure 5.60 Scanning electron microscope images of PAN nanofibers at various magnifications.

The nanofiber in the middle image does not contain carbon black nanoparticles, the one on the right contains carbon black nanoparticles (Weickgenannt et al. 2011a). Reprinted with permission. Copyright (2011) by the American Physical Society.

Nanofiber mats used in the experiments of Weickgenannt et al. (2011a) were electrospun from polyacrylonitrile, PAN, a partially wettable polymer with a water contact angle on a cast sample of about 30–40 ◦ C, and from PAN containing carbon black nanoparticles (CB), which tends to increase roughness of individual nanofibers. Nanofiber mats were electrospun on stainless steel foils attached to a grounded electrode. Typical scanning electron microscope images of the nanofibers used in these experiments are shown in Fig. 5.60. A schematic diagram of the experimental setup for drop impact is shown in Fig. 5.61. It consists of the following main elements: drop generation system, impingement surface (stainless steel foil covered by nanofiber mat), heating system, high-speed imaging and illumination system and an infrared imaging system. The initial drop diameter was D = 2 mm ± 0.3 mm. The syringe was fixed at a vertical adjusting spindle, which allowed the height of the needle tip to be varied over the target surface. The varied impact heights were H = 5 cm, 15 cm and 50 cm, which correspond to the following impact velocities, respectively: V0 ≈ 1 m/s, 1.7 m/s and 3 m/s. The stainless steel foils covered by nanofiber mats were heated electrically. A high-speed CCD camera was used to observe the shape of the spreading drop above the hot surface and to measure the initial drop diameter and impact velocity,

Figure 5.61 Experimental setup for drop impact onto nanofiber mats (Weickgenannt et al. 2011a).

Reprinted with permission. Copyright (2011) by the American Physical Society.

216

Drop Impact onto Dry Surfaces with Complex Morphology

t = 0.0004 sec

t = 0.0024 sec

t = 0.006 sec

t = 0.025 sec

t = 0.0004 sec

t = 0.0024 sec

t = 0.006 sec

t = 0.025 sec

Figure 5.62 Drop impact on a bare steel foil and on a nanofiber mat at a foil temperature of 60 ◦ C.

Top: bare steel; bottom: steel foil covered with PAN nanofiber mat (mat thickness h = 1.05 mm) (Weickgenannt et al. 2011a). Reprinted with permission. Copyright (2011) by the American Physical Society.

while an infrared camera was positioned underneath the targets and recorded the temperature distribution at the reverse side of the steel foils. The two cameras, CCD and infrared, were synchronized to achieve simultaneous drop imaging and thermal measurements. One series of experiments was performed at initial foil temperatures ranging from 60 ◦ C to 120 ◦ C with an increment of 20 ◦ C. The thermal resolution of the infrared camera was limited by a temperature of 120 ◦ C. In another experimental series, at higher foil temperatures (up to 300 ◦ C) a single thermocouple of type K with diameter of 0.5 mm was used for temperature measurements. It was positioned right underneath the impact point of the drop. Figure 5.62 shows typical CCD images of drop impact on a bare steel foil as well as on a steel foil coated with PAN nanofiber mat of thickness h = 1.05 mm. In both cases (bare steel and steel coated with nanofibers) the initial foil temperature was about 60 ◦ C and the drop impact velocity was 1.7 m/s. The drop impact onto a smooth bare steel substrate is followed by spreading and receding of liquid over the surface (see Fig. 5.62, top). After the receding stage the liquid drop reaches a quasi-steady state. The observed behavior during the first few milliseconds after drop impact onto a nanofiber mat is very similar to that in the case of a bare steel foil (see Fig. 5.62, bottom). The drop first spreads on the polymer mat as on a dry and impermeable smooth rigid surface. However, at the end of the spreading stage the contact line of the drop appears to be pinned in the spread-out configuration and does not recede, as observed in the experiments at room temperature (Section 5.5 in the present chapter). Figure 5.63 demonstrates the differences between drop evaporation on a bare steel foil and on a steel foil coated with nanofiber mat. Five typical CCD frames and the corresponding infrared images from below are shown. The nanofiber mat, the initial foil temperature, as well as the impact velocity, were the same as in the example shown in

5.8 Drop Impact onto Hot Surfaces Coated by Nanofiber Mats

217

Figure 5.63 Drop evaporation on a steel foil and on a nanofiber mat: a.1–a.5 bare steel; b.1–b.5

steel covered with PAN nanofiber mat, (mat thickness h = 1.05 mm) (Weickgenannt et al. 2011a). Reprinted with permission. Copyright (2011) by the American Physical Society.

Fig. 5.62. In the case of a bare steel foil, the drop reached a quasi-steady state after the initial liquid motion came to rest, as is seen in Fig. 5.63(a.1). In the following process, the drop contact line remains practically pinned over a significant fraction of the drop lifetime. In contrast to that, the drop height and the contact angle both continually decrease, because of the mass loss due to evaporation. Only near the end of the evaporation process does the contact line show an appreciable shrinkage. The temperature of the cooled area underneath the drop stays nearly constant as long as the evaporation lasts, as it can be seen in Figs. 5.63(a.3)–(a.5).

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Figure 5.63(b.1) shows the end of the spreading stage after drop impact onto the nanofiber mat, when the contact line of the drop is pinned in the spread-out configuration. The corresponding image of the temperature field demonstrates that at this early stage after drop impact, an evenly cooled area underneath the drop has not developed yet. Water penetration through the nanofiber mat thickness is not complete. The entire process can be subdivided into three phases, which were observed for all polymer nanofiber mats used in these experiments. The first phase corresponds to the growth of the cooled area. After the spreading stage is over, the temperature of the back side of the foil is significantly reduced. However, the liquid does not remain at rest after the drop contact line had been pinned but rather starts to spread in the nanofiber mat outside the area encircled by the contact line, as can be seen in Fig. 5.63(b.2). Water spreading inside the mats occurs in an almost axisymmetric manner. The maximum diameter of the cooled area in the case shown in Fig. 5.63 is about 12 times larger than the pre-impact diameter of the drop and reaches the value of about 24 mm. During this process, the temperature of the foil underneath the wetted area of the nanofiber mat continues to change. Namely, the growth of the wetted area inside the nanofiber mats leads to a continuous reduction of temperature of the cooled area. When a maximum size of the wetted spot has been achieved, the temperature attains a constant low value which is nearly uniform over the entire cooled area, as is seen in the image in Fig. 5.63(b.3). At the following second stage the evaporation process is comparable to that taking place on an uncoated foil. For the case shown in Fig. 5.63 the maximal size of the wetted spot does not change during about 20 s and the temperature of the cooled area does not change significantly either. During this time the intensity of the dark colored area visible in the high-speed images in Fig. 5.63(b.3 and b.4) is reduced, which indicates that an additional evaporation cooling occurs over the entire wetted area. The shrinkage of the cooled area, which corresponds to the shrinkage of the wetted spot, signifies the third and last phase of drop evaporation on a nanofiber mat. After reaching a low threshold of moisture content, the wetted area begins to shrink, while the temperature of the cooled area still does not change. The observations at this stage would not have been possible without using an infrared camera. Since the darker coloring of the wetted area in the CCD images fades with increasing time, the complete evaporation process cannot be observed using this as an indicator. The infrared images allow one to observe that the shrinkage process occurs continuously, as documented by a monotonous reduction of the diameter of the cooled area. The shrinkage was practically axisymmetric, which corresponds to the most intensive evaporation over the wetted spot perimeter. If one compares the evaporation process on a bare steel foil and the foil coated with nanofibers, three differences can be observed. First, the cooled area is about four times larger with the nanofiber mat than on the uncoated foil. Second, the minimum temperature at which the evaporation occurs is about 7 ◦ C lower for the nanofiber mat than for the bare steel foil. Third, the evaporation time is about six times shorter on the nanofiber mat for the same drop size. The results are paradoxical: a hot surface is cooled more effectively through a “furry overcoat,” almost 90–95% of which is filled with

5.8 Drop Impact onto Hot Surfaces Coated by Nanofiber Mats

219

Figure 5.64 Spreading diameter versus time for different initial foil temperatures. (a) PAN

nanofiber mat with thickness of h = 0.25 mm; (b) PAN with h = 1.05 mm (Weickgenannt et al. 2011a). Reprinted with permission. Copyright (2011) by the American Physical Society.

air entrapped in the pores in the nanofiber mats. However, this air is easily displaced with water brought by an impacting drop, while nanofibers dramatically improve the contact of water with the hot surface eliminating receding motion and bouncing of the drop. Water penetration and spreading inside nanofiber mats occur for the initial foil temperatures up to 120 ◦ C. As the initial foil temperature increased, the maximum spreading diameter, as well as the evaporation time decreased, since the evaporation rate increased (Fig. 5.64). For h = 1.05 mm at all the initial temperatures, drops spread very rapidly inside the mats and the dependence of the diameter of the cooled area on time had an almost parabolic shape over the entire process, Fig. 5.64b. On the other hand, for a lower mat thickness of h = 0.25 mm, water spreads with a lower speed inside the mats and the maximum spreading diameter dn,max is smaller for the investigated temperatures (Fig. 5.64a). Moreover, not only were the curves for the spreading diameter dn (t ) similar at the increasing foil temperatures but also the wetted spot shapes were similar at the moment of the largest spread-out. Note the difference between the curves dn (t) for the two initial temperatures 60 ◦ C and 80 ◦ C in Fig. 5.64a. At these temperatures the dn (t ) curves are almost identical up to t = 50 s, whereas the evaporation time is significantly shorter for 80 ◦ C. A reduction in the maximum diameter becomes visible only at about 100 ◦ C. This result can be explained by the intensification of the evaporation process at higher temperatures. Note also that, as seen in Fig. 5.65, the rate of water imbibition and evaporation inside nanofiber mats were almost unaffected by the impact velocity/height at different initial foil temperatures. Figures 5.66a and 5.66b show the foil temperatures at the impact axis for different nanofiber mats and the initial foil temperatures 60 ◦ C and 100 ◦ C. It can be seen that the foil temperature underneath the impact point drops almost instantaneously by about 25 ◦ C for the initial foil temperature of 60 ◦ C and by about 45 ◦ C for 100 ◦ C. After that, it stays practically constant during the entire evaporation process. It is emphasized that the minimum temperature at which the evaporation takes place is between 3 ◦ C and 5 ◦ C

220

Drop Impact onto Dry Surfaces with Complex Morphology

Figure 5.65 Spreading diameter versus time for different impact heights. (a) PAN+CB nanofiber mat with thickness h = 0.15 mm; (b) PAN with h = 1.00 mm (Weickgenannt et al. 2011a). Reprinted with permission. Copyright (2011) by the American Physical Society.

lower for all nanofiber mats than in the case of an uncoated steel foil. This is demonstrated in Fig. 5.67a. At higher foil temperatures 220 ◦ C and 260 ◦ C the temperature drop immediately underneath the impacting point can be as high as 140 ◦ C and 180 ◦ C, respectively.

Figure 5.66 Foil temperature under the drop impact point, TA , for different nanofiber mats and different initial temperatures. (a) The initial foil temperature T foil,init = 60 ◦ C; (b) T foil,init = 100 ◦ C; (c) T foil,init = 220 ◦ C; (d) T foil,init = 260 ◦ C (Weickgenannt et al. 2011a). Reprinted with permission. Copyright (2011) by the American Physical Society.

5.8 Drop Impact onto Hot Surfaces Coated by Nanofiber Mats

70 300

(a)

(b)

221

PAN + CB h = 0.15 mm

60

50 PAN +CB h = 0.15 mm PAN h = 0.25 mm

40

PAN h = 1.05 mm PAN h = 1.5 mm Uncoated steel foil

30 60

100 70 80 90 Temperature Tfoil, init [°C]

Evaporation time te [sec]

Temperature Tmin [°C]

PAN h = 0.25 mm PAN h = 1.05 mm

250

PAN h = 1.5 mm Uncoated steel foil

200 150 100 50 0 60

70 80 90 100 110 120 Temperature Tfoil, init [°C]

Figure 5.67 Cooling efficiency of nanofiber mats as compared to the uncoated steel foil. (a) The minimum temperature and (b) the evaporation time versus the initial foil temperature (Weickgenannt et al. 2011a). Reprinted with permission. Copyright (2011) by the American Physical Society.

Furthermore, the evaporation time is significantly shorter for all nanofiber mats, than that for an uncoated steel foil, as is seen in Fig. 5.67b. The evaporation times in the case of the PAN+CB mat with the thickness of h = 0.15 mm are the shortest ones for all temperatures, while at the same time this mat leads to the largest spreading diameters for all temperatures. Polymer nanofibers have a relatively low thermal diffusivity of their own, as well as the individual fibers are pretty smooth. Therefore, a further enhancement of spray cooling through nanofiber mats can be achieved if they are metalized and made as rough as possible, as a means to further increase the effective surface area of nanofiber-coated hot surfaces. In this context, electroplating of electrospun nanofiber mats deposited on high-heat-flux surfaces was used to facilitate their cooling (Sinha-Ray et al. 2011). SEM images of the electroplated nanofiber mats are shown in Figs. 5.68 and 5.69. The individual copper-plated fibers possess thorny (Fig. 5.68c) and grainy (Fig. 5.68d) nanotexture, which makes them reminiscent of the Australian thorny devil lizards. In contrast to the copper-plated fibers, gold-plated fibers are smoother and possess only infrequent spheroidal appendices or their small clusters (Fig. 5.69). The comparison of the images with the same magnification (Figs. 5.68c,d, and 5.69c,d) shows that copper-plated fibers possess the roughest nano-texture uniformly. Moreover, even though only the upper layer of the nanofibers appears to be rough in Fig. 5.68, this is only an artifact related to the electron beam focusing in SEM. The images of a cut copper-plated nanofiber mat show that the fibers at the mat bottom in contact with the underlying substrate are as thorny-devil-like as those near the mat surface in Fig. 5.68. Therefore, the fibers are rough throughout the entire mat depth. The top and side view images of water drop impact from the height of 3.55 cm (impact velocity of V0 = 83.46 cm/s), spreading and evaporating on a copper-plated nanofiber

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Drop Impact onto Dry Surfaces with Complex Morphology

Figure 5.68 SEM images of the thorny-devil copper-plated fibers. (a) The overall view of the

copper-plated fiber mat. (b) The zoomed-in view of the top layer. The individual fibers at different locations: (c) thorny and (d) grainy nano-textures (Sinha-Ray et al. 2011). Reprinted with permission. Copyright (2011) American Chemical Society.

mat, are shown in Fig. 5.70. Figure 5.70a depicts the moment of drop impact (t = 0) onto the nanofiber surface. Figure 5.70b shows that at the moment t = 32.5 ms water boiling is visible at the surface, whereas Fig. 5.70c demonstrates that boiling practically ceased at t = 66 ms and there is no visible activity at the surface anymore. According to Rohsenow et al. (1998), the values of the thermal diffusivity of copper αCu = 1.1 cm2 /s and gold αAu = 1.23 cm2 /s are very close. Nevertheless, identical drops evaporate much faster after an impact onto copper-plated nanofibers than on gold-plated ones (Sinha-Ray et al. 2011). The latter inevitably leads to the conclusion that thorny-devil-like copper-plated fibers are much more effective in drop cooling than the smooth gold-plated ones just because the surface area of the former is dramatically higher than that of the latter (see Figs. 5.68 and 5.69). It is also worth mentioning that the evaporation times on metal-plated nanofiber mats are dramatically shorter than those for the polymer mats, with the proportion of the order of 30 ms versus 30 s. Accordingly, Sinha-Ray et al. (2011) and Sinha Ray and Yarin (2014) reported heat fluxes of the order of 0.6–0.9 kW/cm2 removed by individual drops impacting onto hot surfaces coated with the copper-plated thorny-devil nanofibers.

5.9 Suppression of the Leidenfrost Effect

223

Figure 5.69 SEM images of gold-plated nanofibers. (a) The overall view of the gold-plated

nanofiber mat. (b) The zoomed-in view of the upper layers with the visible appendices scattered over the fibers. (c) and (d) Several individual fibers: almost smooth coatings with some appendices at two different locations (Sinha-Ray et al. 2011). Reprinted with permission. Copyright (2011) American Chemical Society.

5.9

Nano-textured Surfaces: Suppression of the Leidenfrost Effect The dramatic reduction of heat removal rate in the film boiling regime (above the Leidenfrost temperature) is one of the main challenges of spray cooling of high temperature surfaces. One attractive way to enhance heat removal rate is associated with textured substrates, in particular, different types of rough, structured or coated surfaces, which affect the outcome of a drop impact onto cold and hot surfaces (Xu 2007, Sodtke and Stephan 2007, Ojha et al. 2010). One of the recently discovered and very promising methods for controlling hydrodynamics of drop impact and enhancing heat removal from a hot wall to a cold drop is associated with electrospun polymer nanofiber mats (see Sections 5.4–5.6 and 5.8 in the present chapter). Such mats consist of individual polymer or metal-plated fibers of submicron diameters, which are randomly orientated in the mat plane and consist of multiple nanofiber layers. They can be produced on any conductive surface and have a strong adhesion to the hot surfaces even at temperatures as high as 300 ◦ C. Such nano-textured surfaces significantly modify the outcomes of drop impact and dramatically enhance the heat removal rate. They practically eliminate

224

Drop Impact onto Dry Surfaces with Complex Morphology

(a)

at t = 0

(b)

(c)

at t = 32.5 ms

at t = 66 ms

Figure 5.70 Drop impact from the height of 3.55 cm (V0 = 83.46 cm/s) onto a copper-plated nanofiber mat at 150 ◦ C (the display temperature corresponding to 125.6 ◦ C of the sample). (a) 0 ms at the moment of impact; (b) 32.5 ms, (c) 66 ms (Sinha-Ray et al. 2011). Reprinted with permission. Copyright (2011) American Chemical Society.

the receding motion of the contact line and bouncing on cold and hot surfaces. Furthermore, liquid coolants penetrate into nanofiber mats and spread inside them over a very large area, which remains wetted during a period of about 0.1–1 s. As a result, nano-textured surfaces covered with nanofiber mats dramatically increase cooling efficiency of individual drop impacts compared to that of drop impacts on bare metal surfaces below the Leidenfrost temperature (see Section 4.9 in Chapter 4). The aim of the present section is to investigate the influence of nanofiber coatings on drop impact onto very hot surfaces within the Leidenfrost regime. Using the experimental setup shown in Fig. 5.61, Weickgenannt et al. (2011b) studied drop impacts onto hot bare surfaces accompanied by the Leidenfrost effect and its suppression by polymer nanofiber coatings. Bare steel foils under different experimental conditions were used as a reference case in comparison with drop impacts onto nano-textured surfaces. Figure 5.71 illustrates the outcomes of water drop impacts at different initial foil temperatures. In each case the drop evolution after the impact is depicted at the same time instances after the first contact between the drop and the foil surface in order to demonstrate the influence of the foil temperature on hydrodynamics. Figure 5.71a shows the impact of a water drop onto a bare steel foil at an initial foil temperature of 60 ◦ C. The drop impact is followed by spreading and receding of

225

5.9 Suppression of the Leidenfrost Effect

(a)

5 mm

t = 0.2 ms

1.5 ms

5 ms

30 ms

250 s

1.5 ms

5 ms

30 ms

16 s

1.5 ms

5 ms

30 ms

5s

(b)

5 mm

t = 0.2 ms (c)

5 mm

t = 0.2 ms

Figure 5.71 Water drop impact on a bare steel foil at different initial foil temperatures. The initial

foil temperature T foil,init is equal to (a) 60 ◦ C, (b) 220 ◦ C and (c) 300 ◦ C (Weickgenannt et al. 2011b). Reprinted with permission. Copyright (2011) by the American Physical Society.

liquid over the surface. After the receding stage, which is driven by surface tension, the liquid drop reaches a quasi-steady state. Subsequently the drop height and contact angle gradually decrease because of the evaporative mass loss, while the contact line remains completely pinned. Shortly before the end of the evaporation process the contact line de-pins, and the drop shows an additional appreciable shrinkage. In this case, as in general for temperatures below the boiling temperature of the liquid, the outcomes of drop impact are qualitatively comparable with the outcomes of drop impact onto unheated surfaces. Significant differences in the outcomes of drop impacts onto hot bare steel foils in comparison to those for the drop impacts onto unheated bare foils were first observed in these experiments at an initial foil temperature of 220 ◦ C (see Fig. 5.71b). In the first image of the sequence shown in Fig. 5.71b, corresponding to the time instant t = 0.2 ms after drop impact, tiny bubbles at the interface between the drop and hot foil are visible (see the enlarged section), which is typical for nucleate boiling. Then, at the spreading stage, the drop experiences perturbations due to capillary waves excited by boiling. Nevertheless, there is not yet drop fragmentation or levitation at this temperature; the drop stays intact and wets the foil. This is perhaps due to the fact that after drop impact the foil temperature decreases significantly below the temperature range corresponding to significant bubble nucleation. For drop impact onto the bare stainless steel foil at an initial temperature of 300 ◦ C, the drop shatters into secondary droplets and several tiny satellites (see Fig. 5.71c). The atomization process is driven by a high pressure in the vapor below the drop at such high temperatures. Additionally, vapor recoil is probably a source of strong surface

226

Drop Impact onto Dry Surfaces with Complex Morphology

(a)

5 mm

t = 0.2 ms

1.3 ms

5 ms

50 ms

30000 ms

1.3 ms

5 ms

30 ms

1000 ms

1.3 ms

5 ms

15 ms

30 ms

(b)

5 mm

t = 0.2 ms (c)

5 mm

t = 0.2 ms

Figure 5.72 Ethanol drop impact on a bare steel foil at different initial temperatures. The initial

foil temperature T foil,init is equal to (a) 60 ◦ C, (b) 180 ◦ C, and (c) 300 ◦ C (Weickgenannt et al. 2011b). Reprinted with permission. Copyright (2011) by the American Physical Society.

perturbations resulting in breakup into a multitude of tiny droplets. Those droplets are then accelerated away from the primary drop. The core of the primary drop during and after droplet shedding stays at the foil surface. It is emphasized that even though there is still some liquid left on the foil surface after the primary drop scattering, the cooling potential is reduced dramatically in this case. The contact area between the hot surface and the liquid is greatly reduced in comparison with the cases below the Leidenfrost point, the amount of liquid evaporating at the surface is reduced, and as a result, the cooling is much less efficient. Similar drop impact experiments with a bare heated steel foil have been also performed with ethanol as a test liquid. Its saturation temperature, latent heat of evaporation and surface tension are lower than those of water, which leads to the intensification of the shattering effects at high temperature in comparison to those described before. In Fig. 5.72 the outcomes of ethanol drop impact at three initial foil temperatures are illustrated. It is seen that the spreading behavior of an ethanol drop on a stainless steel foil heated up to 60 ◦ C is qualitatively similar to that of a water drop (Fig. 5.71a). The ethanol drop spreads, recedes and stays in a stationary position until it fully evaporates. The most visible differences in comparison to a water drop impact at an initial foil temperature of 60 ◦ C are the reduction of the contact angle and an increase of the spreading velocity and the maximal spread-out contact diameter. As a consequence of the lower saturation temperature (or higher volatility) and lower surface tension of ethanol in comparison to water, the onset of the nucleate boiling occurs already at an initial foil temperature of 140 ◦ C. By increasing the initial foil temperature up to 180 ◦ C, the Leidenfrost regime is nearly reached, as is depicted

5.9 Suppression of the Leidenfrost Effect

227

(a)

5 mm

t = 0.2 ms

1.6 ms

5 ms

15 ms

3000 ms

1.5 ms

5 ms

30 ms

900 ms

1.5 ms

5 ms

30 ms

300 ms

(b)

5 mm

t = 0.2 ms (c)

5 mm

t = 0.2 ms

Figure 5.73 Water drop impact onto PAN+CB nanofiber mat with thickness h = 0.5 mm at

different initial temperatures of the underlying stainless steel foil. The initial foil temperature T foil,init is equal to (a) 60 ◦ C, (b) 220 ◦ C and (c) 300 ◦ C (Weickgenannt et al. 2011a). Reprinted with permission. Copyright (2011) by the American Physical Society.

in Fig. 5.72b. Surfaces at higher initial temperatures supported ethanol drops in the stable film boiling regime, in which the drops are levitated over a thin vapor layer (see Fig. 5.72c, corresponding to the initial foil temperature of 300 ◦ C). In this case cooling is practically impossible due to the absence of a direct contact between the liquid and the hot surface. In the following, the outcomes of water and ethanol drop impacts onto a steel foil coated with a PAN+CB nanofiber mat with a thickness of h = 0.5 mm are illustrated and compared with the observations of the drop impact onto bare foils. Figure 5.73 shows the time sequences for water drop impacts onto the nanofiber-coated foil at different initial foil temperatures. While the first four images of every sequence were recorded with a frame rate of 30 000 fps, the last images were taken with the frame rate of 125 fps under identical experimental conditions. In order to allow comparison with the results of drop impact onto a bare foil, the same initial foil temperatures as in Fig. 5.71 were chosen. Figure 5.73a illustrates the outcome of water drop impact onto a nanofibercoated foil with the initial temperature of 60 ◦ C. Driven by the kinetic energy of impact, the drop spreads over the surface in the early stage of impact, as it would do on an impermeable surface. On the other hand, the receding motion of the drop after impact cannot be observed on a nanofiber mat, as discussed in Sections 5.4–5.6 and 5.8 of the present chapter. The contact line is pinned at the maximum spreading position. After some time the drop starts to spread inside the nanofiber mat. The area of the nanofiber mat impregnated with water beyond the maximal spread-out spot can be seen in the last image in Fig. 5.73a. This area is recognizable as a ring-shaped region which appears lighter than the dry mat at the wetted spot.

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Drop Impact onto Dry Surfaces with Complex Morphology

(a)

5 mm

t = 0.2 ms

10 mm

1.4 ms

5 ms

20 ms

280 ms

(b)

10 mm

5 mm

t = 0.2 ms

1.4 ms

5 ms

20 ms

200 ms

(c)

10 mm

5 mm

t = 0.2 ms

1.4 ms

5 ms

20 ms

250 ms

Figure 5.74 Ethanol drop impact onto PAN+CB nanofiber mat with thickness h = 0.5 mm at different initial temperatures of the underlying foil. The initial foil temperature Tfoil,init is equal to (a) 60 ◦ C, (b) 180 ◦ C and (c) 300 ◦ C (Weickgenannt et al. 2011b). Reprinted with permission. Copyright (2011) by the American Physical Society.

This drop pinning observed on electrospun nano-textured surfaces results from the hydrodynamic focusing effect discussed in Section 5.6 of the present chapter. Due to that effect, water can penetrate into the nanofiber mat pores, irrespective of their wettability and fill the pores under the entire wetted spot visible above the surface. That also explains why a spread-out drop becomes pinned on nano-textured mats: it becomes a circular millipede. Figures 5.73b and 5.73c show the outcomes of water drop impacts onto nanofibercoated foils at initial temperatures of 220 ◦ C and 300 ◦ C, respectively. It is seen that the initial foil temperature has a negligible effect on the outcome of drop impact on nanofiber mats. In both cases water drops spread out as at the lower initial foil temperatures, do not shatter and stay in full contact with the substrate. Breakdown of the spreading lamella, drop fragmentation or levitation as in the case of drop impact onto bare steel foils do not occur on the surface coated with nanofiber mat. One of the noticeable changes in comparison with drop impact onto unheated nanofiber mats is in complete pinning of the drop contact line after reaching the maximum spreading configuration for higher temperatures. A puddle of liquid resting above the nanofiber mat decreases with time and eventually disappears, which happens partially due to its penetration and spreading inside the mat and partially due to evaporation. The time which elapses before the puddle disappears decreases with increasing foil temperature, as can be seen in the last image of every sequence in Fig. 5.73. In Fig. 5.74 the observations of ethanol drop impacts onto nanofiber-coated foils are shown. Slight differences between the behavior of the ethanol drops compared to that of

229

240 (a) 220 200 180 160 140 120 100 80 60 0 5

320

Uncoated steel foil PAN + CB h = 0.5 mm

10 15 20 25 30 35 40 Time t [sec]

Temperature TA [°C]

Temperature TA [°C]

5.9 Suppression of the Leidenfrost Effect

(b)

280 240 200 160 120

Uncoated steel foil PAN + CB h = 0.15 mm

80 0

5

10 15 Time t [sec]

20

25

Figure 5.75 Evolution of foil temperature under the drop impact point, TA , for water drop impact. The initial foil temperature Tfoil,init is equal to (a) 220 ◦ C and (b) 300 ◦ C (Weickgenannt et al. 2011b). Reprinted with permission. Copyright (2011) by the American Physical Society.

the water drops can be seen in the case of the initial foil temperature T foil,init = 60 ◦ C. The ethanol drop spreads wider than the corresponding water drop since the surface tension of ethanol is lower than that of water. Comparison between the third and fourth image of every sequence reveals that the radius of the portion of the liquid which rests over the mats as a sessile drop decreases with time. It is also clearly seen that the time elapsing before the puddle resting over the mat disappears is much shorter for the ethanol drop in comparison with the water drop. It can be seen in Figs. 5.74b and 5.74c that the presence of the nanofiber mat completely eliminates the Leidenfrost effect of ethanol drops at the initial foil temperatures of 180 ◦ C and 300 ◦ C. The overall behavior of ethanol drops after impacts on nanofiber-coated foils at the different initial foil temperatures is practically identical. Neither bubble formation in the nucleate boiling regime, nor drop shattering, nor the Leidenfrost regime are observable. It can be concluded that the presence of nanofiber mats suppresses the Leidenfrost effect. Moreover, when the millipede-like Wenzel state is dynamically imposed by drop impact, nanofiber mats are capable of preserving it in spite of significant surface temperatures, whereas ordinary micro-patterned substrates lose Wenzel state and transfer to the Cassie–Baxter state as in Liu et al. (2011). When a liquid drop comes in contact with a hot surface, the temperature of the surface in the contact area is reduced due to heat conduction between the hot foil and the cold drop and the latent heat of evaporation. At the nano-textured surfaces (coated with nanofibers) the anti-Leidenfrost effect results in a wider wetted spot, suppresses liquid atomization, and thus increases the amount of liquid evaporating in direct contact (through the filled pores) with the underlying hot surface. Both the increase of the area from which the liquid evaporates, and shorter evaporation time increase the heat removal rate and thus, increase the cooling efficiency. This is illustrated by the results of the temperature measurements depicted in Figs. 5.75 and 5.76. In Fig. 5.75 the foil temperature evolution following water drop impact onto a bare steel foil is compared with the temperature evolution resulting from the drop impact onto a PAN+CB nanofiber mat of thickness h = 0.5 mm. The results are presented for the initial foil temperatures of 220 ◦ C and 300 ◦ C.

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Drop Impact onto Dry Surfaces with Complex Morphology

Figure 5.76 Evolution of foil temperature under the drop impact point, TA , for the ethanol drop impact. The initial foil temperature Tfoil,init is equal to (a) 140 ◦ C, (b) 180 ◦ C, (c) 220 ◦ C and (d) 300 ◦ C (Weickgenannt et al. 2011b). Reprinted with permission. Copyright (2011) by the American Physical Society.

Two main effects of the nanofiber mat can be observed in Fig. 5.75a. The first effect is the reduction of the minimum temperature of the foil in the case of the nanofibercoated foil in comparison with the uncoated foil. The minimal temperature is about 15 ◦ C lower in the case of the nanofiber-coated foil compared with that of the bare steel foil. The second effect is the reduction of drop evaporation time on nanofiber mats. It is determined using the time during which the foil temperature is lower than the initial temperature. The evaporation time is about 35 s shorter on the nanofiber mats compared to that of the bare steel foil. The present results show that the main features of drop impact onto nanofiber-coated surfaces do not change, at least up to initial foil temperatures of 300 ◦ C, as it can be seen in Fig. 5.75b. In the latter case the minimum temperature of the foil is about 20 ◦ C lower for the nanofiber-coated foil than that for the uncoated foil. In Fig. 5.76 the foil temperature variation is plotted for the case of the ethanol drop impact onto a bare steel foil and onto a nanofiber-coated steel foil. The foil temperature trends on bare and nanofiber-coated foils for an initial temperature of 140 ◦ C for the ethanol drop impacts (Fig. 5.76a) are similar to the results for water drops. The minimal temperature of the nanofiber-coated foil is lower by about 20 ◦ C compared to the bare foil. The evaporation time on the nanofiber-coated surface is shorter compared to the bare steel surface. The Leidenfrost regime has not been reached yet at this initial foil temperature.

5.10 Bouncing Prevention: Dynamic Electrowetting

231

It has been shown in Fig. 5.72b that at the initial foil temperature of 180 ◦ C the ethanol drop impacting onto the bare foil shatters into smaller droplets which levitate above the foil surface. Only a small residual drop is left on the surface about 1 s after first contact between the drop and the foil surface. Such drop shattering results in a drastic deterioration of the cooling efficiency for drops impacting onto bare foil. Indeed, the minimal foil temperature achieved by the ethanol drop impact at 180 ◦ C is as high as 163 ◦ C (see Fig. 5.76b), whereas with the nanofiber mat coating the minimal temperature is about 110 ◦ C. This result is consistent with the behavior of the impacting ethanol drops (see Fig. 5.74b). The ethanol drop impacts onto the bare foil surface at the initial temperatures of 220 ◦ C and 300 ◦ C exhibit negligible cooling, which is the result of the Leidenfrost effect. Indeed, in these cases there is no direct contact between the hot surface and the drop. In contrast, the ethanol drop impacts onto the nanofiber-coated foil result in a reduction of the initial foil temperature by 95 ◦ C and 115 ◦ C (for the initial temperatures of 220 ◦ C and 300 ◦ C, respectively). This result illustrates that in the case of the ethanol drops the Leidenfrost effect is eliminated on the nanofiber-coated foil surface. Due to the presence of the nanofiber mats direct contact between the liquid and the hot surface occurs and leads to a drastic reduction in the foil temperature. The copper- and silver-plated fibers also reveal suppression of the Leidenfrost effect (Fig. 5.77). For example, after the impact onto a bare copper substrate, the water drop fully bounces back (and practically does not provide any cooling), whereas it spreads, pins itself, penetrates the pores and reaches the substrate on nanofibers. As a result, drop evaporation time te on such metal-plated thorny-devil or cactus-like mats is reduced by more than two orders of magnitude compared to that on polymer mats (te is of the order of 50 ms in Fig. 5.77 for the copper-plated fibers versus 5–10 s for polymer fibers). This explains the tremendous cooling rate of about 0.6–0.9 kW/cm2 recorded after water drop impacts onto copper substrates with a bonded 30 µm-thick copper thorny-devil nanofiber mat at its surface (Sinha-Ray et al. 2011, Sinha-Ray and Yarin 2014).

5.10

Bouncing Prevention: Dynamic Electrowetting Drop bouncing on impact onto a rigid wall can be prevented by means of the dynamic electrowetting-on-dielectric (DEWOD) introduced in Lee et al. (2013). This phenomenon can be used to control spray cooling, painting and coating, as well as for drop transport in microfluidics. Wettability of solid surfaces with liquids is of great importance in fundamental research and practical applications of adhesion and microand nanofluidics (Karniadakis et al. 2005, Yarin et al. 2009, Wong et al. 2011). Control of surface wettability by application of external stimuli has drawn significant attention due to a wide range of potential applications. Among various methods for wettability switching, electrowetting (EW) or electrowetting-on-dielectric (EWOD) is arguably one of the most flexible methods to actively control liquid wettability on partially wettable or completely non-wettable surfaces. EW is an electrically induced alteration in material wettability and has long been known to affect the equilibrium contact angle of drops

232

Drop Impact onto Dry Surfaces with Complex Morphology

Figure 5.77 (a) An individual copper-plated thorny–devil fiber. (b) A layer of silver-plated

nanofibers. (c)–(f ) Water drop impact onto a bare copper substrate (left) and the same substrate with a bonded 30 µm-thick mat of copper thorny-devil nanofibers. In both cases the target temperature is 172.2 ◦ C (Sinha-Ray et al. 2011). Reprinted with permission. Copyright (2011) by the American Chemical Society.

without changing the chemical composition of the contacting phases (Beni et al. 1982, Karniadakis et al. 2005). EW results from the reduction in the solid–liquid interfacial energy due to the free charges (ions) accumulating at the dielectric surface when voltage is applied. In turn, the equilibrium contact angle of drops on such dielectric surfaces is affected according to the Young–Lippmann equation (see below), and thus the surface wettability is changed. The apparent equilibrium contact angle can be dramatically altered (often by 50% or more), which has dynamic implications as well. Therefore, EW offers an excellent alternative for moving tiny amounts of liquid over hydrophobic surfaces or making the surfaces wettable; the effect is precise, repeatable and rapid. Drop manipulation by EW has very useful diverse applications such as digital microfluidics (Zhao and Cho 2007), electronic displays (Hayes and Feenstra 2003), paint drying (Welters and Fokkink 1998), responsive cooling (Garrod et al. 2007), adjustable focusing lenses (Kuiper and Hendriks 2004), among others (Yang et al. 2003, Dubois et al. 2006). Recent needs for the construction of low-power and efficient EW systems have triggered studies aimed at understanding the mechanism and limitations of EW. The control over the wetting properties of the solid surfaces by EW already has numerous

5.10 Bouncing Prevention: Dynamic Electrowetting

233

practical applications, yet many aspects of electrowetting behavior are not fully understood. The main difficulty encountered in static electrowetting is that, after some initial decrease in contact angle, even very large external voltages fail to achieve complete wetting (Karniadakis et al. 2005, Shapiro et al. 2003). Several complementary hypotheses have been expressed in that respect, such as dielectric charging (Kilaru et al. 2007), asymmetric (polarity-dependent) electrowetting responses (Fan et al. 2007), gas ionization in the vicinity of the contact line, contact line instability (Vallet et al. 1999), droplet resistance (Shapiro et al. 2003) and saturation phenomena (Wang and Jones 2005). Numerous EW phenomena on micro-/nano-structured surfaces involve an enormous change in contact angle and a fast dynamic tuning of the wetting (Krupenkin et al. 2004, Zhu et al. 2006, Wang et al. 2007, Han et al. 2009). Generally, wetting behavior is strongly dependent on both surface chemistry (i.e. surface energy) and surface topography (i.e. surface roughness); see Section 1.8 in Chapter 1. Superhydrophobic surfaces have been produced mainly in two ways: (i) by creating a rough structure on a hydrophobic surface, and (ii) by modifying a rough surface by a material with low surface free energy. For water to wet the superhydrophobic surfaces, external work has to be done to overcome the energy barrier to achieve the transition from the superhydrophobic to hydrophilic state (He et al. 2003, Lembach et al. 2010, Sinha-Ray et al. 2011, Weickgenannt et al. 2011a, Weickgenannt et al. 2011b, Sahu et al. 2012). The EW method offers a powerful, non-destructive, on-demand and selective way to achieve this. For example, when a liquid drop is placed on a superhydrophobic surface, it stays at the top of the roughness, leading to a small contact area, a large contact angle (>150◦ ), and a small roll-off angle [the Cassie–Baxter state; Cassie and Baxter (1944), see Section 1.8 in Chapter 1]. Once a voltage is applied, the drop is impaled through the surface texture leading to a contact angle lower than 150◦ and a high roll-off angle [the Wenzel state; Wenzel (1936), also see Section 1.8 in Chapter 1]. It was demonstrated that the use of EW can switch surface wetting properties from the superhydrophobic to nearly completely superhydrophilic (Krupenkin et al. 2004). That was achieved by fabricating nanopillars with a diameter of 350 nm and a height of 7 µm by dry etching a Si wafer. Each pillar had a conductive core of Si covered by a thermally-grown insulating SiO layer and a hydrophobic top coating. Surfaces can be efficiently made wettable and water pumped through superhydrophobic aligned multi-walled nanotube membranes by application of a small EW voltage (Wang et al. 2007). The Cassie–Baxter-to-Wenzel transition by EW was demonstrated on the nanocomposite superhydrophobic surface prepared using spherical amorphous carbon nanoparticles (≈100 nm thick) capped on carbon nanotubes (Han et al. 2009). Behavior of water drops on aligned carbon nanotube surfaces under the influence of an EW voltage was experimentally investigated and the limits of contact angle change on such surfaces were established (Zhu et al. 2006). In EWOD devices, the applied electric field is confined in the dielectric layer, so that EW is determined by the geometrical parameters and dielectric permittivity of the dielectric. The main requirement for an efficient EW is the low contact angle hysteresis on the hydrophobic dielectric layer in conjunction with high initial contact angle. One of the most widely used materials relevant to EW is poly(tetrafluoroethylene) (PTFE),

234

Drop Impact onto Dry Surfaces with Complex Morphology

Figure 5.78 Dynamic electrowetting experiment with drop impact onto Teflon layer. The upper

panel shows the sketch, the lower one the photographic images of the setup (Lee et al. 2013). Reprinted with permission. Copyright (2013) American Chemical Society.

known as Teflon. It was reported that flat fluoropolymer Teflon reveals the equilibrium contact angle of water of around 105 ± 15◦ (Gao and McCarthy 2008). A further increase in the equilibrium contact angle of flat Teflon is preferable for most microfluidic devices, since it will not only facilitate drop motion, but also extend the functional range of the contact angle. An easy and effective method to prepare superhydrophobic Teflon is to change the density of the fibrous crystals on its surface by stretching. In the present section, following Lee et al. (2013), the dynamic electrowetting-ondielectric (DEWOD) performance of unstretched (hydrophobic) and stretched (superhydrophobic) Teflon on Fluorine-doped tin oxide-coated (FTO) substrates is described. The dynamic electrowetting effect is associated with water drop impact and bouncing, and the situation is significantly different from standard EW phenomena with sessile drops attached to an electrode. The experimental setup is shown in Fig. 5.78. The commercially available Teflon tape was used as the dielectric material. The tape was cut into 50–60 mm (length) × 50–52 mm (width) × 100 µm (thickness) pieces and stretched to different strains [strain = (L − L0 )/L0 × 100(%), where L0 is the initial length and L is the stretched length] of 50, 100, 150, 200 and 250%. The thickness of the unstretched Teflon was 100 µm, whereas after stretching to 200%, the thickness diminished to 70 µm. The relative dielectric permittivity of the Teflon samples was ε = 2.1. The unstretched and stretched Teflon pieces were fixed on FTO-coated glass substrate as shown in the lower panel in Fig. 5.78 and used for static and dynamical drop impact measurement (with and without EW voltage applied). The experiments were carried out in the following way: a voltage between 0 and 6 kV was applied directly to the FTO substrate on which either unstretched or stretched

5.10 Bouncing Prevention: Dynamic Electrowetting

235

(a) 160

200%-stretched

(b) Unstretched

Contact angle [deg]

150 140

Orthogonal-dir. Parallel-dir. CA=96° CA=105°

130 120

(c) 200%-stretched 110 100

Orthogonal-dir. Parallel-dir. CA=147° CA=150°

Unstretched

90 0

50

100

150

200

250

300

Stretching ratio [%] Figure 5.79 (a) Static water contact angle (SWCA) as a function of the stretching ratio of Teflon

tape. The insets show the corresponding optical images of the water drops observed orthogonally to the direction of the main surface ridges; SWCA of the unstretched Teflon is 96◦ ± 4◦ , whereas of the 200%-stretched one is 147◦ ± 5◦ . The average values were obtained by measuring at five different positions on the same sample, and the variance is reflected by the error bars. (b) Contact angle of a water drop on the unstretched surface observed orthogonally to the direction of surface ridges and parallel to them. The left-hand side image corresponds to the image of the drop on the unstretched Teflon surrounded by a small rectangular solid frame in panel (a). (c) Contact angle of a water drop on the 200%-stretched surface observed orthogonally to the direction of surface ridges and parallel to them. The left-hand side image corresponds to the image of the drop on the 200%-stretched Teflon surrounded by a small rectangular dashed frame in panel (a) (Lee et al. 2013). Reprinted with permission. Copyright (2013) American Chemical Society.

Teflon was fixed. Spherical DI water drops 1.92 mm in diameter were generated and dripped under gravity onto the substrate. The impact velocity was ≈62 cm/s. The distance between the nozzle tip and the substrate was fixed at h = 5 cm. A high-speed camera captured the magnified images of DI water drops. The equilibrium contact angle of water drops can be dramatically raised by increasing the surface roughness of Teflon by its stretching. With increasing the stretching ratio parallel to the fibrous crystals of Teflon from 0 to 200%, the static water contact angle (SWCA) increases from 96 ± 4◦ to 147 ± 5◦ (Fig. 5.79a) when observed in the direction orthogonal to that of the main surface ridges (Figs. 5.80 and 5.81), i.e. the surface changes its wetting properties from hydrophobic to superhydrophobic. The same conclusion is reached when the contact angle is observed in the direction parallel to the main surface ridges (Figs. 5.79b and c). Note that the latter two figures show some effect of the surface anisotropy on the contact angle. To link the wettability change to the surface morphology, scanning electron microscope (SEM) images of the unstretched and stretched Teflon (strain = 200%) were taken (Fig. 5.80). Teflon is a semi-crystalline polymer, where the crystals are aligned in a regular parallel array of relatively intact ridges before stretching (Fig. 5.80a). Some loosely

(a)

50 μm

10 μm

50 μm

10 μm

stretching direction

(b)

Figure 5.80 SEM images of Teflon surface (a) unstretched and (b) 200%-stretched. Multiple

micro-cracks appear across the ridges in the stretched sample (Lee et al. 2013). Reprinted with permission. Copyright (2013) American Chemical Society.

nm 500 200 0 −250 −500 −750

stretching direction

nm (b) 500 200 0 −250 −500 −750

8

μm

μm 12

12

16

0 16 2

20

(a)

20

8

20

18

18 12

4

4

12 8

μm

8

μm

4

0

0

4

0

(c) nm

0

(d) nm 800

200

600

0

400

−200

200

−400

0 0

4

8

μm

12

16

20

0

4

8

12

16

μm

Figure 5.81 Atomic Force Microscope (AFM) images of Teflon surface (a) unstretched and

(b) 200%-stretched. The surface profiling line is shown by the dashed line. The surface elevations corresponding to the unstretched and stretched samples are shown in panels (c) and (d), respectively (Lee et al. 2013). Reprinted with permission. Copyright (2013) American Chemical Society.

20

5.10 Bouncing Prevention: Dynamic Electrowetting

237

Table 5.2 RMS roughness (Rq ) and the surface area ratio R. The averaged values of Rq were obtained using the data from 16 different areas. Rq (nm)

(a) Unstretched (b) 200%-stretched

20 × 20 µm2

5 × 5 µm2

R

209.9 245.6

157.8 223.1

1 8.0

bound PTFE fibers are clearly seen lying in between the ridges. When the Teflon tape is stretched, the forces to be overcome are not the strong chemical bonds in the polymer chains, but the van der Waals attractions between PTFE crystals. Compared with chemical bonds, van der Waals attractions are so weak that they can be easily overcome by stretching. The SEM image (Fig. 5.80b) shows that the stretched ridges are crisscrossed by micro-cracks resulting from stretching. This increases the surface area ratio R compared to the unstretched Teflon and is responsible for the superhydrophobicity. The three-dimensional surface topography images of the unstretched and stretched Teflon (# = 200%) obtained by atomic force microscope (AFM) are shown in Figs. 5.81a, c and 5.81b, d, respectively. The roughness of the unstretched Teflon measured by AFM was 209.9 nm (when measured using a 20 × 20 µm2 sample) or 157.8 nm (when measured using a 5 × 5 µm2 sample), whereas it was higher, 245.6 nm or 223.1 nm, respectively, in case of the stretched Teflon. The surface area ratio R = cos θ0,stretched / cos θ0,unstretched of the stretched Teflon found from the Wenzel equation [see Eq. (1.45) in Section 1.8 in Chapter 1; θ0 is the contact angle measured without the electric field] is also listed in Table 5.2. Drop impact experiments with and without application of the electric field were performed to investigate the ability of the unstretched and stretched Teflon (strain = 200%) to repel or impale impacting water drops. During these experiments, various wetting regimes such as complete rebound, partial rebound, non-rebound and deposition of drops were observed. With no voltage applied at 0 kV, for both the unstretched (Fig. 5.82a) and stretched Teflon (Fig. 5.83a) it was found that when a water drop impacts onto a surface, it first deforms and flattens into a pancake shape. After that, the receding motion begins when the radius of the pancake diminishes. Finally, a drop can rebound off the surface. The drop remains completely intact during the impact and does not break into smaller water droplets. However, the rebounding heights hmax of water drops at V = 0 kV were different in the case of the unstretched and stretched Teflon (Fig. 5.84a). The rebounding height was higher for the stretched Teflon (hmax ∼ 5.13 mm) and lower for the unstretched one (hmax ∼ 4.16 mm). After applying a potential of 2 kV to the unstretched Teflon (Fig. 5.82b), it was found that although the spreading process was similar to that at 0 kV (Fig. 5.82a), the rebound changed and the drop separated into two parts. The rebound height at 2 kV

238

Drop Impact onto Dry Surfaces with Complex Morphology

(a)

Rebound hmax

(b)

(c)

Partial wetting

Stick

Figure 5.82 Snapshots of drop impact onto the unstretched Teflon at (a) V = 0 kV, (b) V = 2 kV, (c) V = 6.0 kV. Time interval between the frames is 15 ms, scale bar is 2 mm (Lee et al. 2013). Reprinted with permission. Copyright (2013) American Chemical Society.

(hmax ≈ 3 mm) was diminished compared to the one at 0 kV (hmax ≈ 4.16 mm). Then the voltage was systematically increased up to 6.0 kV. At 6.0 kV (Fig. 5.82c), there was no rebound on the unstretched Teflon and the water drop stayed pinned to the surface having the equilibrium contact angle of 78 ± 3◦ . So the voltage of 6.0 kV is considered to be the EW voltage for the unstretched Teflon. Hence, EW decreased the equilibrium water contact angle on the unstretched Teflon from 96 ± 4◦ to 78 ± 3◦ at the EW voltage of 6.0 kV. After applying a potential of 2 kV to the stretched Teflon (Fig. 5.83b) it was found that the impacting water drop spreads first, then recedes and rebounds off the surface, while a tiny part of it is still pinned to the surface. Even though the water drop rebounds (a)

Rebound

(b)

Partial wetting

(c)

Stick

Figure 5.83 Snapshots of drop impact onto the 200%-stretched Teflon at (a) V = 0 kV, (b) V = 2 kV, (c) V = 3 kV. Time interval between the frames is 15 ms, scale bar is 2 mm. The inter-electrode line was parallel to the direction of the sample stretching (Lee et al. 2013). Reprinted with permission. Copyright (2013) American Chemical Society.

239

5.10 Bouncing Prevention: Dynamic Electrowetting

(a) 6

(b) Unstretched

4 3 2

4 3 2

1

1

0

0 0

1

2

3

4

Voltage [kV]

5

Unstretched (orthogonal-dir.) Unstretched (parallel-dir.) 200%-stretched (orthogonal-dir.) 200%-stretched (parallel-dir.)

5

200%-stretched

hmax [mm]

hmax [mm]

5

6

6

0

1

2

3

4

5

6

Voltage [kV]

Figure 5.84 Maximum rebound height of a drop after impact onto the unstretched and the

200%-stretched Teflon surfaces. The experimental data are shown by symbols spanned by solid lines. (a) The inter-electrode line was in the direction orthogonal to the main ridges seen in Figs. 5.80 and 5.81. The theoretical predictions (discussed after the experimental findings) are shown by dashed lines. For the unstretched surface P = 0.098 and for the 200%-stretched one, P = 0.064. (b) The data sets acquired with the inter-electrode line orthogonal to the direction of the main ridges seen in Figs. 5.80 and 5.81 (filled symbols) and in the direction of the main ridges seen in Figs. 5.80 and 5.81 (open symbols) (Lee et al. 2013). Reprinted with permission. Copyright (2013) American Chemical Society.

off the surface, the rebound height is reduced from hmax ∼ 5.13 mm at V = 0 kV to hmax ∼ 1.8 mm at V = 2 kV (Fig. 5.84a). At 3 kV (Fig. 5.83c) water drops strongly stick to the surface without any rebound, so this voltage is considered as the EW voltage for the stretched Teflon. Such water drops possess the equilibrium contact angle of 67 ± 5◦ . Hence, EW decreased the water contact angle of the stretched Teflon from 147 ± 5◦ to 67 ± 5◦ at the EW voltage of 3 kV. In comparison, a higher EW voltage (∼6.0 kV) was required to reduce the equilibrium water contact angle on the unstretched Teflon from 96 ± 4◦ to 78 ± 3◦ , and the corresponding hydrophobic to partially hydrophilic switching angle difference was only 18 ± 4◦ . On the other hand, a comparatively lower EW voltage (∼3 kV) was required to reduce the equilibrium water contact angle of the stretched Teflon from 147 ± 5◦ to 67 ± 5◦ , and the superhydrophobic to partially hydrophilic switching angle difference was significantly larger ∼80 ± 5◦ . It should be emphasized that the directional structure of the unstretched and stretched Teflon seen in Figs. 5.80 and 5.81 does not have any significant effect on drop bouncing from these surfaces at any voltage applied, as the comparison of the results corresponding to the two orientations of the inter-electrode line relative to the ridges reveals in Fig. 5.84b. This is probably partially related to the fact that the static contact angle is also only weakly affected by the surface directionality, as Figs. 5.79b and 5.79c show. Electrowetting phenomena arise due to the change of the equilibrium contact angle when a voltage V is applied. In standard electrowetting experiments a sessile drop is located on a dielectric substrate and attached to an electrode. When a potential difference V exists between the bottom of the dielectric layer and the drop, its equilibrium

240

Drop Impact onto Dry Surfaces with Complex Morphology

contact angle depends on V and is given by the Young–Lippmann formula (Karniadakis et al. 2005, Mugele and Baret 2005) cos θeV = cos θe0 +

Cs 2 V 2γ

(5.63)

where θeV and θe0 are the equilibrium contact angles when voltage V is applied, or no voltage is applied, respectively, and γ is the surface tension of liquid. In addition, the capacitance Cs of the dielectric layer of thickness d in the standard EW experiment with a sessile drop is given as Cs =

ε 4π d

(5.64)

where the Gaussian (CGS) units are used, and ε is the dielectric permittivity of the dielectric layer underneath the drop. According to Eq. (5.63), at a non-zero voltage V the equilibrium contact angle of a drop diminishes, which means that it is stretched over the dielectric layer. This stems from the fact that the counter-ions always present in water almost immediately create the attached Stern layer (Russel et al. 1989) at the dielectric surface when voltage is applied, which reduces the interfacial energy of the solid/liquid interface, and thus affects the horizontal force balance expressed by Young’s equation (1.44) in Section 1.8 in Chapter 1, which results in Eq. (5.63). However, in the present experiments a dynamic, rather than a static situation corresponding to drop impact is involved, as sketched in the upper panel in Fig. 5.78. A drop of a volume-equivalent radius Re rapidly spreads after a normal impact at the surface and the maximum spread radius ai of the liquid footprint at the surface is given by the following expression (Scheller and Bousfield 1995, Yarin 2006)   We 0.166 ai = 0.61 (5.65) Re Oh where the Weber and Ohnesorge numbers We and Oh are defined as We =

2ρReV0 2 , γ

Oh =

μ (2ργ Re )1/2

(5.66)

with ρ and μ being liquid density and viscosity, respectively, and V0 the impact velocity [see Eq. (4.84) in Section 4.4 in Chapter 4 and Eq. (1.1) in Section 1.2 in Chapter 1]. It should be emphasized that Eq. (5.65) is valid for inertial rather than wettabilitydriven drop spreading and therefore, does not include the contact angle. The following values of the dimensionless groups are typical of the present experiments: We = 10 and Oh = 0.004. Due to the fact that spread-out drops are subjected to surface tension forces and the tendency to restore the equilibrium contact angle, receding motion sets in. After a sufficiently strong impact, this motion can even result in drop bouncing due to the significant inertia and low viscous dissipation (Yarin 2006), as sketched in the upper panel in Fig. 5.78. (Note that the ratio of the kinetic energy to viscous dissipation is of the

5.10 Bouncing Prevention: Dynamic Electrowetting

241

Z

−b

−a

a

b dielectric

+U

−U −id

Figure 5.85 Planar electrostatic problem: complex z-plane corresponding to the dielectric layer

ABCC B A (Lee et al. 2013). Reprinted with permission. Copyright (2013) American Chemical Society.

order of the impact Reynolds number Re = ρ2ReV0 /μ = We1/2 /Oh which is close to 828; thus dissipation is negligible in the present case.) These phenomena proceed on the scale of the order of τH = 10–100 ms. The hydrodynamic time scale τH is much longer than the charge relaxation time τC = ε/(4π σe ), with σe being the electric conductivity of liquid, since τC is of the order of 1 µs (Reznik et al. 2004). Since τC  τH , the counter-ions arrive at the surface of the polarized dielectric underneath the drop almost immediately, and the electric situation at the drop/dielectric interface during the spreading and receding stages does not differ from that in the standard static electrowetting experiment. However, the electric connection of the dielectric layer depicted in Fig. 5.78 is different from the one in the standard static electrowetting, and thus the capacitance given by Eq. (5.64) is inapplicable. To calculate the capacitance corresponding to Fig. 5.78, consider the planar electrostatic problem depicted in Fig. 5.85. We use complex variable z = x + iy, where i is the imaginary unit. The strip ABCC B A corresponding to the dielectric layer in Fig. 5.85 is mapped onto the upper half-plane in the complex plane t = ξ + iη by the following conformal mapping function z=−

d ln t π

(5.67)

such that point A is mapped to t = ∞, point B to t = 1, and point C to t = 0, which is depicted in Fig. 5.86. A drop would occupy in Fig. 5.86 a domain in the upper halfplane y > 0, which is much larger than the thickness of the dielectric layer d, as well as the inter-electrode distance 2a, or the electrode size b − a. We are interested in the formation of the Debye layer (Russel et al. 1989) at the drop bottom in contact with the dielectric layer at y = 0. Since the thickness of the Debye layer is on the 10 nm scale (Russel et al. 1989), while the sizes a, b, and d are on the 100 µm scale, and the drop size is on the 1 mm scale, one can effectively assume that liquid would occupy the entire upper half-plane y > 0 and the Debye layer stretches over −∞ < x < ∞.

242

Drop Impact onto Dry Surfaces with Complex Morphology

η t t−b A′

t−a −1 ta B′

tb

0

1

C′ C

B

ξ A

Figure 5.86 Complex upper half-plane corresponding to the dielectric domain of Fig. 5.85 (Lee

et al. 2013). Reprinted with permission. Copyright (2013) American Chemical Society.

According to Fig. 5.85, the boundaries of the left-hand side electrode raised to potential U correspond to z−b = −b – id and z−a = −a – id, whereas the boundaries of the right-hand side electrode at potential –U correspond to za = a – id and zb = b – id (note that b > a). According to Eq. (5.67), these boundaries are mapped to t−b = −exp(π b/d ), t−a = −exp(π a/d ), ta = −exp(−π a/d ) and tb = −exp(−π b/d ), respectively, as shown in Fig. 5.86. Conformal mapping of Eq. (5.67) is equivalent to the following coordinate transformation     πy πy −π x −π x cos , η = −exp sin . (5.68) ξ = exp d d d d The electric potential φ in the dielectric domains in Figs. 5.85 and 5.86 is found from the Laplace equation having the similar form in both planes due to the conformal mapping, namely, ∂ 2φ ∂ 2φ + 2 = 0 (in z), 2 ∂x ∂y

∂ 2φ ∂ 2φ + 2 = 0 (in t ). 2 ∂ξ ∂η

(5.69)

At the domain boundary φ = 0 everywhere, except the electrodes, where φ = U at the left-hand side electrode and φ = −U at the right-hand side electrode. The solution of the Laplace equation in the upper half-plane t is given by Poisson’s integral formula (Polya and Latta 1974), which for the present boundary conditions yields ηU φ(ξ , η) = π



t−a

t−b

dζ (ζ − ξ )2 + η2

 − ta

tb

dζ (ζ − ξ )2 + η2

 (5.70)

where ζ is a dummy variable. Evaluating the elementary integrals in Eq. (5.70), after several transformations and returning to the coordinates x and y using Eqs. (5.68), we arrive at the following expression for the potential        t−a − ξ t−b − ξ tb − ξ U arctan − arctan − arctan φ(x, y) = π η η η   (5.71) ta − ξ . + arctan η An example of the potential field (5.71) is shown in Fig. 5.87.

243

5.10 Bouncing Prevention: Dynamic Electrowetting

POTENTIAL 0.875 0.75 0.625 0.5 0.375 0.25 0.125 0 −0.125 −0.25 −0.375 −0.5 −0.625 −0.75 −0.875

2

y

1

0

−1 −3

−2

−1

0 X

1

2

3

Figure 5.87 Potential distribution according to Eq. (5.71). Potential φ is rendered dimensionless

by U , coordinates x and y – by the dielectric layer thickness d. The following electrode parameters are used: a/d = 0.5 and b/d = 2 (Lee et al. 2013). Reprinted with permission. Copyright (2013) American Chemical Society.

Calculating the derivative ∂φ/∂y at y = 0 using Eq. (5.71), one can find the surface density of the free charges σion on the conducting side (counter-ions accumulated at the drop bottom) in contact with the top side of the dielectric as σion

 ε ∂φ  ε 2U e−πx/d [cosh(π b/d ) − cosh(π a/d )] = = 4π ∂y y=0 4π d ×

[e−2πx/d

+1+

2e−πx/d

(1 − e−2πx/d ) . cosh(π b/d )][e−2πx/d + 1 + 2e−πx/d cosh(π a/d )] (5.72)

The surface density of the free charges predicted by Eq. (5.72) and rendered dimensionless as S = σion /{2U (ε/4π d )[cosh(π b/d ) − cosh(π a/d )]} is illustrated in Fig. 5.88. It shows that anions are located at the drop bottom part above the positive electrode (see Fig. 5.85), i.e. at x < 0, whereas cations are located at the drop bottom part above the negative electrode, i.e. at x > 0. If one assumes sufficiently small drop size, 2π x/d < 1, the latter expression is simplified as σion =

πx ε [cosh(π b/d ) − cosh(π a/d )] U 4π d [1 + cosh(π b/d )][1 + cosh(π a/d )] d

(5.73)

and, accordingly, the electric energy stored in the drop bottom becomes Eel = −

π |x| [cosh(π b/d ) − cosh(π a/d )] ε U2 . 4π d [1 + cosh(π b/d )][1 + cosh(π a/d )] d

(5.74)

Drop Impact onto Dry Surfaces with Complex Morphology

0.0015 0.0010 0.0005 S

244

0.0000 −0.0005 −0.0010 −0.0015 −3

−2

−1

0 x

1

2

3

Figure 5.88 The surface density of the free charges at a/d = 0.5 and b/d = 2 (Lee et al. 2013). Reprinted with permission. Copyright (2013) American Chemical Society.

The electric energy stored per unit area of the drop bottom found from Eq. (5.74), then becomes π ai [cosh(π b/d ) − cosh(π a/d )] ε U2 4π d [1 + cosh(π b/d )][1 + cosh(π a/d )] d ε π ai U 2 4(e−πa/d − e−πb/d ) ≈− 4π d d 2

Eel = −

(5.75)

where it was used that b/d > a/d and assumed that these ratios are sufficiently larger than 1. Equation (5.75) shows that in the drop impact experiments, the “static” equations for the equilibrium contact angle [Eqs. (5.63) and (5.64)] are replaced by the following “dynamic” equations for the unstretched Teflon cos θeV = cos θe0,unstretched +

Cd 2 V 2γ

(5.76)

where V = 2U , and the capacitance in the present experiment is given by Cd =

ε π ai (e−πa/d − e−πb/d ) . 4π d d

(5.77)

Since b/d > a/d and both ratios are expected to be larger than 1, Cd  Cs , as the comparison of Eqs. (5.64) and (5.77) shows. Therefore, in the drop impact experiments the voltage values corresponding to the visible electrowetting effect should be significantly higher than in the standard static experiments. In the case of stretched Teflon, one can expect that an additional surface roughness will appear, as the experimental data in Figs. 5.80 and 5.81 shows. Therefore, when calculating the chemical and electric energy of the drop bottom, one should account for

5.10 Bouncing Prevention: Dynamic Electrowetting

245

the increased effective surface area. Then, Eq. (5.76) is replaced by the following one cos θeV = cos θe0,stretched +

Cd RV 2 . 2γ

(5.78)

Equations (5.76)–(5.78) imply that during drop impact water easily penetrates all the grooves and cracks at the surface, i.e. the Wenzel state is realized in the spread-out wet spot, as it was demonstrated in Lembach et al. (2010), Sinha-Ray and Yarin (2014), Weickgenannt et al. (2011a), Weickgenannt et al. (2011b) and Sahu et al. (2012); Section 5.7 in the present chapter. At the moment of the maximum spread-out the restoring force Fr acting at the drop is applied at the contact line and equal to Fr = γ (cos θi − cos θeV )2π ai

(5.79)

where the spread-out contact angle θi is found below. If one approximates the spread-out drop shape as a spherical segment of radius Ri , the latter is equal to Ri =

41/3 Re 1/3

[(1 − cos θi )2 (2 + cos θi )]

(5.80)

due to the mass conservation. On the other hand, due to the geometric considerations ai = Ri sin θi

(5.81)

and thus, according to Eqs. (5.65), (5.80) and (5.81), we arrive at the following equation which determines the spread-out contact angle θi resulting from drop impact   We 0.166 41/3 sin θi = 0.61 . (5.82) 1/3 Oh [(1 − cos θi )2 (2 + cos θi )] It should be emphasized that in the approximate theory considered above, the spread-out contact angle θi plays the role of the receding contact angle; its value is close to 20◦ . The potential energy available for drop receding from the spread-out shape is of the order of Fr ai . If one neglects viscous dissipation during the receding motion (which is possible in the present case), the height h at which the drop bounces is found from the energy balance Fr ai ∼ mgh, where m is the drop mass, and g is gravity acceleration. Using Eqs. (5.79)–(5.81) and the energy balance, one finds the bouncing height of the drops as

2   We 0.166 γ (cos θi − cos θeV ) (5.83) 0.61 h=P ρgRe Oh where P is the dimensionless factor which is less than 1 to account for the viscous losses and the kinetic energy of the oscillatory motions in the bouncing drops. Note that, as shown above, the viscous losses are practically negligible in the present case and the value of P is predominantly determined by the oscillatory motions in the bouncing drops. In Eq. (5.83) the angle θi is found from Eq. (5.82), and the angle θeV , from

246

Drop Impact onto Dry Surfaces with Complex Morphology

Eqs. (5.76) and (5.77) in the case of the unstretched Teflon. In the case of the stretched Teflon, the only change is that Eq. (5.76) is replaced by Eq. (5.78). Equation (5.83) shows that as the voltage U increases, and thus the angle θeV decreases and cos θeV increases, the height h decreases. The largest bouncing height is achieved at V = 0, when θeV = θe0 where in the cases of the unstretched and stretched Teflon, θe0 = θe0,unstretched or θe0 = θe0,stretched , respectively. The theoretical predictions are compared to the experimental data in Fig. 5.84a. The calculations were done with We = 10.43, Oh = 0.0039, εd = 2.1, γ = 70.76 mN/m, d = 100 µm and a/d = 2.60 (for the unstretched Teflon), and d = 70 µm and a/d = 2.83 (for the stretched Teflon). Also, it was taken ρ = 1000 kg/m3 and Re = 0.00096 m, and assumed that the ratio b/d is sufficiently large compared to a/d to have a negligible effect on the results [see Eq. (5.77)]. The dimensionless factor P in Eq. (5.83) was chosen as P = 0.098 for the impacts on the unstretched Teflon, and P = 0.064 for the impacts on the 200%-stretched Teflon. The values of P were determined from the corresponding cases when no voltage was applied, or a non-zero voltage was applied. They are of the same order of magnitude but the value of P for the 200%-stretched surface is lower than the one for the unstretched surface. This is presumably due to the fact that the rougher stretched surface slightly increases viscous losses, which are still small. The comparison of the experimental results with the predictions reveals a reasonable agreement. The values of P of the order of 0.1 show that only about 10% of the surface energy stored in the spread-out drop is spent for bouncing. The remaining 90% is split between viscous losses (relatively small for water) and the kinetic energy of the vigorous drop oscillations in flight visible in the snapshots in Figs. 5.82 and 5.83. Since viscous losses are negligible in the present case, the value of P can be evaluated using the equation for the distribution of the supplied potential energy, namely mgH = mgh + mVosc 2

(5.84)

where H is the height from which the original drop was released and Vosc is of the order of the magnitude of the speed of the oscillatory motion in the bounced drop. Then, P≈

Vosc 2 h ≈1− 2 H V0

(5.85)

where use is made of the fact that V02 ∼ gH . The analysis of the video recordings corresponding to Figs. 5.82 and 5.83 reveals typical values of Vosc ≈ 0.59–0.6 m/s in drops which bounced off the unstretched and stretched surfaces, while the impact velocity V0 = 0.62 m/s. Then, according to Eq. (5.85), one obtains P = 0.0635 to 0.0944, which is in very good agreement with the values of P = 0.064 and 0.098 used in the theoretical predictions in Fig. 5.84a. Summarizing the findings of the present section, note that hydrophobic Teflon was converted into superhydrophobic Teflon by simply stretching it. The dynamic electrowetting (EW) depicted in Fig. 5.78 does not involve an electrode directly attached to the drop as in the case of the static EW. As a result, the capacitance is dramatically reduced in the present case and the EW phenomena can be observed only in

5.11 References

247

a much higher voltage range (in the kV range instead of the V range) compared to the ordinary static electrowetting. A sharp EW response of the 200%-stretched Teflon was found experimentally and confirmed theoretically, with the full EW with the complete suppression of rebound of water drops achieved at the voltage of 3 kV. Accordingly, the equilibrium contact angle was changed from 147 ± 3◦ (superhydrophobic) at V = 0 kV to 67 ± 5◦ (partially hydrophilic) at 3 kV. The EW response was weaker for the unstretched Teflon, with the full EW with the complete suppression of rebound of water drops achieved at the voltage of 6.0 kV. Accordingly, the equilibrium contact angle was changed from 96 ± 4◦ to 78 ± 3◦ at 6.0 kV.

5.11

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Yarin, A. L. (2006). Drop impact dynamics: splashing, spreading, receding, bouncing . . . , Annu. Rev. Fluid Mech. 38: 159–192. Yarin, A. L., Chase, G. G., Liu, W., Doiphode, S. V. and Reneker, D. H. (2005). Liquid drop growth on a fiber, AIChE J. 52: 217–227. Yarin, A. L., Pourdeyhimi, B. and Ramakrishna, S. (2014). Fundamentals and Applications of Micro- and Nanofibers, Cambridge University Press. Yarin, L. P., Mosyak, A. and Hetsroni, G. (2009). Fluid Flow, Heat Transfer and Boiling in MicroChannels, Springer, Heidelberg. Zhao, Y. and Cho, S. K. (2007). Micro air bubble manipulation by electrowetting on dielectric (EWOD): transporting, splitting, merging and eliminating of bubbles, Lab Chip 7: 273–280. Zheng, Q.-S., Yu, Y. and Zhao, Z.-H. (2005). Effects of hydraulic pressure on the stability and transition of wetting modes of superhydrophobic surfaces, Langmuir 21: 12207–12212. Zhu, L., Xu, J., Xiu, Y., Sun, Y., Hess, D. W. and Wong, C.-P. (2006). Electrowetting of aligned carbon nanotube films, J. Phys. Chem. B 110: 15945–15950.

Part II

Drop Impacts onto Liquid Surfaces

6

Drop Impacts with Liquid Pools and Layers

In this chapter drop impacts onto a liquid layer of the same liquid as in the drop are considered. The chapter begins with consideration of such weak drop impacts on a liquid layer that they result only in capillary waves propagating over the surface. An interesting feature of these waves is that they are self-similar (Section 6.1). In the following Section 6.2 crown formation in strong (high-velocity) impacts onto thin liquid films is considered. Normal and oblique impacts of a single drop onto a wet wall are studied, as well as crown–crown interaction in sprays impacting the wall. Also, the evolution of the free rim on top of the crown is described. Then, in Section 6.3 drop impacts onto a thick liquid layer are considered and the dynamics of the crater formation is explained. Drop impacts onto a wet wall leave a residual liquid film on the wall which is addressed in Section 6.4. Drop impacts onto deep liquid pools produce a plethora of interesting morphological structures considered in Section 6.5. In the following Section 6.6 bending instability of a free rim is considered and the splashing mechanism is discussed. Splashing resulting from impacts of drop trains one-by-one is discussed in Section 6.7, where its physical mechanism and the link to splashing of a single drop impacting onto a liquid layer are elucidated. Several other regimes of drop impact are also mentioned.

6.1

Drop Impact onto Thin Liquid Layer on a Wall: Weak Impacts and Self-similar Capillary Waves Consider patterns of capillary waves propagating over the free surface of a thin liquid film from the point where it was impacted normally by a tiny droplet or a stick (Fig. 6.1), as an example of a relatively weak (low-velocity) impact. For scales of the order of several millimeters the gravity effect on these waves is negligibly small, and for time scales of the order of several milliseconds viscosity effects can also be neglected. The dynamics of the wave propagation is then governed by the following set of quasi-onedimensional equations [cf. Eqs. (1.36) and (1.37) in Section 1.7 in Chapter 1] ∂rh ∂rhV + = 0. ∂t ∂r   2   ∂V ∂ ∂ h ∂V +V =σ h 2 . ρh ∂t ∂r ∂r ∂r

(6.1) (6.2)

256

Drop Impacts with Liquid Pools and Layers

Figure 6.1 Top view of a pattern of capillary waves taken 5 ms after impact of a copper stick onto

an ethanol film of 1.4 mm thickness. The entire picture covers an area of about 25 mm × 35 mm (Yarin and Weiss 1995). Reprinted with permission.

Equation (6.1) is the continuity equation and Eq. (6.2) expresses the momentum balance, which describes the competition of the inertial forces and surface tension. The waves are assumed to be axisymmetric and propagating outwards along the radial coordinate r reckoned from the center of impact. The fluid velocity is denoted by V , the film thickness by h, the density by ρ, the surface tension coefficient by σ , and time by t. If one considers the waves as small perturbations propagating over a liquid layer initially at rest, and of unperturbed thickness h0 , then, linearizing Eqs. (6.1) and (6.2) for small perturbations of the film thickness χ (r, t ), such that h = h0 [1 + χ (r, t )],

(6.3)

one obtains the following equation for χ  3  ∂ 2χ ∂ χ a2 ∂ r 3 = 0. + 2 ∂t r ∂r ∂r The parameter a combines all the given physical parameters as per   σ h0 1/2 a= ρ

(6.4)

(6.5)

and fully determines the wave propagation. Its units are: [a] = m2 /s. The initial impactor is assumed to be pointwise, which means that the wave pattern is considered at distances r much larger than its diameter. Since no length scale is given, an arbitrary one, L, can be used. The dimensionless wave pattern should be of the form (Yarin 2007, 2012)   t r , . (6.6) χ= f L L2 /a However, in the final result an arbitrary length scale L should automatically disappear, which means that the dimensionless function f in Eq. (6.6) should depend not on its two

6.2 Thin Liquid Layer on a Wall: Crown Formation

257

variables separately, but on their specific combination, namely (Yarin 2007), η=

r/L [t/(L2 /a)]1/2

=

r (at )1/2

.

(6.7)

The corresponding self-similar wave pattern χ = F (η) should satisfy Eq. (6.4), which yields F IV +

1 η2 3 F + F + ηF = 0. η 4 4

(6.8)

The solution should correspond to the following initial perturbation t = 0, χ = 4π S

δ(η) 2π η

(6.9)

where the dimensionless factor S corresponds to the impact intensity, and δ(η) is the delta function. Using the solution of Eq. (6.8) and the initial condition (6.9), one finds that for large η the axisymmetric waves are described by the following expression      1 2 1 1 1 2 1 2S cos η + π + sin η + π (6.10) χ= (1/4) η3/2 4 8 4 8 where the gamma function of the argument 1/4 equals 3.6256. The corresponding fluid velocity is given by       a 1/2 S 1 2 1 1 2 1 1 η π + sin η π . (6.11) + + cos V = t (1/4) η1/2 4 8 4 8 The self-similar capillary waves (6.10) with η = r/(at )1/2 found in Yarin and Weiss (1995) are in good agreement with the experimental data of that work. Moreover, the solution shows that wave patterns similar to those of Fig. 6.1 shot at different time moments can be collapsed onto a single self-similar pattern given by Eq. (6.10) which is shown in Fig. 6.2. Indeed, the theory predicts the η(r) dependence corresponding to the depressions of the wave profile, in the form of a straight line (6.7). For a time delay after impact of t ≈ 5 ms, one obtains from Eq. (6.7) η/r = 1/(at )1/2 = t −1/2 (σ h0 /ρ)1/4 = 995 m−1 (σ = 23 mN/m, ρ = 790 kg/m3 , h0 = 1.4 × 10−3 m, as in the experiment). The predicted value of the slope of 995 m−1 is in good agreement with the one found experimentally and listed in the caption of Fig. 6.2, 937 m−1 .

6.2

Strong Impacts of Drops onto Thin Liquid Layer: Crown Formation The recent progress in the research on the drop and spray impact on a wetted wall can be attributed to rapid development of experimental techniques, allowing one to obtain high-quality images of impacting drops and to collect detailed information about the splashing threshold (Walzel 1980, Wang and Chen 2000), drop shape, crown propagation, fingering of the rim, etc. (Levin and Hobbs 1971, Macklin and Metaxas 1976, Yarin and Weiss 1995, Cossali et al. 1997, 1999), as well as to measure distributions of

258

Drop Impacts with Liquid Pools and Layers

15

10 η 5

0 5

10 r (mm)

15

Figure 6.2 Capillary waves: comparison between the experiment and theory. Dark rings in Fig. 6.1 were measured at the radii r corresponding to the vertical lines; they correspond to η-values where the profile (6.10) has maximum curvature (horizontal lines). The intersections of these two families of lines lie on a straight line (which is already a manifestation of self-similarity) with the slope η/r ≈ 937 m−1 (Yarin and Weiss 1995). Reprinted with permission.

the secondary drops in the impinging spray using the phase Doppler technique (Mundo et al. 1998, Roisman et al. 1999) or a photocamera (Shin and McMahon 1990). Several reviews (Rein 1993, Prosperetti and O˜guz 1993, Yarin 2006, Josserand and Thoroddsen 2016) can be recommended for further details of the phenomena. Detailed data about the above-mentioned phenomena, including the velocity field in the drop and film, can also be obtained using direct numerical simulations of single drop impact (Harlow and Shannon 1967, Schelkle et al. 1999, Weiss and Yarin 1999, Rieber and Frohn 1999) or even of multiple drop impacts (Böhm et al. 1999). These data can be especially important at the initial stages of drop impact, at times of order of t1 = D/V0 , when the drop shape is complicated and the theoretical analysis of the problem is difficult. Here D and V0 are the initial drop diameter and impact velocity, respectively. However, experiments show that the duration of the crown evolution is usually much larger than t1 . Moreover, at large times after drop impact, the radius of the crown is much larger than the thickness of the crown or the thickness of the film on the wall. Therefore, the computations of the drop impact require a fine mesh, a large computation domain, and are very time consuming. On the other hand, the small thickness of the crown and the film on the wall at large times, t t1 , makes the phenomena very attractive for theoretical modeling as remote asymptotics at distances much larger than D. Strong impacts of drops correspond to high impact velocity V0 and are prone to produce crowns. In the theoretical work of Yarin and Weiss (1995) the impact of a drop onto a liquid film was considered. The crown-like sheet formed by the impact was described as a kinematic discontinuity of the liquid film. The velocity fields in the liquid film on the wall and the expansion of the base of the crown in the axisymmetric case of a normal

6.2 Thin Liquid Layer on a Wall: Crown Formation

259

drop impact were expressed analytically. Following their theory, the base of the crown can be considered as a kinematic discontinuity of radius RB , subdividing the liquid film on the wall into two parts: an outer unperturbed film of constant film thickness h f and radial velocity V f = 0; and an inner part of thickness hl and radial velocity Vl , where at large times V l (r, t ) =

r , t +τ

hl (r, t ) =

η , (t + τ )2

RB (t ) = β(t + τ )1/2

(6.12)

(see subsection 4.1.2 in Chapter 4). It should be emphasized that Eqs. (6.12) are found as the solutions of the quasi-one-dimensional equations of the lamella flow (1.36) and (1.37) in Section 1.7 in Chapter 1. Here the overbars denote dimensionless values rendered dimensionless with the drop initial diameter D used as a length scale and its impact velocity V0 as a velocity scale. The radial coordinate is denoted r, and the time reckoned from the impact moment is denoted t. Parameters β, τ and η are some dimensionless constants determined by the parameters of the impact: the Reynolds number, the Weber number and the dimensionless initial film thickness h f . Equations (6.12) also follow from the general solution for the lamella motion in subsection 2.1.1 and the jump conditions at the kinematic discontinuity, established in Section 2.2 in Chapter 2. The expression (6.12) for the radius of the base of the crown RB is in good agreement with the experimental data of Yarin and Weiss (1995) and Cossali et al. (1999); see Section 6.7 in this chapter. Moreover, the predictions of RB by Eq. (6.12) describe very accurately the results obtained by Rieber and Frohn (1999) using the three-dimensional numerical solution of the Navier–Stokes equations. In the work of Trujillo and Lee (2001) the theory of Yarin and Weiss (1995) for the propagation of the kinematic discontinuity was generalized taking into account the effect of the viscous forces. The model further improves the prediction of the propagation of the crown radius. However, the differences with the theory of Yarin and Weiss (1995) are not large. This indicates that in the case of high impact velocities the main factor affecting crown formation is the liquid inertia. In the study of polydisperse spray impingement (Tropea and Roisman 2000) it has been shown that the resulting flux of splashed droplets is not a simple superposition of single-drop impact events arising from the primary drops of the impacting spray. One of the reasons for this is the interaction of neighboring crowns. Such a situation is shown in Fig. 6.3. In this image, recorded with a high-speed camera, the impact of a polydisperse water spray onto the north pole of a steel spherical target is shown. Two neighboring splashing crowns can be clearly seen, as well as a drop impacting in the same region. The parameter characterizing the probability of crown interactions at the wall depends on the number flux of the impacting drops, on the rate of change of the crown diameter and the total time of the crown propagation. This lifetime of the crown is determined by the motion of the rim on top of it: its initial elevation and its descent due to the surface tension and gravity. The instant when the rim reaches the wall corresponds to the total time of the crown propagation, and is one of the key parameters for modeling a dense polydisperse spray impact.

260

Drop Impacts with Liquid Pools and Layers

crowns

Figure 6.3 Crowns formed due to polydisperse spray

Figure 6.4 Sketch of the crown produced by the drop

impact onto a rigid wall and their interactions. Image made using a high-speed camera (Roisman and Tropea 2002). Reproduced with permission.

impact and of the regions considered analytically (Roisman and Tropea 2002). Reproduced with permission.

The main aim of the present section is the description of the expansion of a crown ejected from the wetted wall due to impact of a liquid drop following Roisman and Tropea (2002) and Roisman et al. (2007). The impacting drop and the crown are shown schematically in Fig. 6.4. Four main regions are considered: the liquid lamella on the wall inside the crown (region 1 in Fig. 6.4), the undisturbed film on the wall outside the crown (region 2), the sheet, i.e. the crown body (region 3) and the free rim bounding the crown from above (region 4). The crown is ejected from the boundary between regions 1 and 2, a kinematic discontinuity, where the film thickness and the velocity of the liquid both experience a jump. The radius-vectors XB , XJ and XR (see Fig. 6.4) correspond to the front of the kinematic discontinuity, to the wall of the crown and to the centerline of the free rim, respectively. The crown is formed due to impact with a relatively high initial drop velocity V0 . Thus, in such cases, the Reynolds number Re = V0 D/ν and the Weber number We = ρDV02 /σ are much larger than unity; ν, ρ and σ are the kinematic viscosity, density and surface tension of the liquid, respectively. Moreover, the Reynolds number Reh = V0 h f /ν based on the thickness h f of the undisturbed film is also assumed to be large. Therefore, the phenomenon of the crown formation and propagation is assumed to be inertia dominated and viscosity effects are neglected in the present analysis. Note however, that even at the high values of We, Re and Reh the velocity gradients in the drop at the initial stage of the impact can be so high that the effect of viscosity cannot be neglected yet. The effect of the viscosity becomes apparent in the viscous boundary layer near the wall. The velocity of the liquid outside the boundary layer is of the order of the initial drop impact velocity V0 . The velocity field u in the viscous boundary layer near the wall can be estimated using the velocity distribution in the unsteady boundary layer corresponding to Stokes’ first (or Rayleigh) problem   z (6.13) u = V0 erf √ 2 νt

6.2 Thin Liquid Layer on a Wall: Crown Formation

261

where erf(•) is the error function, z is the coordinate normal to the wall. The shear stress √ √ at the wall can be thus estimated as τw = V0 ρ ν/ πt. The effect of the viscosity can be neglected when the inertial terms in the momentum balance of the liquid are much larger than the viscous drag forces (ρh f V02 Dτw ). The latter condition can be rewritten in the form t tν =

νD20 . π h2f V02

(6.14)

It is convenient to express time tν in the dimensionless form tν =

1 2

π h f Re

(6.15)

where the initial drop diameter D is used as a length scale, and the ratio D/V0 as a time scale. The theory is valid for the cases when time tν is smaller than the time of the initial drop deformation, which is of the order of D/V0 , or in the dimensionless form, when t ν < 1. In subsection 6.2.2 the theory of Yarin and Weiss (1995) for the dynamics of the liquid lamella on the wall and the propagation of the kinematic discontinuity is generalized for an arbitrary non-axisymmetric case. After that, the equations of the crown and rim motion are formulated. Also, an analytic expression for the shape of the crown is obtained for the case when viscosity and surface tension are negligibly small. The applications of the theory to various drop impact morphologies are considered in the following order. The normal impact of a single drop, oblique impact of a single drop and the interaction of two crowns are considered in subsections 6.2.1, 6.2.2 and 6.2.3, respectively. The results of the theoretical predictions are discussed in subsection 6.2.4. See also Roisman and Tropea (2002) for more detail.

6.2.1

Normal Impact of a Single Drop onto a Wetted Surface Consider a liquid drop of diameter D impacting with the impact velocity V0 onto a liquid film of constant thickness h f at rest. If the impact velocity is high enough, the initial deformation of the drop and of the film is followed by formation of a crownlike sheet. The geometry under consideration is axisymmetric, therefore, the base of the crown is an expanding circle with the center at the point of impact. The components of the velocities in the film normal to the kinematic discontinuity are simply the radial velocities, while the tangential components vanish. Consider the cylindrical coordinate system {r, φ, z} with base vectors {er , eφ , ez } and the origin fixed at the point of impact. Assume that the impact velocity is so high that the viscous and surface tension effects are negligibly small in comparison to the liquid inertia. The theory of Yarin and Weiss (1995) describes the velocity inside the kinematic discontinuity and the thickness of the film at times sufficiently larger than D/V0 in the dimensionless form (6.12), whereas the film of the thickness h f outside the kinematic discontinuity remains undisturbed. It can be shown that the solution of Eq. (2.55) in Section 2.2 in Chapter 2 yields in the axisymmetric case the radius of the kinematic

262

Drop Impacts with Liquid Pools and Layers

discontinuity RB in the form of (6.12). The velocity and the thickness of the sheet at the kinematic discontinuity can be obtained in dimensionless form by substituting Eq. (6.12) into Eqs. (2.71) and (2.72) in Section 2.2 in Chapter 2 and neglecting the terms associated with the surface tension   βη1/2 1/2 1/2 η e + h (t + τ )e ) (6.16) VB (t ) = r z f (t + τ )1/2 [η + h f (t + τ )2 ] η hB (t ) = + hf . (6.17) (t + τ )2 Here the drop diameter D is used as a length scale, the impact velocity V0 as a velocity scale and D/V0 as a time scale. Here and hereinafter overbars over dimensionless variables are omitted for brevity. On the right-hand side of the equations of motion of the crown, Eqs. (6.16) and (6.17), the gravity effects are not taken into account. If they are included and the equations of motion are integrated, the analytic solution for the shape of the crown takes the form

η $ (t − tB ) er # XJ (tB , t ) = β(tB + τ )1/2 1 + (tB + τ ) η + h f (tB + τ )2 + β(tB + τ )1/2

η1/2 h1/2 f η + h f (tB +

τ )2

(t − tB )ez −

Fr−1 (t − tB )2 ez 2 (6.18)

where the Froude number is defined as Fr =

V02 . gD

(6.19)

The solution for the thickness of the crown is obtained by substituting Eq. (6.18) into Eq. (2.81) in Section 2.2 in Chapter 2, taking the azimuthal angle ϕ as a parameter ξ and accounting for the axial symmetry of the problem  2 2 RB (tB ) G1 (tB , tB ) + G2 (tB , tB )  (6.20) hJ (tB , t ) = hB (tB ) RJ (tB , t ) G2 (t , t ) + G2 (t , t ) 1 B

2 B

where the radius of the crown wall and the functions G1 and G2 are defined as

η 1/2 $ (t − tB ) # 1+ RJ (tB , t ) = β(tB + τ ) (tB + τ ) η + h f (tB + τ )2

(6.21)

1 4(t − tB )η2 η(3tB − 5t − 2τ ) G1 (tB , t ) = τ + tB + # (6.22) $2 + 2(τ + tB )3/2 η + h f (tB + τ )2 η + h f (tB + τ )2 # $ (t − tB ) (ηh f )1/2 h f tB3 − (3tB + 2τ )(η + h f τ 2 )+t(η−3h f (tB +τ )2 ) + . G2 (tB , t ) = # $2 β Fr 2(τ + tB )1/2 η + h f (τ + tB )2 (6.23)

6.2 Thin Liquid Layer on a Wall: Crown Formation

263

Consider now the rim bounding the top of the crown. We assume the centerline of the rim to be a circle with the radius-vector x = XR (t ), see Fig. 6.4. The rim belongs to the crown, therefore the rim location can be expressed in the form XR (t ) = XJ (tR , t )

(6.24)

where parameter tR is a function of time t. Expression (6.24) means also that a material point located at a time instant tR at the kinematic discontinuity reaches the rim at an instant t. Denote by WR the total volume of liquid ejected from the film to the sheet (the crown wall) at the kinematic discontinuity during the time interval from t = 0 to tR tR WR (tR ) = 2π

RB (t )Q(t ) dt.

(6.25)

0

The static volumetric flux into the discontinuity is denoted as Q(t ) (see Section 2.2 in Chapter 2). Using the mass balance, at time instant t > 0 this entire volume WR (tR ) is accumulated in the rim or ejected from the rim as the jets and (via the capillary breakup of these jets) into the secondary droplets, see Fig. 1.6 in Section 1.10 in Chapter 1. If one assumes that the rim moves with the velocity (2.159) in Section 2.7 in Chapter 2 relative to the liquid in the crown, the total volume flux can be obtained as dWR = 2π RR (t )hR (t )UR (t ) dt

(6.26)

where RR is the dimensionless radius of the centerline of the rim and UR is the rim velocity. On the other hand, differentiating Eq. (6.25) with respect to tR , one obtains dWR = 2π RB (tR )Q(tR ). dtR

(6.27)

Equations (6.26) and (6.27) lead to the following differential equation for the time tR RR (t )hR (t )UR (t ) dtR (t ) = . dt RB (tR )Q(tR )

(6.28)

Equation (6.24) with the help of Eqs. (6.12), (2.55) in Section 2.2 in Chapter 2 and (6.28) yields  dtR (t ) 8 RR (t )h1/2 R (t ) = (6.29) dt We β 2 hB (tR ) where the radius of the centerline of the rim and the thickness of the sheet at the rim are defined as RR (t ) = XJ (tR , t ) · er , hR (t ) = hJ (tR , t ) respectively.

(6.30)

264

Drop Impacts with Liquid Pools and Layers

6.2.2

Oblique Impact of a Single Drop In this subsection the general theory described in Section 2.2 in Chapter 2 is applied to the case of impact of a single drop onto a uniform film moving with a constant velocity Vτ . The impact velocity V0 is directed normally to the wall. The initial diameter of the drop is denoted D. The impact angle θ is defined here as tan θ = Vτ /V0 = V τ . The axisymmetric case considered in subsection 6.2.1 corresponds to a zero angle of inclination. Consider the Cartesian coordinate system {x, y, z} with the base vectors {ex , ey , ez } fixed at the point of impact, with the axis x being directed against Vτ , and the axis z being normal to the wall. The velocity Vl in the lamella inside the crown [see Eq. (6.12)], is assumed to be unaffected by the outer flow. Then Vl and the velocity V f in the outer undisturbed film are expressed in the dimensionless form as Vl =

1 (xex + yey ), t +τ

V f = −Vτ ex

(6.31)

where, as in subsection 6.2.1, D is used as a length scale and V0 is used as a velocity scale. Here and hereinafter overbars denoting the dimensionless variables are omitted for brevity. The thickness hl of the inner film is expressed in the form Eq. (6.12), whereas the thickness h f of the outer film is constant. The solution of the equation describing propagation of the base of the crown, Eq. (2.55) in Section 2.2 in Chapter 2, can be obtained with the help of Eqs. (6.31) and (6.12) in the form √ √ (6.32) XB = [−Vτ (t + τ ) + C1 t + τ ]ex + C2 t + τ ey where C1 and C2 are the integration constants which are found from the initial conditions. Assuming an elliptical initial shape of the kinematic discontinuity just after the initial deformation of the drop, at the time of order t ≈ 1, the solution (6.32) yields the shape of the kinematic discontinuity in the following parametric form √   √ √ √ t +τ ex XB (ϕ, t ) = βx cos ϕ t + τ − Vτ (t + τ ) + Vτ 1 + τ t + τ + x0 √ 1+τ √ + βy sin ϕ t + τ ey (6.33) where ϕ ∈ [−π , π ] is the circumferential parameter (the azimuthal angle of polar coordinate system), x0 is the displacement of the center of the initial ellipse at the instant t ≈ 1; βx > 0, βy > 0 and τ are dimensionless constants. It is interesting to note that the shape of √ the kinematic√discontinuity (6.33) remains elliptic. The half-axes of the ellipse are βx 1 + τ and βy 1 + τ , and the ratio between these two half-axes is constant: γ = βy /βx . The unit vector n normal to the ellipse XB is expressed now in the form γ cos ϕex + sin ϕey . n=  γ 2 cos2 ϕ + sin2 ϕ

(6.34)

6.2 Thin Liquid Layer on a Wall: Crown Formation

265

Therefore the strength of the sink Q supplying liquid from the film on the wall into the kinematic discontinuity can be obtained in dimensionless form using Eq. (2.53) of Section 2.2 in Chapter 2 and Eqs. (6.31), (6.33) and (6.34) as (h1 + h2 )(βx + B cos ϕ)γ  , Q= √ 2 2 2 2 t + τ γ cos ϕ + sin ϕ

(6.35)

where the film thicknesses at the wall h1 and h2 are shown in Fig. 6.4 and √ x0 . B = Vτ 1 + τ + √ 1+τ

(6.36)

Note that the flux into the sink Q, defined in Eq. (6.35), is positive everywhere on the kinematic discontinuity only if βx > B. In this case the crown base is a full ellipse expressed by Eq. (6.33). Otherwise, the sheet is formed only on the part of the ellipse corresponding to ϕ ∈ [−ϕ ∗ , ϕ ∗ ] where ϕ ∗ = arccos(−βx /B).

(6.37)

The velocity of the crown sheet at the kinematic discontinuity is expressed as VB (ϕ, t ) =

h f Vτ hl XB (ϕ, t ) − ex (hl + h f )(t + τ ) hl + h f   hl h f (β cos ϕ + B)2 + γ 2 β 2 sin2 ϕ + ez . √ (hl + h f ) t + τ

(6.38)

The crown in the case of the oblique impact is a three-dimensional expanding surface. It can be expressed in the parametric form using Eqs. (6.33), (6.38) and the parameter ϕ instead of ξ in Eq. (2.55) in Section 2.2 in Chapter 2. In the present analysis we do not consider the motion of the rim, which in the case of oblique impact is a complex three-dimensional body. Nevertheless, the shape of the crown obtained can be used as an asymptotic shape for the case We → ∞.

6.2.3

Interaction of Two Crowns Consider two drops of diameter D1 and D2 = kd D1 impacting normally onto a wetted wall with the impact velocities V1 and V2 = kuV1 . The film has a constant thickness h f before impact and its velocity is V f = 0. Consider also the Cartesian coordinate system {x, y} in the wall plane with the base vectors {ex , ey }. The first drop impacts at the origin of the coordinate system (0, 0), the second one at point (x, 0). Denote by t the time interval between the impacts. Each impacting drop disturbs the film on the wall and produces a spot with the radially expanding flow. This flow is analyzed in the theory of Yarin and Weiss (1995) and is defined in Eq. (6.12). In the present coordinate system the velocities V1 and V2 , the thicknesses h1 and h2 of the film at these spots, as well as the crown radii RB1 and RB2

266

Drop Impacts with Liquid Pools and Layers

can be expressed in the following dimensionless form xex + yey t + τ1 (x − x)ex + yey V2 = t + τ2 η1 h1 = (t + τ1 )2 η2 h2 = (t + τ2 )2 RB1 = β1 (t + τ1 )1/2 V1 =

RB2 = β2 (t + τ2 )

(6.39) (6.40) (6.41) (6.42) (6.43)

1/2

(6.44)

where D1 is used as the length scale and V1 as the velocity scale. Note, that the dimensionless parameters β1 , τ1 and η1 are, in general, functions of the Reynolds number, Re1 = ρV1 D1 /μ, the Weber number, We1 = ρV12 D1 /σ and the dimensionless film thickness h f β1 = β(Re1 , We1 , h f ),

τ1 = τ (Re1 , We1 , h f ),

η1 = η(Re1 , We1 , h f ). (6.45)

The parameters β2 , τ2 and η2 are defined as  β2 = ku kd β(Re2 , We2 , h f /kd ) ku τ2 = τ (Re2 , We2 , h f /kd ) − t kd k3 η2 = d η(Re2 , We2 , h f /kd ) ku

(6.46) (6.47) (6.48)

with Re2 = ρV2 D2 /μ and We2 = ρV22 D2 /σ . Functions β, τ and η correspond to a single drop impact. The initial instant t0 of the intersection of the bases of the two crowns can be found from the condition RB1 + RB2 = x

(6.49)

which, with the help of Eqs. (6.43) and (6.44), yields t0 =

(β12 + β22 )x2 − β14 τ1 − β24 τ2 + β12 β22 (τ1 + τ2 )  2 2 β1 − β22  2β1 β2 x x2 − (β12 − β22 )(τ1 − τ2 ) −  2 2 β1 − β22

(6.50)

if β1 = β2 , or t0 = otherwise.

x4 + β14 (τ1 − τ2 )2 − 2β12 x2 (τ1 + τ2 ) 4β12 x2

(6.51)

6.2 Thin Liquid Layer on a Wall: Crown Formation

267

After the intersection, each crown has a circular part of radii RB1 and RB2 and the common curve defined as x = XB . Denote as Xi and Yi the coordinates of the point of intersection of the circular parts of the crowns at the time instant ti > t0 (see also the definition of the point Xi in Fig. 6.10 in Section 6.3 in this chapter). These coordinates can be found from the geometrical conditions Xi2 + Yi2 = R2B1 ,

(x − Xi )2 + Yi2 = R2B2 .

(6.52)

The solution of the system (6.52) is Xi (ti ) = Yi (ti ) =

R2B1 (ti ) − R2B2 (ti ) + x2 2x  R2B1 (ti ) − Xi2 (ti ).

(6.53) (6.54)

At time t > ti the material point located at the time instant ti at (Xi , Yi ) belongs to the interface between the two crowns. Equation (2.55) in Section 2.2 in Chapter 2 solved with the help of Eqs. (6.39)–(6.43) subject to the initial conditions t = ti :

XB (t, ti ) = Xi (ti )

(6.55)

yields the following expression for the shape XB (t, ti ) of the interface given in parametric form: √ √ t + τ1 t + τ2 Yi (ti ) (6.56) YB (t, ti ) = √ √ ti + τ1 ti + τ2 and XB (t, ti ) =

√  √  √ √ Xi (ti ) t + τ1 ti + τ1 x t + τ1 t + τ2 √ + −√ √ √ τ1 − τ2 ti + τ1 ti + τ2 ti + τ2 t + τ2 (6.57)

with the parameter ti ∈ [t0 , t]. In the case τ1 = τ2 the solution of Eq. (2.55) in Section 2.2 in Chapter 2 takes the form   t + τ1 x t − ti t + τ1 ex + Yi (ti ) XB = Xi (ti ) − ey . (6.58) ti + τ1 2 ti + τ1 ti + τ1 In the present case the analytical expressions for the velocity VB of the uprising sheet at the interface XB , and the shape XJ of the sheet (the crown body) can be also determined by using Eqs. (6.56) and (6.57) or (6.58). These expressions will be further developed in subsection 9.1.1 for the specific application of two drops impacting simultaneously on a dry surface.

6.2.4

Examples of Crown Shapes As in the model of Yarin and Weiss (1995), consider the initial phase of drop deformation when it becomes a disc of radius R0 and thickness h f . The dimensionless duration of this first period is of order t ≈ 1. Neglecting the momentum loss in liquid during the

268

Drop Impacts with Liquid Pools and Layers

Figure 6.5 The height of the crown after normal impact of a single drop onto a film on the wall.

Comparison of the theory with the experimental data (Cossali et al. 1999), h f = 0.29; (a) We = 297, (b) We = 484, (c) We = 667, (d) We = 842 (Roisman and Tropea 2002). Reproduced with permission.

drop deformation, invoking the mass balance of the drop, and considering the initial conditions at t = 1, yields the following expressions for the dimensionless parameters involved in the solution of the problems in subsections 6.2.1–6.2.3  β=

3h f 2

−1/4

,

1 τ= − 1, 24h f

η=

1 24

(6.59)

with h f being the dimensionless initial film thickness. The results of the theoretical predictions of the dimensionless height of the rim, ZR = XR (t ) · ez , are shown in Fig. 6.5 for the four different Weber numbers. The dimensionless film thickness is h f = 0.29. The results of the theoretical predictions are compared with experimental data of Cossali et al. (1999), with the agreement being rather good. The analytically predicted crown shapes are shown in Fig. 6.6 at different time instants. The median surface of the crown XJ is defined by Eq. (6.18). The outer and

269

6.2 Thin Liquid Layer on a Wall: Crown Formation



Figure 6.6 Predicted shapes of the crown at different instants after normal drop impact. The

parameter values are h f = 0.29, We = 842 and Fr−1 = 0, the dimensionless time instants are: t = 5, 10, 20 and 30 (Roisman and Tropea 2002). Reproduced with permission.

the inner surfaces of the crown are located at the distances ±hJ /2 from the median surface, where the local sheet thickness hJ is defined in Eq. (6.20). These two surfaces are reconstructed beginning from the point corresponding to XJ = XB , z = 0. The smooth connection of these inner and outer surfaces of the crown with the free surface of the film on the wall is not imposed by the present solution. The predicted crown shapes are nearly cylindrical, similar to the crowns observed in Cossali et al. (1997) or seen in Fig. 6.3. Note that the predicted crown shapes were obtained by neglecting the surface tension and viscosity effects. However, the resulting forces applied to the element of the surface of a free film associated with the surface tension, are directed normally to the film. Considering the near-cylindrical shape of the crown, the curvature of the crown is of order κ ∼ 1/RB . The thickness of the crown is of order hJ ∼ h f . Therefore, the term associated with the surface tension on the right-hand side of the momentum balance equation (2.78) in subsection 2.2.2 in Chapter 2 can be estimated in the dimensionless form as 2(h f WeRB )−1 . Therefore the deviation RB of the crown radius due to the surface tension can be estimated as RB ∼ 2(h f WeRB )−1 t where t is the time which a material element spends in the crown. This time is estimated as a ratio of the crown height to the vertical component of the sheet velocity. Using Eqs. (6.12) and (6.59) one obtains Z2 RB ∼ r . RB We

(6.60)

270

Drop Impacts with Liquid Pools and Layers

Note also that crown formation and splashing take place in the case of high values of the Weber number, when the surface tension effect is small. Specifically, in the cases shown in Fig. 6.5, the Weber numbers range from We = 297 to We = 842, whereas the height of the crown is of order Zr ∼ 1. Therefore, the error RB /RB due to neglected surface tension in these cases is negligibly small. The shape (6.18) of the crown is obtained using an assumption of a simple ballistic trajectory for liquid elements – the same as the one used in Peregrine (1981). However, the present results are very different from those obtained in Peregrine (1981). The first main difference is in the use of the square-root of time dependence (6.12) for the radius RB and not a linear dependence as implied in Peregrine (1981). Second, the prediction of the crown thickness is added here, which allows a description of the propagation of the rim at the edge of the crown. The shape of the base of the crown in the case of an oblique impact at long times t 1 is predicted in (6.33). This shape is an ellipse with the constant ratio of the halfaxes γ . One can only speculate regarding the value of γ in the absence of experimental data. However, extrapolating the assumption of a constant γ to the time interval t < 1 and noting that the initial shape of the drop is spherical, and thus the initial spot created on the wall is circular, one can conclude that the value of γ must be close to unity. Therefore, βx ≈ βy = β, and the shape of the kinematic discontinuity is a circle of radius √ β t + τ. The initial displacement x0 of the center of the circle at the instant t ≈ 1 is assumed to be small and neglected. The latter assumption together with the assumed circular shape of the initial spot means that the influence of the outer flow Vτ in the film during the short first phase of the impact (t < 1) can be assumed to be negligibly small. Therefore, the parameters β, τ and η are the same as in the case of the normal impact of a single drop and are determined in Eq. (6.59). It is more convenient to present the results on oblique impact in the coordinate system moving with the outer film with the velocity −Vτ ex . This coordinate system corresponds to the oblique impact of a drop with an initial tangential velocity Vτ ex onto a steady uniform film of thickness h f . Following Eqs. (6.33) and (6.59) the x-coordinate xc of the center of the circle and its radius RB are   √  √ 3h f −1/4 √ 1+τ t +τ −τ , RB = t +τ (6.61) xc = Vτ 2 and the angle ϕ ∗ defined in Eq. (6.37) becomes ϕ ∗ = arccos(−2/Vτ ).

(6.62)

Therefore, the minimal obliquity angle θ ∗ between the drop velocity vector and the normal to the wall, at which the base of the crown is no longer a closed circle, can be estimated as θ ∗ = arctan(2) ≈ 63.4◦ . The predicted shapes of the base of the crown are shown in Fig. 6.7 for three different values of the dimensionless tangential velocity Vτ of the drop. The base of the crown is shown only at points where the source term Q defined in Eq. (6.35) is positive, meaning that at these points an inclined sheet is produced. It is shown that if the impact angle

6.2 Thin Liquid Layer on a Wall: Crown Formation

271

− − −

− − −

− − −

Figure 6.7 Shape of the base of the crown in the oblique impact case at time instants: t = 5, 10, 15. The dimensionless thickness of the film is h f = 2.9. (a) normal impact, (b) Vτ = 1, (c) Vτ = 2 (Roisman and Tropea 2002). Reproduced with permission.

272

Drop Impacts with Liquid Pools and Layers

Figure 6.8 Shape of the crown produced due to

Figure 6.9 Shape of the crown produced due to

the oblique impact with h f = 0.29, Vτ = 1 and We → ∞ (Roisman and Tropea 2002). Reproduced with permission.

the oblique impact with h f = 0.29, Vτ = 3 and We → ∞ (Roisman and Tropea 2002). Reproduced with permission.

θ is non-zero (θ = π /4 on Fig. 6.7b), the shape of the crown can be presented as a moving, and expanding circle. If the impact angle is smaller than some critical value [θ = arctan(3) ≈ 1.249 ≈ 71.6◦ in Fig. 6.7c], the shape of the base of the crown is no longer a closed curve. A similar behavior of the crown after an oblique drop impact was observed in the experimental study of Lavergne and Platet (2000). The shapes of the crowns predicted using Eqs. (6.33) and (6.38) are shown in Figs. 6.8 and 6.9 for two different obliquity angles: θ = π /4 < θ ∗ (Fig. 6.8), and θ = arctan(3) ≈ 71.6◦ > θ ∗ (Fig. 6.9).

6.3 Thick Liquid Layers on a Wall: Cavity Formation

273

In the case of impact of two drops of diameters D1 and D2 with the impact velocities V1 and V2 , the parameters β1 , τ1 and η1 are determined in Eq. (6.59). The parameters β2 , τ2 and η2 can be obtained using Eqs. (6.59) in Eq. (6.46) in the following form & %  −1/4 kd ku 1/2 3/4 3h f  , τ2 = − 1 − t (6.63) β2 = ku kd 2 kd 24h f η2 =

kd3 24ku

(6.64)

where h f is the dimensionless film thickness with the diameter D1 used as a length scale. The results of the theoretical predictions of the shape of the interface between two crowns on the wall are shown in Figs. 6.10 and 6.11 at different time moments t after impact. It should be emphasized that in the expression for the curve outlined by the radius-vector XB determined in Eqs. (6.56) and (6.57) or (6.58) and corresponding to this interface, the component of the velocity parallel to the curve is not considered. This component does not influence the shape of the curve. However, in some cases it can lead to the extension of the theoretical line beyond the space occupied by the crowns on the wall, which has no physical meaning. In Figs. 6.10 and 6.11 only the part of the curve outlined by XB is shown which belongs to the discs of radii RB1 and RB2 . In Fig. 6.10 a symmetric case is shown, where two drops of the same diameter and the same impact velocity impact onto the film simultaneously. The dark discs correspond to the inner fields of the two crowns, while the white line is the interface between these crowns. Due to the symmetry, the interface outlined by XB is a straight line coinciding with the symmetry axis. In Fig. 6.11 the interaction of two crowns produced by impacting drops of different initial diameters and impact velocities is depicted.

6.3

Drop Impact onto Thick Liquid Layers on a Wall: Cavity Expansion The dynamics of crown motion described in Section 6.2 in this chapter, approximates it as propagation of a kinematic discontinuity in the liquid film. The expression for the dimensionless radius of the crown, Rcr , is obtained in Yarin and Weiss (1995) in the form  (6.65) Rcr = β (t − τ ) where β is a constant. Note, that Eq. (6.65) expresses essentially the same square-root dependence on time as Eq. (6.2) in Section 6.2. This constant is determined by the initial phase of the drop deformation and penetration into the liquid film. Generally speaking it must depend on the Reynolds and Weber numbers, as well as on the dimensionless initial film thickness. In the case of inertia dominated drop impact (like the drop impact considered in the present section) the effects of the viscosity and surface tension are

274

Drop Impacts with Liquid Pools and Layers

(a)

(b)

(a)

(b)

(c) (c)

Figure 6.10 Symmetric interaction of two

Figure 6.11 Interaction of two drops of

drops. h f = 0.29, x = 5, t = 0, kd = 1, ku = 1. The dimensionless times t after impact are: 5, 10 and 15 (Roisman and Tropea 2002). Reproduced with permission.

different diameter and impact velocity. h f = 0.29, x = 5, t = 2, kd = 2, ku = 0.8. The dimensionless times t after impact are: 5, 10 and 15 (Roisman and Tropea 2002). Reproduced with permission.

negligibly small and the parameter β is determined solely by the initial thickness of the undisturbed liquid film, Yarin and Weiss (1995). The expression (6.65) for the crown radius can be associated with the crater radius, when a crater in a thick film is formed due to strong drop impact. As a drop crater radius

6.3 Thick Liquid Layers on a Wall: Cavity Formation

275

it is supported by numerous experimental data of Cossali et al. (2004), and by the results of numerical simulations of drop impact (Rieber and Frohn 1999). The expression (6.12) in Section 6.2 in this chapter accurately describes the film thickness resulting from drop impact onto a dry spherical target (Bakshi et al. 2007) if the film is thicker than the viscous boundary layer. However, Sivakumar and Tropea (2002) observed that in the case of drop impact onto a layer formed by spray impact, the crater diameter reaches a certain maximum and even starts to diminish. Such behavior cannot be described by Eq. (6.65). In Sivakumar and Tropea (2002) the deviation from the square-root law (6.65) was attributed to the crown– crown interaction. Note that such a deviation can also be explained by the influence of the surface tension and gravity which are not accounted for in Eq. (6.65). An example of drop impact onto a thick liquid layer is presented in Fig. 6.12. The images shown in this figure were obtained using a CCD digital camera (Roisman et al. 2008). The time sequence of drop impact was obtained by varying the delay time of the camera trigger at fixed impact parameters. This method is based on the high repeatability of the drop impact events. In Fig. 6.12 the impact of a water drop is compared with the impact of a 1-Propanol drop. The difference in the outcomes of these two impacts and the rate of crater formation is caused by the difference in the surface tension of both liquids, as well as the difference in their viscosity and density. Namely, the Weber number of the impact of the 1-Propanol drop (shown in the left-hand side column in Fig. 6.12) is higher than that of the water drop impact (in the right-hand side column). The Froude numbers in these two experiments are similar, therefore, the gravity effect is almost the same in these cases. Also, the Reynolds numbers in both experiments are rather high. It is known that at such high impact Reynolds numbers the effect of viscosity on the crater propagation is insignificant (Trujillo and Lee 2001). The effect of viscosity at such high Reynolds numbers is expressed by the appearance of a thin viscous boundary layer which is unable to affect the outer flow and thus the propagation of a cavity. The mechanism of drop impact onto a liquid layer can be subdivided into several phases (see Fig. 6.12): r drop impact onto the surface of the liquid layer and its initial deformation, r prompt splash leading to an almost immediate formation of very small jets and secondary drops on the periphery of the drop impact, r formation of a cavity (crater) and emergence of a crown-like sheet bounded by a rim (Taylor 1959b, Roisman and Tropea 2002), r expansion of the crater, reaching the maximum diameter, r appearance of capillary waves on the surface of the cavity, r receding and merging of the cavity, r emergence of a central jet. In all the experiments the crater diameter reaches its maximum diameter and then recedes. Therefore, its propagation cannot be described by a square-root-of-time dependence. For a fixed relative initial thickness of the liquid layer (rendered dimensionless

_ t

_ t

_ t

_ t

_ t

_ t

_ t

Figure 6.12 Single drop impact onto a thick liquid layer. Comparison of the craters formed in

1-Propanol (We = 527, Fr = 330, Re = 2080, in the left-hand side column) and distilled water (We = 343, Fr = 280, Re = 8700, in the right-hand side column). Drop impact evolution is shown at various time instants. The dimensionless thick film thickness is H = 2.0. The impact parameters are D = 2.2 mm and V0 = 2.7 m/s for 1-Propanol and D = 3 mm and V0 = 2.9 m/s for distilled water (Roisman et al. 2008). Reprinted with permission. Copyright (2008) by the American Physical Society.

277

6.3 Thick Liquid Layers on a Wall: Cavity Formation

z

4

Rcr* R*2 *

h

H

u

*

1

2

*

r 3

Figure 6.13 Sketch of a kinematic discontinuity propagating towards the liquid layer: 1 is the

target, 2 the inner region, 3 the outer static undisturbed region, 4 the uprising crown-like sheet (Roisman et al. 2008). Reprinted with permission. Copyright (2008) by the American Physical Society.

by the drop size) the overall time of the cavity propagation increases for higher values of the Weber number.

6.3.1

Propagation of Kinematic Discontinuity in the Case of Thick Liquid Films If drop impact velocity is high enough, i.e. the Reynolds and the Weber numbers are much larger than one, the flow associated with the crater expansion is governed mainly by inertia, surface tension and gravity. The crater expansion is associated with propagation of a kinematic discontinuity (Yarin and Weiss 1995) which divides the liquid film into the inner region of thickness h∗ (r, t ) and the outer undisturbed film of constant thickness H ∗ at rest. Here and hereinafter the dimensional parameters are denoted by asterisks. The velocity averaged over the film cross-section is denoted u∗ (r, t ). The sketch of the propagating kinematic discontinuity is shown in Fig. 6.13. It can be shown that the remote asymptotic solution (6.12) of Section 6.2 in this chapter exactly satisfies the mass and the momentum balance equations for the inviscid flow even if the capillary forces and gravity are significant. The velocity of the kinematic discontinuity propagation can be found by applying the quasi-stationary modified Bernoulli equation (Roisman and Tropea 2002). In the present case this equation is modified accounting for the average pressure drop associated with gravity, ∗−1 ∗ p∗g ≈ ρg(H ∗ − h∗ )/2, and surface tension, p∗γ ≈ −γ (R∗−1 cr + R2 ). Here R2 is the characteristic curvature of the crater profile (see Fig. 6.13). The expression for the capillary ∗−1 is pressure is obtained from the Young–Laplace relation, whereas the sum R∗−1 cr + R2 an approximate expression for the total curvature of a surface of revolution associated with the crater. The modified Bernoulli equation yields the following equation for the velocity of the kinematic discontinuity propagation Ucr∗ ρ(u∗ − Ucr∗ )2 ρUcr∗2 + p∗γ = + p∗g 2 2

(6.66)

which is associated with the crater widening. The pressure drop due to the viscous drag is neglected in Eq. (6.66). This assumption is valid only for very high impact Reynolds numbers, which typically result in crater formation as in the present experiments. At such high Reynolds numbers the viscosity

278

Drop Impacts with Liquid Pools and Layers

effect is significant only in relatively thin boundary layers. The viscous dissipation in these layers leads to the thickening of the boundary layer but it cannot influence the outer solution and thus does not affect the crown propagation. The solution of Eq. (6.66) is   γ H ∗ − h∗ u∗ 1 1 ∗ −g − + ∗ . (6.67) Ucr = ∗ ∗ 2 2u R2 Rcr ρu∗ Equation (6.67) can be written in the following dimensionless form and simplified accounting for the fact that at large times h∗  H ∗ :   1 u H 1 dRcr 1 = − − . (6.68) + Ucr = dt 2 2uFr R2 Rcr uWe Starting from this equation, all the equations and terms are written in the dimensionless form using the initial drop diameter as a length scale and the impact velocity as a velocity scale. Equation (6.68) is an ordinary differential equation for the crater radius, Rcr , which can be solved numerically if the value of R2 is known. From the geometrical considerations (see the sketch in Fig. 6.13) one can assume that R2 is comparable with the film thickness. In the present model one can assume that R2 ≈ H/2. This assumption is substantiated by the comparison of the model predictions with the experimental data. A more accurate estimate of the value of R2 could be obtained from a detailed analysis of the crater shape, which is beyond the scope of the present section. It can be shown that in the case of We → ∞ and Fr → ∞, Eq. (6.68) has an analytical solution in the asymptotic form given in Eq. (6.65) obtained in Yarin and Weiss (1995). Consider now drop impacts with large finite values of the impact Weber and Froude numbers. In this case the terms corresponding to the gravity and surface tension effects in Eq. (6.68) become significant only at the late stages of crater spreading. The crater radius deviates from the asymptotic square-root behavior (6.65) only when the velocity of the inner region is small enough: u ∼ Fr−1/2 or u ∼ We−1/2 . The crater radius at this stage is comparable with the maximum crater radius Rcr max . Accounting for the smallness of the terms involving 1/Fr and 1/We, Eq. (6.68) can be simplified to the following form   1 u H 2 1 dRcr ≈ − − + . (6.69) dt 2 2uFr H Rcr max uWe The solution of Eq. (6.69) yields   Rcr =

βT −

T = t − τ.

4 H2 + + Rcr max We We Fr 2H



T2 H

(6.70) (6.71)

The instant Tmax at which the crater radius reaches the maximum can be, therefore, expressed in the form  −1 2H 4 H2 βH + + . (6.72) Tmax = 2 Rcr max We We Fr

6.3 Thick Liquid Layers on a Wall: Cavity Formation

279

The value of the maximum crater radius can now be obtained as the positive real root of the equation Rcr (T = Tmax ) = Rcr max

(6.73)

which, with the help of Eq. (6.70), accounting for the large values of We and Fr, yields √ β H H (6.74) Rcr max = √ − GWe 2 G H2 4 + . (6.75) G= We Fr The value of the parameter β is by definition independent of the Weber and Froude numbers. It is, therefore, only a function of the thick film thickness, β = β(H ). In many practical applications, for example under microgravity conditions or in the case of impact of very small drops (10–100 µm drops in sprays), the Froude number is much higher than the Weber number and the effect of gravity is negligibly small in comparison with that of the surface tension. Then, the maximum cavity diameter and the corresponding time Tmax can be expressed using the simplified expressions √ β HWe − H (6.76) Rcr max = 4 √ βH We (β We − H ) Tmax = (6.77) √ 8 (β We + H ) at Fr → ∞. It is non-trivial to perform accurate size and time measurements in the experiments with the impact parameters satisfying the condition Fr We when Eqs. (6.76) and (6.77) are valid. The corresponding drop diameter and the thick film thicknesses must be much smaller than the capillary length. Therefore, more general expressions (6.70) and (6.72) must be used to describe the experiments discussed in the present section. An example of the measured evolution of the crater diameter at the initial stage of the crown expansion is shown in Fig. 6.14. The square of the crater diameter increases linearly with time at 1 < t < 6 confirming the theoretical predictions (6.65). This behavior is rather surprising since the remote asymptotic solution (6.65) is developed for large times t 1 only. At times t > 6 the behavior of the crater diameter deviates from the square-root dependence on time. To predict the crater expansion at the later stages the full theory described in this section must be applied which accounts for gravity and surface tension. In Fig. 6.15 the evolution of the crater diameter is shown for the experiments listed in Table 6.1. In Fig. 6.15a the effect of the initial velocity is illustrated, whereas in Fig. 6.15b the initial film thickness is varied. It is seen that the crater is wider for the impacts with higher impact velocity, since at higher Weber and Froude numbers the inertia is more dominant in comparison with the restraining capillary and gravitational

280

Drop Impacts with Liquid Pools and Layers

Table 6.1 Parameters of selected experiments on crater expansion over a thick layer of liquid after drop impact. The liquid is 1-propanol. Experiment

We

Fr

Re

H

Dcr max

Tmax

Exp1 Exp2 Exp3 Exp4 Exp5 Exp6

194 384 541 539 535 527

133 261 373 369 366 365

12070 17000 20100 20120 20040 19830

0.5 0.5 0.5 1.0 1.5 2.0

3.85 4.5 4.85 5.0 4.6 4.3

8.9 16.2 22.8 21.3 22.5 24.8

effects. Also, an impact onto a thicker liquid layer leads to a narrower expansion of the crater diameter, since the surface tension and the gravity forces increase for thicker layers. Additionally, in Fig. 6.15 the theoretical results for the crater diameter obtained by the numerical integration of the differential equation (6.68) are shown. The parameter τ related to the initial condition is a fitted parameter in these calculations since the detailed description of the initial stage of the crater formation is beyond the scope of the present theory. It is assumed that the parameter τ is a function of the initial film thickness H . The values of the parameter τ are shown in Fig. 6.16. The magnitude of

Figure 6.14 Measurement results of the crater diameter for various experiments at the initial stage

of the crater expansion. The impact parameters are given in Table 6.1. The square of the diameter Dcr is shown as a function of the dimensionless time t (Roisman et al. 2008). Reprinted with permission. Copyright (2008) by the American Physical Society.

6.3 Thick Liquid Layers on a Wall: Cavity Formation

281

(a)

(b)

Figure 6.15 Evolution of the dimensionless crater diameter for various experiments. The impact

parameters are given in Table 6.1. The theoretical predictions (lines) are compared to the experimental data (symbols). (a) The effect of the impact velocity, (b) the effect of the initial film thickness (Roisman et al. 2008). Reprinted with permission. Copyright (2008) by the American Physical Society.

this parameter can be best expressed in the form τ = 0.8H 1.7

at

0.5 < H < 2.

(6.78)

Several adjustable parameters are used in the model, namely, R2 , τ and β. Therefore, it is unclear whether the present theory can be used as a universal predictive tool for

282

Drop Impacts with Liquid Pools and Layers

H Figure 6.16 Fitted parameter τ as a function of the dimensionless initial film thickness H

(Roisman et al. 2008). Reprinted with permission. Copyright (2008) by the American Physical Society.

the entire possible range of impact parameters. Nevertheless, this theory elucidates the deviation of the crater diameter from the square-root law based on the kinematic discontinuity approach and demonstrates that this deviation can be caused by the capillary forces and gravity. The theoretical curves shown in Fig. 6.15 agree with the experimental data for the crater diameter during the expansion phase, as well as accurately predict the magnitude of the diameter maximum and the corresponding time instant. Moreover, despite the fact that the theory has been developed only for the crater expansion, it agrees surprisingly well with the crater diameter data even for the phase of crater receding. There are several reasons for such unexpectedly good theoretical predictions of the receding phase. First, as observed in the experiments, the film thickness in the outer region remains approximately equal to the initial film thickness even during the receding phase. Then, the velocity of the liquid in the outer film during the receding phase is also relatively small, such that the inertial terms in the modified Bernoulli equation (6.66) associated with the flow in the outer film are negligibly small. Finally, the most important conclusion is that the influence of viscosity in the experiments described in the present section is negligibly small. Viscous effects could lead to a decrease in the magnitude of the propagation velocity of the crater. Such a velocity decrease is not observed in the present experiments, since the velocity of receding is comparable with the velocity of spreading. Therefore, as it was mentioned above, the viscosity effect on the crater expansion is negligibly small if the Reynolds number is high enough. One important parameter, which is relevant to the spray impact modeling, is the magnitude of the maximum crater diameter. However, in the impact parameter range

283

6.4 Residual Film Thickness

Figure 6.18 Comparison of the theoretical

Figure 6.17 Comparison of the theoretical

predictions based on Eq. (6.72) of the maximum crater diameter with the experimental data. The impact parameters are in the range given in Table 6.1 (Roisman et al. 2008). Reprinted with permission. Copyright (2008) by the American Physical Society.

predictions of Eq. (6.72) of the dimensionless time Tmax with the experimental data. The impact parameters are in the range given in Table 6.1 (Roisman et al. 2008). Reprinted with permission. Copyright (2008) by the American Physical Society.

explored in the present experiments it is not easy to separate the influence of the capillary forces and gravity, since both are of the same order. In order to model the problem, the experimental data for the maximum crater diameter are expressed using the approximate formula (6.74). One parameter in this expression, β, must be a function of the initial film thickness only. Fitting with the experiments yields the following adjustable function for β β = 0.62H −0.33

at

0.5 < H < 2.

(6.79)

Note also that in order to predict the time instant Tmax , at which the crater diameter reaches the maximum, the adjustable functions β and τ are used in Eq. (6.72). The corresponding theoretical predictions are compared with the experimental data in Figs. 6.17 and 6.18. The agreement is rather good indicating that the assumptions used in the theory are valid.

6.4

Residual Film Thickness In this section the evolution of the thickness of the liquid film below a penetrating and expanding cavity, produced by drop impact onto a pre-existing liquid film on the wall, is discussed. The experimental investigations were performed in van Hinsberg et al. (2010) using the setup shown in Fig. 6.19. Drops were generated using a syringe pump (syringe unit 1) with a preset flow rate. A drop is formed at the tip of the needle and pinches-off at the moment its weight exceeds the net upward surface tension force. After detaching from the needle tip, the drop freely falls down. On its way down, it passes a light barrier before striking normally onto a 130 µm thick borosilicate glass

284

Drop Impacts with Liquid Pools and Layers

Stepper motor

Image processing

Syringe unit 1 – drop generator, adjustable height

HS camera Light source

mm-scale Syringe unit 2 – liquid thickness control Frame Stepper motor

Glass plate with holes Reservoir CHR-sensor

Film thickness detection

Figure 6.19 Experimental setup for measurement of residual film thickness left after drop impact

(van Hinsberg et al. 2010). Reprinted with permission from Elsevier.

plate of 20 mm diameter. This plate was chemically treated to make it hydrophilic. Twenty two laser-drilled holes of 0.5 mm diameter, arranged within an annular area in the diameter range from 16 mm to 18 mm, allowed the liquid to be supplied from the reservoir below to form a preset initial liquid film on the glass plate. The diameter of the holes was sufficiently small not to cause any perturbations in the impingement process and its outcomes. The system was designed to avoid mechanical perturbations of the hydrophilic layer and allowed the thickness of the film on the surface to be finely adjusted using the syringe unit 2 in combination with the reservoir. The light barrier activates an electronic circuit that triggers the imaging system, consisting of a Photron Fastcam SA-1 high-speed CCD camera backlit by a continuous light source. This system is used to investigate the interaction between the drop liquid and the liquid film on the surface. To measure the dynamically varying film thicknesses a confocal chromatic sensor (CHR) was positioned below the glass plate. Spectrally broadband light is focused by an optical probe with a defined chromatic longitudinal aberration onto the liquid layer surface. For the wavelength in which an interface is in focus, the reflected light is maximized. If two interfaces exist within the measurement region, sharp signals appear at two wavelengths. From the spectral distance of these two wavelengths and the refractive index of the liquid, the film thickness is determined. The uncertainty of the film thickness measurements is less than 0.4 µm.

6.4 Residual Film Thickness

285

Figure 6.20 Experimentally and numerically determined thickness of the film below the centerline

of the cavity for a single drop impact onto a layer of deionized water. Top panel: We = 309, Re = 7123, liquid film of dimensionless thickness h/D = 0.07. Lower panel: We = 122, Re = 4772, liquid film of h/D = 0.1. U is the impact velocity. The experimental data are obtained using the CHR system (van Hinsberg et al. 2010). Reprinted with permission from Elsevier.

In this study the CHR-sensor was positioned with its axis aligned with the vertical axis of the impinging drop. With this technique the film thickness evolution in time h(t ) below the center of the expanding and receding cavity is obtained at a sample rate of 4 kHz over a duration of 6 s. Both the camera and the CHR-sensor were synchronized and triggered by the same signal coming from the computer. The fluid of the drop and the liquid surface film was deionized water. Two other liquids, isopropanol and a mixture of 30% glycerine and 70% distilled water, were used to vary the viscosity and surface tension. In Fig. 6.20 the experimental data for the evolution of the film thickness at the impact axis are compared with the numerical simulations based on the volume-of-fluid method (van Hinsberg et al. 2010). The experimental data are shown only for the film thickness range below 600 µm, which is determined by the applicability range of the CHR instrument. The thickness of the film below the expanding cavity, generated by drop impact, starts to follow the remote asymptotic solution Eq. (4.5) in subsection 4.1.2 in Chapter 4 and

286

Drop Impacts with Liquid Pools and Layers

Figure 6.21 Values of A estimated from the numerical simulations as a function of the initial film

thickness on the wall (symbols) in comparison with the empirical correlation Eq. (6.81) (dashed line). Symbols correspond to the results of the numerical simulations (van Hinsberg et al. 2010). Reprinted with permission from Elsevier.

decreases as t −2 , if the inertia of the liquid flow is strong enough. As a result of the effect of viscosity, the flow of the liquid film is damped and a minimum film thickness below the cavity is reached, corresponding to the plateau-region seen in Fig. 6.20. In the case of drop impingement onto a thin liquid film with a relatively high drop impingement velocity (corresponding to high values of the Weber and Reynolds numbers), a near-wall viscous boundary layer appears at the impingement instant. Its thickness grows as a square root of time. An analytical self-similar solution of the full Navier–Stokes equations for this flow is obtained in Section 4.2 in Chapter 4 for the case of a drop impingement onto a dry substrate. It is clear that the flow behavior in the case of drop impact onto a pre-existing liquid film on the wall is similar, since Fig. 6.20 demonstrates that the dimensionless film thickness on the wall h behaves as h ∼ 1/t 2 , similar to Eq. (4.5) in subsection 4.1.2 in Chapter 4 obtained for drop impact onto a dry wall. On the other hand, the dimensionless thickness of the viscous boundary layer hbl increases in time as hbl ≈ t 1/2 Re1/2 (Roisman 2009). Following the same considerations as in subsection 4.2.6 in Chapter 4, one can imply that the expression for the residual film thickness on a pre-wetted wall takes the same form as in Eq. (4.80) in Section 4.2 in Chapter 4 for drop impact onto a dry wall, namely hres = A(h)Re−2/5

(6.80)

where h is the initial dimensionless film thickness, scaled by the drop diameter. The coefficient A is obtained by fitting the numerical results as A ≈ 0.79 + 0.098h4.04

(6.81)

as shown in Fig. 6.21. It is interesting to note that for small values of h, the expression for A approaches the value predicted in Eq. (4.80) in Section 4.2 in Chapter 4 for the case of drop impact onto a dry, flat, solid substrate.

6.5 Drop Impact onto a Deep Liquid Pool

287

Figure 6.22 Drop impact onto a deep liquid pool in the splashing regime (water, D = 2.8 mm, V0 = 4.2 m/s, We = 683, Fr = 620 and Re = 14 638). Panel (d) shows the central Worthington jet [see Worthington (1908)] (Bisighini et al. 2010). Reprinted with permission. Copyright (2010) by the American Physical Society.

6.5

Drop Impact onto a Deep Liquid Pool: Crater and Crown Formation, the Worthington Jets and Bubble Entrapment When a single drop impacts onto a deep pool of liquid with high velocity, it creates an expanding, nearly spherical cavity. The cavity first penetrates into the liquid pool and then recedes under the action of the surface tension and gravity. Crater formation by drop impact has been studied, according to Prosperetti and O˜guz (1993), because of its relation to underwater noise of rain, which can be detected by remote acoustic sensors. Phenomena of drop impacts onto liquid layers are related to various engineering applications in which secondary spray atomization by collision with a wall, spray deposition and coating, and spray cooling play an important role. The wall flow generated by spray impact is rather complicated and is still not fully understood (Panão and Moreira 2005, Roisman et al. 2006, Schneider et al. 2010). In order to model and predict the outcome of spray/wall interaction an accurate prediction of the typical sizes of a crater formed by a single drop impact and the characteristic time of its formation and collapse is necessary. Moreover, a simplified description of the liquid flow generated by drop impact is required in order to reliably model the heat transfer associated with spray cooling (Visaria and Mudawar 2008). A typical time sequence of a drop impact is shown in Fig. 6.22. It should be emphasized that in this case the crater and the crown reach their maximum size at about the same time. Then the crown begins to descend, generating capillary waves which travel to the bottom of the receding crater (Berberovi´c et al. 2009), after which a central jet is ejected. At higher impact velocities the thin liquid sheet forming the crown may ascend further and possibly neck in, forming a dome above the crater, as shown in Fig. 6.23. A downward jet is then formed, which moves toward the raising crater and may or may not intersect it and encapsulate one or more air bubbles in the target liquid. Successively the crater and the liquid sheet reach a configuration with a shape similar to a bubble, which is sustained briefly until it is broken by instabilities. In this case capillary waves have not been observed on the crater wall.

288

Drop Impacts with Liquid Pools and Layers

Figure 6.23 Drop impact onto a deep liquid pool with formation of a bubble above the pool

surface and the entrapment of a small bubble in the pool (acetic acid, D = 2.9 mm, V0 = 4.4 m/s, We = 2177, Fr = 691, Re = 12 642 and Bo = 1.25) (Bisighini et al. 2010). Reprinted with permission. Copyright (2010) by the American Physical Society.

Pumphrey and Elmore (1990) have classified various types of bubble entrainment resulting from drop impact onto a deep liquid pool as r irregular (the Franz type) entrainment, which has a random nature (Franz 1959) with the outcome being unpredictable and depending significantly on the phase of the drop oscillations at the collision instant; r regular entrainment, leading to formation of a single bubble at the bottom of a cavity by propagation of capillary waves along the cavity surface. The condition for a regular entrainment has been analyzed in O˜guz and Prosperetti (1990) and can be expressed in the form 37.3Fr1/5 < We < 47.7Fr1/4 .

(6.82)

r entrainment of large bubbles which are formed by a significant deformation of the cavity, leading to its deviation from a nearly spherical shape. The rear part of the cavity starts to merge, while the front part continues to expand and penetrate the liquid pool. Large bubble entrainment in water corresponds to 5–6 mm drops impacting with the velocity around 1 m/s (Pumphrey and Elmore 1990, Wang et al. 2013). r the Mesler entrainment (Blanchard and Woodcock 1957, Carroll and Mesler 1981, Esmailizadeh and Mesler 1986) which is caused by formation of a thin air layer between the pool liquid and the deforming liquid drop, or by a breakup of an unstable primary toroidal bubble formed by capillary waves at the cavity surface (O˜guz and Prosperetti 1989). The instability and collapse of this air layer leads to generation of a large number of fine bubbles. Some existing models of drop impact available in the literature deal with predictions of the maximum crater depth. All the models, which are usually based on the consideration of the energy balance, are formulated with the assumption of a hemispherical crater shape, centered at the impact point (Prosperetti and O˜guz 1993, Fedorchenko and

6.5 Drop Impact onto a Deep Liquid Pool

τ τ

289

τ τ

Figure 6.24 Typical stages of drop deformation and cavity formation (Bisighini et al. 2010).

Reprinted with permission. Copyright (2010) by the American Physical Society.

Wang 2004, Leng 2001, Macklin and Metaxas 1976, Engel 1966, Pumphrey and Elmore 1990). Engel (1967) found an analytic expression for the crater evolution in time, using the energy balance equation and assuming a kinematically admissible irrotational flow around the expanding crater. The assumption of an irrotational flow around the crater was confirmed later by O˜guz and Prosperetti (1990), who pointed out that when the drop impacts onto the pool surface, very little vorticity is generated by the impact. During the subsequent development of the crater, vorticity production at the free surface (see the viscous boundary layer near the free surface in Section 1.6 in Chapter 1) is minimal in view of the fact that the Reynolds number is of the order of thousands. A simple flow generated by a pointwise source is, in fact, widely used in the studies of impact craters (Leng 2001, Holsapple and Schmidt 1987, Merzhievsky 1997). Such a flow has been used in Berberovi´c et al. (2009) for the description of the pressure field around the crater. The equations of the crater expansion are obtained from the kinematic and dynamic boundary conditions at the free surface of a spherical crater. An analytical expression for the crater radius and depth is then obtained, which is in good agreement with the experimental data for the relatively early stages of crater penetration. This model does not predict the maximum penetration depth of a crater and its receding, since the surface tension and gravity are not taken into account. There are two main disadvantages of almost all the existing theoretical models for cavity expansion: they are often based on energy balance, which can be unreliable due to severe assumptions on the flow kinematics inevitably involved. On the other hand, simplified hydrodynamic models have been successfully applied in penetration mechanics for description of similar problems of high velocity penetration and deformation of shaped-charge jets, rods and projectiles into metal targets (see Chapter 13). In order to develop a simplified model for crater formation, the process is subdivided into two main phases illustrated in Fig. 6.24. During the first phase, τ < τ ∗ (with τ being a dimensional time rendered dimensionless by D/V0 ; D is the drop diameter, V0 is the impact velocity), the drop deforms, generating a thin, radially expanding liquid layer. The material interface between the drop liquid and the target liquid is invisible

290

Drop Impacts with Liquid Pools and Layers

with the same liquid being on both sides. The motion of this interface is governed by the balance of the stresses generated by the flow of the drop and pool liquids. The velocity of penetration of the drop/target interface at the time period τ < τ ∗ is approximately half of the impact velocity [see Eq. (13.2) in Chapter 13]. This result is well known from penetration mechanics (Birkhoff et al. 1948, Yarin et al. 1995) and was previously used for the description of drop impact (Fedorchenko and Wang 2004, Berberovi´c et al. 2009). The approximate dimensionless rate of drop erosion is, therefore, approximately equal to 1/2. The typical dimensionless time of drop deformation is thus, τ ∗ ≈ 2. At times τ > τ ∗ a thin residual liquid layer of the drop material is formed on the crater surface. During this phase the cavity shape can be well approximated by the shape of the drop/target interface. The inertial effects associated with the flow in this layer are negligibly small and the dynamics of the cavity expansion can be analyzed considering the boundary conditions at the free surface. The crater expansion is governed by the flow inertia in the liquid pool and decelerated by the capillary forces and gravity. At some time instant the cavity reaches a maximum penetration depth and starts to recede. The receding phase is influenced significantly by the capillary waves generated by the falling crown (Berberovi´c et al. 2009). Moreover, the interaction of these waves leads to the creation of the central Worthington jet (see Fig. 6.22d). These phenomena affect significantly the dynamics of cavity collapsing. In the absence of gravity a spherically symmetric flow and stress fields can be found which satisfy the boundary conditions at the crater surface. The pressure gradients associated with gravity destroy the spherical symmetry of the pressure field. Therefore, gravity effects cannot be precisely accounted for using the simple flow associated with the expansion of a spherical cavity with a fixed center. Let us approximate the cavity by an expanding sphere of radius a(t ) with the center at depth zc below the undisturbed level of the liquid layer. The irrotational flow past an expanding translating sphere, sketched in Fig. 2.6, is considered in subsection 2.3.1 in Chapter 2. The velocity field past a crater is expressed in Eq. (2.90) and the pressure at the sphere surface in the absence of gravity is expressed in Eq. (2.94). In the present case the part of the pressure associated with the gravity is significant. The pressure distribution at a cavity surface becomes, accordingly   9 2 3a˙2 3 aU˙ 1 − sin θ + cos θ + + aa¨ + aU ˙ cos θ + g(zc + a cos θ ) 4 2 2 2 (6.83) where θ is the polar (or zenith) angle, U = z˙c is the velocity of penetration of the cavity center, a˙ is the rate of growth of the cavity radius, g is gravity acceleration, ρ is the liquid density. The expression for the pressure (6.83) contains two unknown functions, the radius a(t ) and the penetration depth of the cavity center zc , which should be determined from the dynamic boundary conditions at the crater surface accounting for the capillary forces and gravity. At large times the pressure gradient in the thin drop spreading on the expanding crater is negligibly small, as shown in the sketch in Fig. 6.24. The Young–Laplace equation applied to the crater surface, ps + 2σ /a = 0, cannot be satisfied exactly over the entire U2 ps = ρ 2

6.5 Drop Impact onto a Deep Liquid Pool

291

3.5 We = 19000, Fr = 7000 We = 258.5, Fr = 287 We = 54.5, Fr = 60.6 Asymptotic solution

3.0 2.5

zc

2.0 1.5 1.0 0.5 0.0

0

2

4

6

8

10

tU/D Figure 6.25 Drop impact onto a deep pool: comparison of the asymptotic solution (6.89) for the

penetration depth zc =  of the cavity with the experimental data from Berberovi´c et al. (2009). Reprinted with permission. Copyright (2009) by the American Physical Society.

cavity surface. On the other hand, Eq. (6.83) can be linearized near the cavity bottom, θ → 0. The dynamic boundary condition can be written then in the following form, 2σ 3a˙2 7U 2 + gzc + + aa¨ + 4 2 ρa   2 ˙ aU 3aU ˙ 9U + ga + + cos θ + O(U 2 θ 4 ). + 4 2 2

0=−

(6.84)

It should be emphasized that at large times U  a˙ since in all the cases of interest the Froude number is small. Therefore, the last term in Eq. (6.84) is negligibly small in comparison with the other terms. Denote the dimensionless crater radius and axial coordinate of the center of the sphere as α and ζ , respectively, both scaled by the initial drop diameter D. The dimensionless penetration depth zc (scaled by D) is expressed as  = ζ + α. Condition (6.84) yields a system of ordinary differential equations for α(τ ) and ζ (τ ) which can be written in the following dimensionless form, 2 1 ζ 7 ζ˙ 2 3α˙ 2 − 2 − + 2α α We Fr α 4 α ˙ 9 ζ˙ 2 α ˙ ζ 2 ζ¨ = −3 − − . α 2 α Fr

α¨ = −

(6.85) (6.86)

The evolution of the crater can be now evaluated by numerical integration of the system of ordinary differential equations (6.85) and (6.86) subject to the initial conditions which will be considered later. It can be shown that in the limiting case We → ∞, Fr → ∞ the system (6.85) and (6.86) reduces to α¨ = −

3 α˙ 2 , 2 α

α˙ ζ˙ 9 ζ˙ 2 ζ¨ = −3 − α 2 α

(6.87)

292

Drop Impacts with Liquid Pools and Layers

Figure 6.26 Superposition of the theoretical model and the experimental images. The open circle

symbols correspond to the center of the circle fitting the observed cavity. The square symbols correspond to the center of the predicted crater. Water drop of diameter D = 2.8 mm with the impact velocity V0 = 4.2 m/s (Bisighini et al. 2010). Reprinted with permission. Copyright (2010) by the American Physical Society.

with a solution found as α = C1 (τ − C2 )2/5 ,

ζ =0

(6.88)

where C1 and C2 are the constants determined by the initial conditions. In Berberovi´c et al. (2009) the constants in (6.88) were estimated leading to the following expression for the crater radius α = 2−4/5 (5τ − 6)2/5 .

(6.89)

Note that in the present case zc =  = α, and the predicted penetration depth is shown in Fig. 6.25 in comparison with experimental data. At the early stages of a strong (high velocity) drop impact the crater expansion is determined exclusively by inertia. Therefore, it almost does not depend on viscosity or the surface tension of the liquid. The tip of the crater penetrates with an almost constant penetration velocity α˙ + ζ˙ ≈ 0.44 at τ < 2 (Bisighini et al. 2010). It has been shown that at the early stages of impact α = 0.77 + 0.17τ,

ζ = −0.77 + 0.27τ,

for

τ < 2.

(6.90)

Finally, to predict the evolution of the crater radius and the penetration depth, the system of ordinary differential equations (6.85) and (6.86) should be integrated numerically subject to the initial conditions at τ = 2, obtained with the help of Eq. (6.90): α(2) = 1.11, α(2) ˙ = 0.17, ζ (2) = −0.23 and ζ˙ (2) = 0.27.

6.6 Mechanism of Splash

293

3.5 3 2.5

Δ

2 1.5 water, We = 170, Fr = 262 water, We = 226, Fr = 489 water, We = 406, Fr = 569 water, We = 683, Fr = 620 acetic acid, We = 2190, Fr = 694

1 0.5 0

0

10

20

30

τ

40

50

60

Figure 6.27 The dimensionless depth of crater penetration as a function of the dimensionless time. Comparison between the experimental data (symbols) and the theoretical predictions (curves) for various impact parameters (Bisighini et al. 2010). Reprinted with permission. Copyright (2010) by the American Physical Society.

Figure 6.26 shows a superposition of the crater predicted by the theoretical model onto the experimental images. The agreement between the theoretical predictions and the experimental shape of the cavity is good at the bottom region of the crater where the model is valid. Some discrepancy appears when the shape of the crater is deformed by the capillary waves. Further comparisons between the model prediction and experiments on the evolution of the penetration depth are shown in Fig. 6.27. The agreement is good for various impact conditions. As expected, the difference between the experimental data and the predicted values increases at the last part of the receding phase. Some discrepancy between the predictions and the experiments can be explained by the influence on the flow from the propagating capillary waves (not considered in the theory) and by the crater deformation at the bottom part, leading to the formation of the central Worthington jet. In Fig. 6.28, the results of the theoretical predictions of the dimensionless maximum crater depth are shown as a function of its experimental value for a wide range of the Weber and the Froude numbers. A straight line corresponds to a perfect agreement. The agreement between the theoretical predictions and the experiments is very good.

6.6

Bending Instability of a Free Viscous Rim on Top of the Crown: Mechanism of Splash The evolution of free thin liquid sheets is relevant for many industrial applications like polymer or foam processing, cooling, coating, washing, as well as in various hydrodynamic and rheological experiments. The dynamics of free liquid sheets is determined by the inertia, viscous stresses and the surface tension. The description of liquid flow in the sheet can be significantly simplified by accounting for the smallness of the sheet

294

Drop Impacts with Liquid Pools and Layers

Figure 6.28 Maximum crater depth. Comparison of the theoretical predictions with the

experiments (Bisighini et al. 2010, Engel 1966, Leng 2001, Macklin and Metaxas 1976, Olevson 1969, Brutin 2003, Fedorchenko and Wang 2004). Reprinted with permission. Copyright (2010) by the American Physical Society.

thickness in comparison with the typical lengths in the other two directions. Such a simplifying approach has been used in the past to predict the shape of steady-state liquid bells (Boussinesq 1869a,b, Taylor 1959a, Yarin 1993 and Clanet 2001), and the instability of free planar sheets (Taylor 1959b,c). Entov (1982) developed a quasi-twodimensional theory describing the dynamics of thin films of viscous Newtonian liquids and polymer solutions. At the free edges of a free, uniform, liquid sheet the surface tension can be balanced only by the inertia of the liquid. Capillary forces are responsible for the emergence of a free rim propagating toward the liquid sheet with a finite velocity given by Eq. (2.159) in Section 2.7 in Chapter 2 (Taylor 1959b, Culick 1960). Expression (2.159) is valid for low-viscosity liquids (Savva and Bush 2009). The theoretical model of Entov et al. (1986) for the stationary rim bounding a planar sheet accounts for the flow and the internal stresses in the rim and in the free film (Yarin 1993). Clanet and Villermaux (2002) have applied the rim dynamic equations to describe the stationary shape of the rim that arises when a radially expanding liquid sheet collides with an obstacle. Many physical phenomena can be associated with the propagation of the free rim, among them the spreading and receding of a drop impacting onto a dry partially wettable substrate (Roisman et al. 2002, Rozhkov et al. 2002, Bartolo et al. 2005) [see Section 4.5 in Chapter 4], dewetting of a dry substrate (Brochard-Wyart et al. 1987, Brochard-Wyart and de Gennes 1997), aerodynamic drop deformation by a shock wave (Hsiang and Faeth 1992, 1995), drop binary collisions (Ashgriz and Poo 1990, Brenn et al. 2001, Roisman 2004) and the interaction of two jets (Bush and Hasha 2004). In some cases the rim becomes unstable, which leads to the emergence of finger-like jets which subsequently break up into drops. The jets appear usually in the plane of the film (see Fig. 1.6 in Section 1.10 in Chapter 1). One of the most important

6.6 Mechanism of Splash

295

Rim

(a)

(b)

(c)

(d)

Figure 6.29 Unstable liquid sheets, jets and secondary drops generated by spray impact.

phenomena related to the rim instability is liquid atomization by a fan spray sheet (Clark and Dombrowski 1972) or pressure swirl atomizers, as well as secondary atomization by spray/wall or drop/substrate interaction (Yarin and Weiss 1995). Comprehensive reviews on the topics of liquid fragmentation, breakup and atomization can be found in Lin and Reitz (1998), Villermaux (2007), Gorokhovski and Herrmann (2008), Eggers and Villermaux (2008) and Eggers (1997). The rim instability is often related to the capillary instability of a free infinite cylindrical jet (e.g. Deegan et al. 2007, Zhang et al. 2010, Agbaglah and Deegan 2014). Lord Rayleigh (1878) obtained the characteristic equation for an inviscid capillary instability problem in the form of Eq. (1.71) in subsection 1.10.1 in Chapter 1, while Weber (1931) accounted for the effects of viscosity, Eq. (1.73), found in the same subsection. The jet stability and the wavelength of the most unstable mode can be significantly influenced by longitudinal stretching (Frankel and Weihs 1985, Khakhar and Ottino 1987, Kolbasov et al. 2016) or by hydrodynamic forces applied from the surrounding fluid (Weber 1931, Tomotika 1935, Entov and Yarin 1984, Brenner et al. 1996). Rim dynamics and stability are influenced by the flow entering the rim from the free sheet and by the capillary forces. These two factors often lead to rim stabilization. Fullana and Zaleski (1999) have performed numerical simulations of rim propagation and deformation using the volume-of-fluid method. In their simulations rim radius perturbations do not grow significantly and do not lead to rim breakup. In the numerical study of Bagué et al. (2007) a rim starts to propagate toward a uniform stationary liquid film. It is accelerated by the surface tension and deforms. The deformation of the rim centerline in some cases leads to the appearance of cusps (regions at which the radius of curvature of the rim centerline vanishes), ejection of finger-like jets and drop formation. The nonlinear phenomenon of cusp formation was predicted by Yarin and Weiss (1995) and is discussed at the end of this section in more detail. The flow in finger-like jets appearing as the result of rim instability is usually almost parallel to the plane of the free liquid sheet and is directed nearly normal to the rim centerline (see Figs. 1.6b and 1.6c in Section 1.10 in Chapter 1). Consider for example the shape and the breakup of an unstable rim bounding a liquid sheet emerging as a result of spray impacting onto a rigid wall, as shown in Fig. 6.29. A stability analysis

296

Drop Impacts with Liquid Pools and Layers

Figure 6.30 Splash produced by a single drop impact onto a smooth, rigid, wetted substrate.

of a straight, cylindrical, infinite jet cannot describe the emergence of the finger-like jets and this type of rim breakup. Therefore, the Rayleigh capillary instability alone cannot be responsible for splash in spite of the fact that this analysis predicts the experimental data on the rim breakup length in certain cases where the initial perturbations are regular and the effect of the rim stretching is negligibly small (Deegan et al. 2007, Zhang et al. 2010). It should be emphasized that one of the conclusions of the Rayleigh analysis is that the fastest growing perturbation of an infinite cylinder is axisymmetric. However, the observed rim deformation is never really axisymmetric. For the same reason the Richtmyer–Meshkov instability (Krechetnikov and Homsy 2009) can hardly be considered as the main reason for splash. Moreover, the exponentially growing solutions implied by the Rayleigh capillary instability can be a valid solution only in the cases without longitudinal stretching of the rim, which is frequently not the case for rims on top of the propagating crowns. The linear stability of the inviscid rim has been analyzed in Roisman et al. (2006), accounting for the factors associated with the film flow. The linearized equations of motion for a nearly straight rim were derived from the mass, momentum and angular momentum balance equations. Both the axisymmetric perturbations of the radius of the rim cross-section and the transverse disturbances of the rim centerline were considered. The growth rate of the rim perturbations increases significantly with the rim acceleration toward the film and decreases with the relative film thickness. The wavelength of the most unstable mode is very similar to the wavelengths obtained by Lord Rayleigh (1878) and Weber (1931) for a wide range of the parameters. In the study of spray impact by Roisman et al. (2007) the effect of film stretching was investigated. It was shown that strong film stretching can lead to the appearance of long stable portions of the rim (Kolbasov et al. 2016). The corresponding experimental evidence confirming these theoretical predictions was provided in Roisman et al. (2007) and Kolbasov et al. (2016). It should be emphasized that the reason for the rim instability and the mechanisms of splash are still under discussion and most probably under different impact conditions the reasons are different, or there are combinations of the reasons simultaneously. One important topic which is not considered in the previous theory of rim bending in Roisman et al. (2006, 2007) is the shape of the rim. These studies have concentrated solely on the shape of the centerline whose deformation could lead to cusp formation, as suggested by Yarin and Weiss (1995) and discussed below. In fact, at the instant of the emergence of the finger-like jets the perturbations of the rim centerline are frequently not clearly visible. One such example is shown in Fig. 6.30. The jets originate from the bumps distributed along the rim and are growing in the direction of motion of the rim centerline. One possible explanation for the bump formation is

6.6 Mechanism of Splash

297

x,t

V0(x,t)

a x ,t

y V(x,t) U(x,t) Y0(t)

Rim

VS(y,t) /2

z

hS(y,t)

x

Figure 6.31 Sketch of the disturbed rim (Roisman 2010). Reproduced with permission.

related to the instability of the internal flow inside the rim leading to the deformation of the rim cross-section. Below it is shown that this phenomenon can be explained in the framework of the long-wave approximation of the quasi-two-dimensional theory of rim dynamics. The main topic of the present section is the development of a theoretical model for the evolution of a rim bounding a free viscous sheet. The theory combines the quasi-onedimensional approach of the dynamics of free liquid jets and the quasi-two-dimensional approach of the dynamics of free, thin liquid sheets (Yarin 1993). In the following the governing equations for the propagation of the rim centerline, growth of the size of the rim cross-section and for the internal liquid flow in the rim are obtained from the mass, momentum and moment-of-momentum balance equations accounting for the capillary forces, internal viscous stresses in the rim and in the sheet, as well as the inertia of the flow in the rim and of the flow entering the rim from the liquid sheet. Then the theory is applied to the linear stability analysis of a nearly straight rim. The discussion of the results elucidates the effect of viscosity on the maximum growth rate of the perturbations and on the corresponding wavelength. The predicted shape of the perturbed rim is similar to the forms observed in experiments.

6.6.1

Linear Stability Analysis of an Infinite, Straight, Viscous Rim Consider a stationary Cartesian coordinate system {x, y, z} with base unit vectors {ex , ey , ez } and a rim bounding a thin planar liquid sheet of thickness hS (y, t ) and velocity VS (y, t ), whose median surface lies in the plane {x, y}, as shown in Fig. 6.31. Consider also a stretching rate S of the free liquid sheet only in the y-direction defined as S(y, t ) = ∂VS /∂y. The stretching leads to the emergence of viscous stresses in the sheet which then influence the dynamics of the rim. In the present case the effect of the rim formation on the film flow is assumed negligibly small, which is typical for lowviscosity fluids (Yarin 1993). The film stretching is therefore an independent parameter in the present case. The stress tensor in a free thin planar liquid sheet can be found as (Yarin 1993) σS = 2μ[Sex ⊗ ex + 2Sey ⊗ ey ]

(6.91)

using the condition of vanishing of the z-component (normal to the free surfaces of the sheet) of the stress tensor.

298

Drop Impacts with Liquid Pools and Layers

Base Solution The base solution for the undisturbed rim was obtained by Taylor (1959b) and is also available in Yarin (1993) and Roisman et al. (2006). Here it is written to account for the viscous stresses emerging in the liquid film dY0 (t ) = V0 (t ) dt dA0 (t ) = [VS (Y0 , t ) − V0 (t )]hS (Y0 , t ) dt 4μShS (Y0 , t ) dV0 (t ) 2σ = [VS (Y0 , t ) − V0 (t )]2 hS (Y0 , t ) − − A0 (t ) dt ρ ρ

(6.92) (6.93) (6.94)

where Y0 (t ) is the undisturbed position of the rim centerline, A0 (t ) is the rim crosssectional area, and V0 is the rim transverse velocity. It can be shown that solution Eqs. (6.92)–(6.94) satisfies the equations for rim dynamics given in Section 2.7 in Chapter 2.

Long-Wave Approximation of a Disturbed Rim In order to analyze the rim stability, consider only small perturbations of the rim centerline y = Y (x, t ) and the cross-sectional radius a(x, t ). For very small perturbations (∂Y/∂x  1 and ∂a/∂x  1) the system of governing equations (2.169), (2.171) and (2.173) given in Section 2.7 in Chapter 2 can be linearized and written in the coordinate system {x, y, z}. The transverse and longitudinal components of the liquid velocity in the disturbed rim are denoted as V (x, t ) ≡ ∂Y/∂t and U (x, t ), respectively. The balance equations for the mass and momentum in the x- and y-directions and for the moment of momentum in the z-direction become ∂A ∂AU + − hS (VS − V )(1 + κa0 ) = 0 ∂t ∂x ∂P ∂U ∂Y − + ρhS (VS − V )U − 2σ =0 ρA ∂t ∂x ∂x ∂Q ∂V − Pκ − + [2σ − ρhS (VS − V )2 + 4μShS ](1 + κa0 ) = 0 ρA ∂t ∂x ∂I0  ∂M − − Q + ρhS a0 (VS − V0 )U = 0 ρ ∂t ∂x

(6.95) (6.96) (6.97) (6.98)

with the expressions for the forces and moments of stresses obtained with the help of Eqs. (2.191) and (2.193), and the linearized curvature and the angular velocity obtained with the help of Eq. (2.167) in Section 2.7 in Chapter 2 in the following form ∂ 2a ∂U − μhS (VS − V )(1 + κa0 ) + 3μA ∂x2 ∂x ∂ κI0 ∂V M = −ρI P + 3μI0 + ∂t ∂x A0 ∂V ∂ 2Y . , = κ= ∂x2 ∂x P = π σ a + σ A0

(6.99) (6.100) (6.101)

6.6 Mechanism of Splash

299

In Eqs. (6.95)–(6.101) A is the rim cross-sectional area, P is the magnitude of the longitudinal force in the rim, and Q and M are the shearing force and the moment of stresses in the rim cross-section, respectively. The linearized balance equations (6.95)–(6.98) were obtained by Roisman et al. (2006) for the inviscid case, whereas the expressions (6.99) and (6.100) for the force and moment of force applied in the rim cross-section, include the additional terms associated with the viscous stresses. Consider small perturbations of the rim centerline, of its cross-sectional radius and of its velocity in the form Y = Y0 (t ) + #(x, t ),

a = a0 (t ) + α(x, t ),

U = u(x, t ).

(6.102)

The system of linearized equations for these small perturbations can be obtained from Eqs. (6.95)–(6.98) in the following form 2π a0 α,t + A0 u,x + h(#,t − S#) − W0 a0 (#,xx − α/a20 ) = 0

(6.103)

−ρA0 u,t + π σ α,x + σ A0 α,xxx + 3μA0 u,xx − μW0 a0 #,xxx + μhS (#,tx − S#,x ) − ρW0 u + 2σ #,x = 0 −ρA0 #,tt − π ρa0V˙0 (2α − 2ρI0 #,xtt +

a20 #,xx )

(6.104)

+ π σ a0 #,xx + μW0 #,xx

+ Q,x − 2ρW0 (#,t − S#) 3 ρ I˙0 #,xt + π ρa0V˙0 α,x − 3μI0 #,xxxt π σ a30 a2 #,xxx + ρa0W0 u − Q + 0 μW0 #,xxx − 4

4

=0

(6.105)

=0

(6.106)

where W0 = h(VS0 − VR0 ),

I˙0 ≡

a2W0 dI0 = 0 , dt 2

V˙0 ≡

dV0 dt

(6.107)

with W0 being the volumetric flow rate per unit length of the liquid entering the rim.

Immediate Loss of Stability Coefficients a0 , A0 and I0 corresponding to the base solution are functions of time and the exact solution of the system of equations (6.103)–(6.106) is non-exponential in time and not-trivial, albeit the system is linear. Nevertheless, an expected fast exponential growth of the rim perturbations in the case of its instability allows one to obtain a solution assuming “frozen” coefficients a0 , A0 and I0 , which immediately implies exponentially growing solutions. Assume small rim perturbations of the form # = #0 exp(ωt + iξ x),

α = α0 exp(ωt + iξ x)

(6.108)

u = u0 exp(ωt + iξ x),

Q = q0 exp(ωt + iξ x),

(6.109)

where ω is the growth rate of the perturbations and ξ = 2π / is the wavenumber ( being the perturbation wavelength).

300

Drop Impacts with Liquid Pools and Layers

The linearized system of equations, (6.103)–(6.106), can now be reduced to the following form A·b=0 where



2π a0 ω + W0 /a0 ⎜ π σ iξ − σ A0 iξ 3 A=⎜ ⎝ −2π ρa0V˙0 π ρa30V˙0 iξ b = (α0

#0

u0

A0 iξ A22 0 ρa0W0

A13 A23 A33 A43

(6.110) ⎞ 0 0 ⎟ ⎟ iξ ⎠ −1

(6.111)

q0 )T

A13 = h(ω − S) + a0W0 ξ

(6.112)

2

A22 = −ρW0 − ρA0 ω − 3A0 μξ

(6.113) 2

(6.114)

A23 = μW0 a0 iξ + μhiξ (ω − S) + 2σ iξ 3

A33 = −ρA0 ω − 2ρW0 (ω − S) − 2

(π ρa30V˙0

(6.115) + μW0 + π σ a0 )ξ

(6.116)

− ωiξ + 3I0 μωiξ 3 + iξ 3 . (6.117) 2 4 The characteristic equation corresponding to the system of equations (6.103)–(6.106) is obtained as A43 = 2ρI0 ω2 iξ +

ρa20W0

π a30 σ

2

C4 ω4 + C3 π ω3 + C2 ω2 + C1 ω + C0 = 0

μW0 a20

(6.118)

where the coefficients C0 , C1 , C2 , C3 and C4 are functions of the dimensionless parameters of the problem. Obtaining the expressions for these parameters is a straightforward algebraic operation. These expressions are rather cumbersome and as such are omitted here. All the overbarred variables and parameters starting from Eq. (6.118) are dimen

sionless, with ρa30 /σ taken as the time scale, and a0 as the length scale. The main parameters of the problem are expressed through the dimensionless variables as   σ σ σ ω= V˙ 0 ω, S= S, V˙0 = (6.119) ρa20 ρa30 ρa30  σ a0 hS = ha0 , W0 = W 0 ξ = ξ /a0 , . (6.120) ρ The relation between W 0 and V˙0 can be found from the base solution (6.92)–(6.94) as  W 0 = h(2 + π V˙0 + 4hOhS) (6.121) where dot denotes time differentiation and Oh is the Ohnesorge number [cf. Eq. (1.1) in Section 1.2 in Chapter 1]. The characteristic equation, Eq. (6.118) is an algebraic equation for ω and has four roots. The presence of a root with a positive real part indicates the instability of the rim. The rim acceleration V˙0 = dV0 / dt can be found from Eq. (6.94) at any instant in time. For a given flow in the film its value is determined by the initial conditions and by the history of the rim propagation. The acceleration does not vanish even if at some

6.6 Mechanism of Splash

301

Figure 6.32 Dimensionless rate of growth of the rim perturbations as a function of the

dimensionless wavenumber. Oh = 0.1, V˙ 0 = −0.1. (a) S = 0, the dimensionless sheet thickness correspondence for line 1 h = 0, 2 h = 0.1, 3 h = 0.2, 4 h = 0.3; (b) h = 0.1, the sheet stretching rate varies: the lines correspondence is: 5 S = 0.5, 6 S = 0.3, 7 S = 0.2, 8 S = 0.1 (Roisman 2010). Reproduced with permission.

instant of time the velocity gradient in the film is zero. Therefore, in the present analysis V˙0 = dV0 / dt is considered as an independent parameter.

Rate of Growth of Small Perturbations of the Rim The values of the maximum positive root of Eq. (6.118) are shown in Fig. 6.32 as functions of the dimensionless wavelength ξ for various values of the governing parameters. The value of ω is real and positive at small values of the wavenumber and reaches a maximum at ξ ≈ 0.6–0.8 which is of the same order as the wavelength of the most dangerous mode of the capillary instability of a free cylindrical jet found by Lord Rayleigh (1878), which is ξ = 0.698 (see subsection 1.10.1 in Chapter 1, where the notation χ for the dimensionless wavelength corresponds to the present ξ ). In Fig. 6.32a the effect of the film thickness is investigated. In Fig. 6.32b the values of ω are shown for various values of the dimensionless sheet stretching rate S. In the range of S considered, the sheet stretching rate has almost no influence on the maximum growth rate ω at the wavenumbers ξ > 0.5, whereas the growth rates of the longest waves (with ξ < 0.3) increase significantly with S. This means that in the presence of the sheet stretching, relatively stable long parts of the rim can appear. Such long parts of the rim have been reported in Roisman et al. (2007) and also predicted there theoretically for an inviscid rim. Using the characteristic equation (6.118) it is possible to calculate the maximum growth rate of the rim perturbations ω∗ and the corresponding wavenumber ξ ∗ . The values of ω∗ are shown in Fig. 6.33 for different situations. In all the cases rim deceleration leads to a significant increase in ω∗ , whereas viscosity and a larger relative film thickness stabilize the rim. These results are in agreement with the evident stabilizing role of viscosity, as in the case of capillary instability studied by Weber (1931) [see Eq. (1.73) in subsection 1.10.1 in Chapter 1]. The values of the dimensionless wavelength ∗ = 2π /ξ ∗ of the most unstable mode are shown in Fig. 6.34. In all the cases the variation of ∗ is relatively small and remains

302

Drop Impacts with Liquid Pools and Layers

Figure 6.33 Maximum growth rate as a function of the rim acceleration at various values of the

governing parameters at S = 0. (a) effect of the sheet thickness at Oh = 0.1: the lines’ correspondence is 1 h → 0, 2 h = 0.05, 3 h = 0.1, 4 h = 0.3; (b) the effect of the Ohnesorge number at h = 0.1; the lines’ correspondence is 5 Oh = 0, 6 Oh = 0.05, 7 Oh = 0.15, 8 Oh = 0.3 (Roisman 2010). Reproduced with permission.

in the 8 <  < 11 range for a wide range of the governing parameters used in the analysis. The rim deceleration leads to a small decrease in the value of ∗ , whereas viscosity causes an increase in ∗ . This result is confirmed by the experimental data for the relative distance between the jets emerging from the rim due to its instability. These distances can be estimated from the images of splash produced by a single drop impact (Yarin and Weiss 1995, Cossali et al. 2004, Vander Wal et al. 2006, Roisman et al. 2006) and of spray impact (Roisman et al. 2007). In most of the cases the inter-jet distance is approximately 10 rim radii.





































Figure 6.34 The most unstable wavelength ∗ as a function of the rim acceleration at S = 0. (a) effect of the dimensionless sheet thickness at Oh = 0.1: the lines’ correspondence is 1 h = 0, 2 h = 0.1, 3 h = 0.3; (b) the effect of the Ohnesorge number at h = 0.1: the lines’ correspondence is 4 Oh = 0, 5 Oh = 0.05, 6 Oh = 0.15, 7 Oh = 0.3 (Roisman 2010). Reproduced with permission.

303

6.6 Mechanism of Splash





































Figure 6.35 The dimensionless ratio u0 /α 0 and the ratio #0 /α0 at S = 0.1, h = 0.1. (a) the amplitude of the velocity u0 of the rim deflection #0 rendered dimensionless by the perturbations of the rim radius α0 at Oh = 0.1; (b) the ratio #0 /α0 at various Ohnesorge numbers (Roisman 2010). Reproduced with permission.

The eigenvector of the system (6.103)–(6.106) is determined, which allows the ratios of the amplitudes #0 /α0 and u0 /α0 to be predicted (#0 is the rim deflection amplitude, and u0 is the amplitude of the longitudinal velocity of liquid in the rim). The value of #0 /α0 is dimensionless, whereas the ratio u0 /α0 is rendered dimensionless as  u0 σ u0 =i . (6.122) α0 ρa30 α 0 The imaginary unit i appears in Eq. (6.122) to ensure the real values of the ratio u0 /α 0 . In Fig. 6.35 the dimensionless amplitude ratios u0 /α 0 and #0 /α0 at the maximum growing mode (corresponding to ω∗ and ξ∗ ) are shown as functions of the dimensionless rim acceleration V˙ 0 . The Ohnesorge number (Oh) has a significant effect on the ratio #0 /α0 , while variation of various other governing parameters of the problem only leads to minor changes in the predicted values of u0 /α 0 and #0 /α0 .

Distance between Finger-like Jets As mentioned above, one of the most important parameters which can be used to validate the theory is the distance between the finger-like jets appearing due to the rim instability. It is difficult to design an experiment in which the flow in the sheet and in a nearly straight rim would be perfectly controlled. In order to observe the initial stages of the rim instability and to measure its typical wavelength and growth rate, an axisymmetric spreading of a free sheet resulting from jet or drop impact onto a rigid surface is the most convenient. Such impacts were investigated in Clanet and Villermaux (2002) and Rozhkov et al. (2002). In the case of the radial expansion of the rim centerline, the additional stresses appear in the rim cross-section associated with the term μVn κ, i.e. with the transversal motion of the bent rim. They vanish in the case of the nearly straight rim considered in the present case.

304

Drop Impacts with Liquid Pools and Layers

Figure 6.36 Probability density function of the dimensionless inter-jet distances 2/Drim for the perturbed rim diameters between 70 and 100 µm (Roisman 2010). Reproduced with permission.

Moreover, the inter-jet distances in the experiments may not be necessarily exactly equal to the most unstable wavelength, since their values are affected by the spectrum of the initial natural perturbations. On the other hand, high-speed video observations of spray impact (several sequences of which are shown in Fig. 6.29) provide a large number of data which can be analyzed. Some of the results of the measurements of the inter-jet distances in a water spray have been presented in Roisman et al. (2007). The spray was generated by a full-cone pressure swirl atomizer. The volumetric flux was varied from 0.25 to 0.5 l/min. The average drop velocity was in the 5 to 11 m/s range. In Fig. 6.36 this data is analyzed in detail, namely the probability density function of the distribution of the dimensionless inter-jet distances  = 2/Drim is shown for rim diameters between 70 and 100 µm. Each point in the histogram is obtained by counting the values of  from the interval  = ±1. The histogram is then normalized to obtain the probability density function. Several peaks in the probability density function can be identified, which indicates that the mechanism of the rim breakup is rather complicated. These peaks are clearly marked in Fig. 6.36. In Fig. 6.37 the values of 2/Drim corresponding to the first two peaks are shown as functions of the rim diameter. In the majority of the cases the most probable inter-jet distance is comparable with the theoretically predicted values shown in Fig. 6.34. The Ohnesorge number is in the range from 0.01 to 0.03. Therefore, the effect of viscosity on the rim instability in these experiments is negligibly small. For smaller rim diameter the experimental values of  corresponding to the second peak increase and approach the double wavelength of the most unstable mode,  ∼ 2∗ . In the case of spray impact the initial perturbations are non-uniform along the rim due to the interactions with other drops and due to the drop interactions with the fluctuating

6.6 Mechanism of Splash

305

Figure 6.37 Spray impact onto a rigid wall. Most probable dimensionless inter-jet distances as a

function of the rim diameter. The Ohnesorge number is in the range from Oh = 0.01 to Oh = 0.03. The theoretically predicted values for the inter-jet distances range from 2/Drim = 7.6 (at Oh = 0, V˙ 0 = −0.4) to 2/Drim = 8.7 (at Oh = 0.05, V˙ 0 = 0) corresponding to lines 4 and 5 in Fig. 6.34b. The error bars are defined by the corresponding class intervals for Drim and 2/Drim (Roisman 2010). Reproduced with permission.

liquid film formed on the substrate by spray impact. Therefore, the finger-like jets do not appear from the rim simultaneously. Some of the inter-jet distances thus correspond to multiple lengths of the most unstable wavelength, as predicted theoretically. Moreover, the resolution of the camera was 20 µm, which is comparable with the smallest rim diameters shown in Fig. 6.37. In the case of such small rim diameters some bumps cannot be clearly identified, which also leads to statistically larger inter-jet distances. Other possible reasons for some deviation of the measured values of  from the theoretically predicted values is due to non-stationary effects, rim stretching and compression during breakup. However, it should be emphasized that the predicted values of  are of the same order of magnitude as the measured values.

Role of the Moment-of-momentum Equation The rim theory described in the previous subsections can be characterized as a momentless theory. In the absence of the rim acceleration and in the absence of the moment of momentum in the flow entering the rim, the capillary forces stabilize the rim bending. In this case the equation for the centerline bending Eq. (6.105) is independent of the perturbations of the rim radius. It can be shown that the rim centerline is then stable. The more general stability analysis, which accounts for the moment of momentum, shows that in the absence of the acceleration, i.e. for V˙ 0  1, the amplitude of the rim bending does not vanish, but it is much smaller than the perturbation amplitude of the rim radius, #0 /α0  1. Therefore, in this regime the rim bending can be neglected

306

Drop Impacts with Liquid Pools and Layers

and the rim deformation is almost axisymmetrical. The mechanism of this deformation is explained by the Rayleigh capillary instability of a free infinite cylindrical jet. This result indicates also that the effect of the terms associated with the moment of momentum is rather small for various sets of the governing parameters of the problem. The effect of the moment of momentum of the rim can be significant if the components of the rim angular velocities, defined in Eq. (2.167) in subsection 2.7.1 in Chapter 2, are high. However, the examples considered revealed that the effect of the moment of momentum is negligibly small. Therefore, the following nonlinear theory of rim evolution will be momentless.

Nonlinear Rim Deformation Consider, as an example, a rim bounding a uniformly stretching sheet with a uniform velocity gradient S(t ) in the y-direction. The dimensionless nonlinear equations of the rim motion in the plane {x, y} can be obtained from the mass and momentum balance equations in the x- and y-directions, derived from Eqs. (2.169) and (2.171) in subsection 2.7.2 in Chapter 2 and written in the Eulerian form as (6.123) (λA),t + (λAU ),x = hS (VS − Y,t )(1 + κa) # $ −1 ρλA(U,t + UU,x ) = (λ P),x + 2σY,x + 2μhS S − U hS (VS − Y,t ) (1 + κa) (6.124) ρλA(V,t + UV,x ) = [ρhS (VS − Y,t )(VS − V ) − 2σ − 4μShS ](1 + κa)

(6.125)

where the liquid velocity V (x, t ) in the y-direction, the curvature κ (x, t ) and the term λ(x, t ) are defined as  (6.126) V = Y,t + UY,x , λ = 1 + Y,x2 , κ = λ−3Y,xx . The longitudinal and the normal velocities in the rim can be written as Vn = λ−1Y,t ,

Vτ = λU + λ−1Y,xY,t .

(6.127)

The stretching force P acting in the rim cross-section is defined in Eq. (2.191) in subsection 2.7.3 in Chapter 2, with 1 (6.128) δ = − (λ−1Vτ,x − κVn ), W = hS (VS − Y,t )(1 + κa). 2 The velocity VS and the thickness hS of a free stretching liquid sheet with a uniform velocity gradient can be obtained from the mass and momentum balance equations as VS =

Y , t + t0

hS =

hS0t0 , t0 + t

S=

1 t + t0

(6.129)

where t0 is a parameter and hS0 is the initial sheet thickness. The results of the numerical integration of the system of equations (6.123)–(6.125) are shown in Figs. 6.38–6.41. √ The boundary conditions are periodic at x = 0 and x = 2a0 π /ξ ∗ , where ξ ∗ = 1/( 2a0 ) is the wavelength of the most unstable mode of an inviscid infinite cylindrical jet. In Figs. 6.38 and 6.39 the evolution of the rim position and shape at several time instants are shown for the case of a stationary (non-stretching)

6.6 Mechanism of Splash

307





− − −

Figure 6.38 Nonlinear rim deformation. The sheet is stationary (non-stretching): t → ∞, VS = 0, ∗ S = 0, hS = 0.3a0 . The initial  rim radius is a = a∗0 [1 + 0.2 √ cos(ξ x)] and the initial velocity of the rim centerline is Y,t = − 2σ /(ρh0 ), where ξ = 1/( 2a0 ). The initial shape of the rim centerline is undisturbed. The Ohnesorge number is Oh = 0.01. The time instants are 4 ×, 6 ×

and 8 × ρa30 /σ from top to bottom. The axes labels are dimensionless, rendered dímensionless by the initial average rim radius a0 (Roisman 2010). Reproduced with permission.

free sheet. Figure 6.38 shows the effect of the initial perturbation of the radius of the rim cross-section, whereas Fig. 6.39 shows the influence of the initial perturbation of the rim centerline. In both cases the capillary forces stabilize the rim bending. The rim cross-section deforms solely due to the Rayleigh capillary instability. This result is in agreement with the numerical simulations of Fullana and Zaleski (1999). In Fig. 6.40 the effect of a relatively weak sheet stretching leading to rim acceleration is shown. This situation is relevant to the flow in the ejecta sheets produced by drop impact onto a liquid film, albeit these sheets are subjected to additional azimuthal stretching (Thoroddsen 2002, Josserand and Zaleski 2003). The velocity gradient in such sheets appears due to the deceleration of the propagation of the base of the crown (Roisman and Tropea 2002). The perturbations of the rim centerline and of the rim radius shown in Fig. 6.40 grow rather quickly leading to the appearance of bulbous regions frequently observed during the initial stage of splashes caused by drop impact (Fig. 6.30). The rim shown in Fig. 6.41 corresponds to a stronger sheet stretching than in the case of Fig. 6.40. The initial perturbations of the rim radius are relatively small, 0.01a0 , while the initial perturbations of the rim centerline and of the axial velocity are in the range obtained from the linear stability analysis (Fig. 6.35). The case shown in Fig. 6.41

308

Drop Impacts with Liquid Pools and Layers





− − −

Figure 6.39 Nonlinear rim deformation. The sheet is stationary (non-stretching): t → ∞, VS = 0, S = 0, hS = 0.3a0 . The  initial rim radius is undisturbed, a = a0 , and the initial velocity of the rim centerline is Y,t = − 2σ /(ρh0 ). The initial shape of the rim centerline is Y = 0.5a0 cos(ξ ∗ x), √ ∗ where  ξ = 1/( 2a0 ). The Ohnesorge number is Oh = 0.01. The time instants are 4 ×, 6 × and

8 × ρa30 /σ from top to bottom. The axes labels are dimensionless, rendered dimensionless by the initial average rim radius a0 (Roisman 2010). Reproduced with permission.

(middle) corresponds to the same problem parameters as in Fig. 6.41 (left), except for a much higher Ohnesorge number, Oh = 0.5. Viscosity diminishes the rate of the instability growth, but cannot prevent it, as shown in Fig. 6.41 (right) for a longer time. For long times and strong rim bending deformations the motion of the rim can be assumed quasi-steady with the relative velocity in the normal direction determined by the Taylor rim velocity given by Eq. (2.159) in Section 2.7 in Chapter 2. Indeed, the free rim at the crown top is formed and propagates over the crown sheet with the speed Vrim = [2σ /(ρh)]1/2 , with h being the sheet thickness (Taylor 1959b, Culick 1960, Savva and Bush 2009). Formation of secondary droplets at the crown top is frequently attributed to the capillary instability of the free rim (see Section 1.10 in Chapter 1), which is considered as a toroidal thread and assumed to be fully detached from the crown (Mundo et al. 1995, Range and Feuillebois 1998) or attached to it (Fullana and Zaleski 1999, Zhang et al. 2010, Agbaglah and Deegan 2014). Yarin and Weiss (1995) argued, however, that this mechanism significantly under-predicts the number of secondary droplets breaking up simultaneously on the crown top in their own and in Levin and Hobbs’ (1971) experiments, and accordingly attributed secondary droplet formation to the bending perturbations of the toroidal rim. These perturbations could be triggered, for example, by the wall roughness at the moment when a liquid element is being

309

6.6 Mechanism of Splash





− −





Figure 6.40 Nonlinear rim deformation, the effect of the sheet stretching leading to the rim

acceleration: t0 = 10 ρa30 /σ , h0 = 0.3a0 . The initial rim radius is a = a0 [1 + 0.2 cos(ξ ∗ x)] and  √ the initial velocity of the rim centerline is Y,t = − 2σ /(ρh0 ), where ξ ∗ = 1/( 2a0 ). The initial shape of the rim centerline is undisturbed. The Ohnesorge number is Oh = 0.01. The time  instants are 4 ×, 6 × and 8 × ρa30 /σ from top to bottom. The axes labels are dimensionless, rendered dimensionless by the initial average rim radius a0 (Roisman 2010). Reproduced with permission.





− − − − −



− −



− −

− −

Figure 6.41 Nonlinear rim deformation, the effect of the film stretching leading to the rim

acceleration: t0 = 2 ρa30 /σ , h0 = 0.1a0 . The initial rim radius is a = a0 [1 + 0.01 cos(ξ ∗ x)], the initial shape ofthe rim centerline is Y = 0.005a0 cos(ξ ∗ x), the initial axial velocity distribution is U =  −0.01 σ /(ρa0 ) sin(ξ ∗ x) and the initial velocity of the rim centerline is √ Y,t = − 2σ /(ρh0 ), where ξ ∗ = 1/( 2a0 ). The axes labels are dimensionless,  rendered dimensionless by the initial average rim radius a0 . Left: Oh = 0.01, t = 9.5 ρa30 /σ , middle:   Oh = 0.5, t = 9.5 ρa30 /σ , right: Oh = 0.5, t = 14.5 ρa30 /σ (Roisman 2010). Reproduced with permission.

310

Drop Impacts with Liquid Pools and Layers

propelled into the crown. Then the crown rim shape y = Y (x, t ) is governed by the eikonal equation (Yarin and Weiss 1995, Roisman and Tropea 2002)

 2 1/2 ∂Y ∂Y = Vrim 1 + (6.130) ∂t ∂x with Y being the coordinate directed along the crown sheet towards the solid surface, and x being the circumferential coordinate over the crown sheet. Solution of Eq. (6.130) shows that a perturbed rim always evolves into a cusped shape at any section where it was initially convex upward. At the cusp sites the two neighboring sections of the free rim impinge onto each other, squeezing a thin jet directed upwards from the crown similar to the photographs in Figs. 1.6b and 1.6c in Section 1.10 in Chapter 1 and also sketched in Fig. 6.44 in the following Section 6.7.

6.7

Impact of Drop Train The present section deals with drop or drop train impacts onto pre-existing liquid films, or films created on the wall by impacts of previous drops. Normal impacts of successive monodisperse ethanol drop trains (with drop diameter D in the 70–340 µm range and the impact velocity V0 up to 30 m/s) onto a solid surface were studied experimentally in Yarin and Weiss (1995). After the first impact, the surface was permanently covered √ by a thin liquid film with thickness of the order of h ≈ ν/ f , with f being the impact frequency and f −1 being the characteristic time of a single impact. For f ∼ 104 s−1 and the kinematic viscosity ν ∼ 10−6 m2 /s the values of h were in the 20–50 µm range, with the ratio h/D being close to 1/6. The film thickness was sufficiently large relative to the mean surface roughness (1 or 16 µm for the two surfaces used). The experiments revealed two characteristic flow patterns on the surface shown in Fig. 6.42. At sufficiently low impact velocities the drops spread over the surface, acquiring the shape of lamellae with a visible outer rim (Fig. 6.42a). At still lower impact velocities practically no rim was visible, and the entire flow pattern can be characterized as spreading. By contrast, at higher impact velocities the lamellae later took the shape of crowns (Fig. 6.42b) consisting of a thin liquid sheet with an unstable free rim at the top, from which numerous small secondary droplets were ejected. The impact patterns of the type shown in Fig. 6.42b are called splashing. Such phenomena as crater formation and liquid ejection from the crater center in the form of the so-called Worthington jet (Worthington 1908) [see Section 1.1 in Chapter 1 and Section 6.5 in this chapter] do not occur after drop impacts onto the pre-existing thin (compared to the drop diameter) liquid films on the wall. Note that such thin films may be the remains of the preceding drop impacts. The experimental threshold velocity for drop splashing in a train of frequency f was established in Yarin and Weiss (1995) as  1/4 σ ν 1/8 f 3/8 . (6.131) V0S = 18 ρ

6.7 Impact of Drop Train

(a)

311

(b)

Figure 6.42 (a) Spreading ethanol drops illuminated stroboscopically. The spreading lamellae at

two different stages can be recognized. Drop diameter D = 279 µm, impact velocity V0 = 7.8 m/s, Weber number We = 588, Reynolds number Re = 1409 and Ohnesorge number Oh = 1.72 × 10−2 . (b) Splashing ethanol drops illuminated by a single flash. Drop diameter D = 276 µm, impact velocity V0 = 12.7 m/s, We = 1542, Re = 2270, Oh = 1.73 × 10−2 (Yarin and Weiss 1995). Reprinted with permission.

Drop spreading occurs at the impact velocities V0 < V0S , whereas at V0 > V0S splashing and formation of a crown and multiple secondary droplets takes place as in Fig. 6.42b. Here and hereinafter the subscript S denotes the splashing threshold. A remarkable finding was that drop diameter D does not affect the splashing threshold, which implies that the crown originates from the liquid lamella at the surface long after the memory of the squashed primary drop has faded. Due to the small scales involved, only inertia and surface tension are significant factors (with viscosity involved only via √ the film thickness h ∼ ν/ f ), while the role of gravity is negligible. By contrast, in hydraulic jumps, even small-scale ones – i.e. in the radially spreading water films created by the impinging jets in kitchen sinks – gravity is the main driving force, although surface tension has to be taken into account (Bush and Aristoff 2003) [see Section 3.3 in Chapter 3]. Splashing and hydraulic jumps are totally distinct phenomena. Single drop impact onto a pre-existing film of the same liquid is an important limiting case of a drop train impact, which are worth considering together. Single drop impacts onto pre-existing films were studied in Cossali et al. (1997), Wang and Chen (2000) and Rioboo et al. (2003). In all these cases, crown formation, i.e. splashing, was recorded at sufficiently high impact velocities. For single drop impacts the ratio D/V0 plays the role of the characteristic impact time, thus replacing f −1 of the case of drop train impacts (Yarin and Weiss 1995). Moreover, for thin pre-existing liquid films the initial film thickness h0 might be expected to play a less important role than that of the layer stopped by the viscous forces at the wall h ∼ (νD/V0 )1/2 . Then the splashing threshold of Eq. (6.131) with f ∼ V0 /D could be expected to hold for a single impact. Using the dimensionless groups described in Section 1.2 in Chapter 1, Eq. (6.131) can be recast as WeS Oh−2/5 ∼ 10 396. This implies that a single dimensionless group K = WeOh−2/5 [see Eq. (1.2) in Section 1.2 in Chapter 1] governs the splashing threshold in the case of a single impact onto a pre-existing liquid film. Accordingly, Cossali et al. (1997)

312

Drop Impacts with Liquid Pools and Layers

Figure 6.43 Radial position of crown’s rim rc as a function of the dimensionless time τ = 2πf t.

Experimental data are shown by symbols; the best fit is shown by the dashed line; the theoretical prediction [Eq. (6.136)] is shown by the bold line. Data obtained with ethanol drops with D = 271 µm, V0 = 12.5 m/s, f = 19.973 kHz, We = 1467, Re = 2194, Oh = 1.75 × 10−2 (Yarin and Weiss 1995). Reprinted with permission.

established the empirical condition of drop splashing on pre-existing liquid films in the following form K > KS = 2100 + 5880H 1.44 ,

0.1 < H < 1,

Oh > 7 × 10−3 ,

(6.132)

where H = h0 /D is the dimensionless pre-existing film thickness. The threshold value on the right-hand side in Eq. (6.132) was established for a specific value of the dimensionless roughness Rd = Ra /D = 5 × 10−5 , with Ra being the roughness amplitude. A higher roughness can significantly affect the flow in the extremely thin liquid lamellae on the surface, and thus, the threshold value of KS decreases (Cossali et al. 1997). The data of Wang and Chen (2000) and Rioboo et al. (2003) for lowviscosity liquids are also in agreement with Eq. (6.132). In splashing regimes crowns formed after a normal drop impact on a thin liquid layer propagate radially outward. The crown motion was recorded in the experiments of Yarin and Weiss (1995) with drop trains and an example of the experimental data is given in Fig. 6.43. The best-fit of the data in Fig. 6.43 (the dashed line) is rc = 1.12(τ − 1.28)1/2 D

(6.133)

where rc is the crown radial position and τ = 2πf t is dimensionless time. The data of Levin and Hobbs (1971) and Cossali et al. (1997) for a single drop impact onto thin pre-existing liquid layers also reveal the square-root dependence of rc on time

6.7 Impact of Drop Train

(a)

313

(b)

Figure 6.44 (a) Sketch of splashing mechanism: 1, residual top of impacting drop; 2, solid

surface; 3, section of the crown-like sheet propagating outwards; 4, cross-section of free rim; 5, secondary droplets formed from the cusps of the free rim; 6, liquid layer on the solid surface. (b) Free rim and secondary droplets magnified; 1, crown-like sheet; 2, free rim at its top edge; 3, cusp; 4, thin jet emerging at cusp; 5, secondary droplets formed due to the capillary breakup of the jet (Yarin 2006).

in close agreement with Eq. (6.133); Cossali et al. (2004) reported the exponent value of 0.43 instead of 1/2. Flash illumination of a crown reveals the details of secondary droplet formation there. The photographs combined in Fig. 1.6 in subsection 1.10.1 in Chapter 1 show a free rim on the crown top with multiple thin jets emerging from the cusps. These jets break up into droplets due to the capillary breakup mechanism. This is a good illustration of the general fact that a two-dimensional liquid sheet is broken by the surface tension, but indirectly, since the surface energy increases only through rearrangement into a free rim at the edge (Taylor 1959b). On being disturbed, a rim on any liquid film inevitably forms cusps (Yarin 1993); also see Eq. (6.130) in Section 6.6 in this chapter. The cusps become the sites where the practically one-dimensional jets are squeezed out thereby undergoing capillary breakup due to the Rayleigh capillary instability discussed in Section 1.10 in Chapter 1. Thus, the mechanism leading to the secondary droplet formation in splashing is, in fact, a multi-stage rearrangement of a two-dimensional continuum (liquid sheet in the crown wall) into a one-dimensional one (thin jets on the crown top). A sketch of the splashing mechanism is shown in Fig. 6.44, with the detailed experimental information on the number of jets and secondary droplet sizes on the crown top available in Cossali et al. (2004). This analysis of rim motion is valid for strong rim bending deformations. Linear analysis of the rim dynamics for small bending deformations of its centerline is discussed in Section 6.6. At first glance, the real pattern of drop splashing in a train appears too complicated for any theoretical tool except the numerical ones. However, the above-mentioned fact that the crown originates from the liquid lamella at the surface after the impinging drop has already been squashed, indicates that the underlying mechanism is relatively simple and its description could be based on a quasi-one-dimensional model. Such a model (Yarin and Weiss 1995; for the details see Section 2.2 in Chapter 2) considers a situation where, after a drop impact, a circular spot near the lamella center has a

314

Drop Impacts with Liquid Pools and Layers

distribution of the initial outward velocity of the order of the impact velocity V0 . The liquid moving outward from this central spot impinges onto the surrounding liquid in the lamella moving slower or being at rest. Since the liquid is incompressible, formation of an ordinary shock wave due to the inward collapse of the material elements (in the gas-dynamical sense) is ruled out. However, if the condition  1/4 σ ν 1/8 f 3/8 , (6.134) V0 ρ holds for a drop train impact, the surface tension is totally dominated by liquid inertia [if f = V0 /D as for a single drop impact, Eq. (6.134) is equivalent to K = WeOh−2/5 1]. As a result, due to the presence of the free surface, a kinematic discontinuity in the velocity distribution emerges, which is reminiscent of a shock wave in an incompressible liquid. Then, the liquid from the central spot moving along the surface and impinging onto the surrounding liquid moving slower or being at rest, is propelled upward as a thin layer (the crown). The discontinuity propagates toward the thicker section of the film at the surface, detaching a part of it in the manner of a cutter and propelling it into the crown. The theoretical splashing threshold (6.134) fully agrees with the experimental one (6.131). The theory [see Section 2.2 in Chapter 2 and Yarin and Weiss (1995)] also predicts the crown position as it moves outward, as per V 1/2 rc = 1/4 1/2 01/8 1/4 3/8 (τ − τ0 )1/2 , D 6 π ν D f

(6.135)

where the dimensionless shifting time τ0 = 2πf t 0 is similar to the “polar distance” required for comparison of self-similar solutions for laminar and turbulent jets with experimental data (Yarin 2007). Note that Eq. (6.134) follows from Eq. (2.52) in Section 2.2 in Chapter 2; see Eq. (6.12) in Section 6.2 accounting for the fact that the parameter involved there is, in fact, negative. For the conditions corresponding to Fig. 6.43, Eq. (6.135) yields rc = 1.29(τ − 1.28)1/2 , D

(6.136)

which is compared to the experimental data and their best fit, Eq. (6.133), in Fig. 6.43 (the bold line). The agreement is fairly close, except that the theoretical values of the ratio rc /D exceed the experimental ones, due to the exclusion of the viscous losses at the moment of impact. For a single drop impact on a thin pre-existing liquid layer of thickness h0 , the crown position is given by  1/4 V01/2 2 rc = (t − t0 )1/2 (6.137) D 3 D1/4 h1/4 0 with t0 being the shifting time. Equation (6.137) describes the experimental data of Levin and Hobbs (1971) fairly well, with an overestimate of about 10%, which is similar to the accuracy of Eqs. (6.135) and (6.136) versus the data in Fig. 6.43.

6.8 References

315

Note also that Cossali et al. (1997) have distinguished two types of the splash: corona splash and prompt splash (when small secondary droplets are ejected from the region where the drop contacts the free surface of the film; see Fig. 4.23 in Section 4.7). In addition to corona splash, many other drop impact outcomes have been observed in experiments, namely sticking, rebound and deposition (Stanton and Rutland 1998). The stick regime occurs when a nearly spherical drop adheres to the film. The condition for the sticking regime, determined from the experiments (Stanton and Rutland 1998, Walzel 1980, Rodriguez and Mesler 1985, Jayaratne and Mason 1964) is We < 5. However, this regime should probably depend also on the viscosity of the ambient gas, since the phenomenon is associated with the squeezing of a gas layer between the impacting drop and the liquid layer. The rebound regime occurs in the Weber number range 5 < We < 10, as found in Rodriguez and Mesler (1985). For the Weber number higher than 10 the impact leads to the deposition of the impinging drop if the impact parameters are below the splash threshold.

6.8

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Part III

Spray Formation and Impact onto Surfaces

7

Drop and Spray Diagnostics

A description of collision phenomena involving drops and/or sprays requires a characterization of the drops before and after the collision as well as information about possible liquid films if impact on a solid surface is involved. The present chapter is devoted to the various techniques used to visualize and characterize drops, sprays and films. Independent of the measurement technique employed, collision phenomena are often described in terms of statistical quantities and Section 7.1 provides some fundamental definitions in common use. The remainder of the chapter, dealing with measurement techniques for drops and sprays, is divided into three sections: non-optical measurement techniques (Section 7.2), direct imaging techniques (Section 7.3) and non-imaging optical techniques (Section 7.4). The measurement of liquid films on a surface is treated separately in Section 7.5. Very general reviews of measurement techniques for drops and sprays can be found in textbooks (Lefebvre 1989, Liu 1999), handbooks (Crowe 2005) and review articles (Bachalo 1994, Chigier 1983, Jones 1977); however, many techniques discussed have been superseded by more recent developments of imaging and non-imaging optical methods. The field of optical diagnostics has developed rapidly in recent years, primarily due to improvements in illumination technology (LEDs, solid-state lasers, etc.) and camera/detector technology, offering higher temporal and spatial resolution visualization of transient phenomena. Perhaps for this reason more recent review articles and handbook entries addressing spray measurement technology concentrate more on developments of optical techniques, e.g. (Bachalo 2000, Fansler and Parrish 2015, Tropea 2011).

7.1

Fundamentals In this section some basic relations expressing the most common quantities necessary to describe impacting drops and sprays onto surfaces – input and outcome – will be presented, with special attention on how these quantities are derived from experimental measurements. The most important fundamental quantities to be acquired are r flux density distributions (e.g. number or diameter flux density) r local concentration (e.g. number or mass concentration)

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Drop and Spray Diagnostics

r local probability density function (PDF) of particle properties (e.g. diameter, velocity, and their moments). Many further quantities of interest can be derived from these, for instance, impacting force, deposited mass, spray pattern, spray angle or, through integration, overall volumetric flow rate. The quantities named above will be discussed individually in the following subsections; however, it is noteworthy that there is not a high degree of uniformity in measurement practice, perhaps related to the relative lack of comprehensive standards concerning measurements in sprays (ASTM-E-1620 2004, ASTM-E-799 1981, ISO-9276 2004, ASTM-E-1260 2003). For certain specific applications of sprays more detailed standards have been developed, for instance for oil pressure atomizing nozzles (DIN-EN299 2009), for agricultural irrigation equipment (ISO-11545 2009), for crop protection (ISO-22856 2008), pesticide drift adjuvants (ASTM-E-2798 2011) or for spray-tip classification according to droplet size (ASAE-S-572 1999), but only recently have efforts been made to unify computational procedures for measuring and comparing spray properties (DIN-SPEC-91325 2015).

7.1.1

Flux Density Vectors and Concentration Perhaps one of the most important quantities characterizing a spray is the local flux density,1 a vector quantity taking the direction normal to the considered plane through which the flux density is evaluated. This plane is typically determined by the measurement technique or measurement device orientation and a transformation to flux density through a different plane, for instance normal to a surface, may be subsequently necessary. The measurement of flux density clearly requires a measurement of velocity, since the direction of drop movement must be known. For most practical situations this requires at least two velocity components, if not three, to be measured, depending to what extent symmetry of the spray configuration can be exploited. The flux density vector q of an arbitrary scalar property P of drops in the direction normal to a surface A is given for an ensemble of NP drops by qP =

N 1 is necessary because not all drops are necessarily detected, and furthermore, not all detected drops are validated, for instance because of non-sphericity constraints, or in the case of the phase Doppler technique, because more than one drop may be present in the detection volume simultaneously. Other reasons for drops not being detected or validated include factors such as low signal-to-noise ratio, or with imaging techniques, when the drop is out of focus or below a detection threshold in intensity. Unfortunately this factor is neither constant for different measurement techniques, nor is it universal for different spray conditions; hence it represents an uncertainty in the final computed quantities. One example of how ηval can be estimated is given in Roisman and Tropea (2001). The flux QP of the property P is obtained by integrating the flux density over the reference area  (7.3) QP = qP · dA. A

Common examples for the scalar property P are Pi = 1

number flux density

diameter flux density Pi = D p,i π ρP 3 D p,i mass flux density Pi = 6 π ρP 3 D |v p |2 kinetic energy flux density Pi = 12 p,i where ρ p is the density of the drop material.

(7.4) (7.5) (7.6) (7.7)

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Drop and Spray Diagnostics

The concentration cP of the scalar property P is a scalar and can be expressed in the form Nval ηval Pi 1 cP = T i=1 Aγ i (D p,i , v p,i )|v p,i |

(7.8)

where |v p,i | is the velocity magnitude of the ith particle/drop. The denominator corresponds to a detection volume, dependent in size on the velocity and size of each drop. The most common concentration used is the number concentration with Pi = 1: cP =

7.1.2

Nval ηval 1 . T i=1 Aγ i (D p,i , v p,i )|v p,i |

(7.9)

Probability Density Functions of Particle Properties Most properties of sprays and drops can only be expressed statistically and the foundations for such descriptions are the probability function, the probability density function and its moments. The probability function of a random variable x, given by P(x) =Prob[x(k) ≤ x], is simply the probability that a particular value x(k) is less than the value x. The probability density function is then   Prob[x < x(k) ≤ x + x] . (7.10) p(x) = lim x→0 x Besides the obvious facts that P(−∞) = 0, P(∞) = 1 and p(x) ≥ 0, the following expressions hold

∞ (7.11) −∞ p(x) dx = 1

x dP(x) P(x) = −∞ p(ξ ) dξ ; = p(x). (7.12) dx Both the probability and probability density functions are continuous in x and must be estimated from a discrete ensemble of values when measurements are performed. For this the histogram density as an analog to the probability density function is used: h(xk < x ≤ xk+1 ) =

Nk 1 N x

(7.13)

where N is the total number of values in the ensemble, k is the class number and Nk is the number of values in the interval xk (xk < x ≤ xk+1 ). The kth moment of the probability density distribution is defined by  ∞ xk p(x) dx (7.14) mk = −∞

and the central moment is obtained by first subtracting the mean of x from each value. The first moment (k = 1) is simply the mean value and the second central moment is the variance of the random variable x.

7.1 Fundamentals

327

Figure 7.2 Visualization of local volume and flux density distributions. The particles which move

through the unit area during time T are projected with dark circles. The positive correlation between particle size and particle velocity leads to a larger mean particle diameter in the flux density distribution than in the volume distribution. Small particles at the back of the unit volume do not cross the unit area in time T , whereas large particles do.

One can differentiate between the probability density distribution of some property (e.g. drop size, drop velocity, etc.) at a particular instant in time and over some volume (called volume or spatial distribution), and a local probability density distribution of all particles crossing a unit area per unit time (called flux density distribution). These two distributions will be different whenever there exists a non-zero correlation between velocity and the particle property being examined. This difference can be illustrated using the following example. Assuming in the visualization in Fig. 7.2 that large particles are faster than small particles (correlation between velocity and particle diameter is non-zero), implies that per unit time more large particles will pass through the reference unit area than small particles; hence the volume distribution of particle size, represented by all particles in the shaded volume, will differ from the flux density diameter distribution, represented by all particles passing the reference plane in a given measurement time interval. The two distributions will be identical if there is no correlation between the velocity and investigated particle property. Volume distributions are typically obtained from high-speed imaging or sensing of diffracted light from a particle ensemble, and flux density distributions are typically obtained by collection techniques or by optical instruments capable of sensing individual particles in flight. Certain sampling methods may provide neither flux nor volume distributions.

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Drop and Spray Diagnostics

Table 7.1 Mean diameters and their applications. Diameter

A

B

Name of mean diameter

Application

D10 D20 D30 D21 D31 D32 D43

1 2 3 2 3 3 4

0 0 0 1 1 2 3

Arithmetic mean diameter Surface area mean diameter Volume mean diameter Surface area – length Volume – length Sauter (SMD) De Brouckere or Herdan

General comparison Monitoring of surface area Monitoring of volume Absorption Evaporation, molecular diffusion Mass transfer, reaction Combustion equilibrium

Spatial distributions may be transformed into flux density distributions by multiplying the number (or fraction) of particles of a given velocity by that velocity, as described in DIN-SPEC-91325 (2015) or ASTM-E-1620 (2004). The above densities and distributions are valid for one position (or volume) of the spray. Distributions for the whole flow field can be computed from these local distributions through a weighted integration, as outlined in DIN-SPEC-91325 (2015). Number concentration distributions for diameter are often modeled using analytic functions containing only a few parameters. The motivation is simply to reduce the number of parameters required to describe a large number of measurement results and to simplify comparisons between measurements taken under different conditions. Typical distributions used in practice are the normal distribution, the log-normal distribution or empirical distributions introduced by Nukiyama and Tanasawa (1939) or Rosin and Rammler (1933). The log-hyperbolic distribution, as introduced by Barndorff-Nielsen (1977) and later applied by Bhatia and Durst (1989) has also been found to provide a good description of measurement results for a wide variety of applications. These authors also compare their results with several other widely used distributions. Villermaux (2007) postulates that many fragmentation phenomena associated with liquid sheets and sprays can be viewed as a sequential cascade of breakups leading to an exponential-like form for the size distribution. Together with occurring aggregation phenomena he finds that these physical processes are well described by a gamma distribution. However, experience shows that there is little universality in any pre-supposed distribution, since each application may exhibit very specific boundary and operating conditions. In describing impacting sprays and drops, several mean diameters are in common use, obtained using the ratio of the Ath and Bth moments of diameter: 1 ⎛ ⎞ A−B D p,max A D p,min D p p(D p ) dD p ⎠ . (7.15) DAB = ⎝ D p,max B p(D ) dD D p p p D p,min The physical meaning of the various mean diameters and their areas of application are summarized in Table 7.1 adapted from Mugele and Evans (1951) and Lefebvre (1989). It is typically one of these mean diameters which is used in specifying the Reynolds or Weber number of impacting drops and sprays.

7.2 Non-Optical Measurement Techniques

329

Table 7.2 Representative diameters characterizing a spray, according to ASTM-E-799 (1981). DN0.5 DL0.5 DA0.5 DV 0.5 DV 0.9

Number median diameter Length median diameter Surface area median diameter Volume median diameter Drop diameter such that 90% of the total liquid volume is in drops of smaller diameter

Further descriptors are used to parameterize the measured probability density distributions, as summarized in Table 7.2. For instance the polydispersed nature of a spray is often quantified using the “relative span” relative span = SP =

7.2

DV 0.9 − DV 0.1 . DV 0.5

(7.16)

Non-Optical Measurement Techniques Liu (1999) and Lefebvre (1989) provide overviews of some non-optical measurement techniques, summarized in Table 7.3. The majority of these techniques were developed prior to the appearance of imaging and non-imaging optical techniques of high resolution and few are in widespread use today. None of the listed techniques provide velocity information about the droplets. Nevertheless, three of the methods are still used for specific applications. One is the simple inspection of spray deposition on surfaces, used widely in the automotive paint industry to confirm the proper functioning of spray nozzles/atomizers. This is a manual variation of the “collection of drops on slides” method; however this method remains qualitative, based on extensive experience of the user. The second non-optical method used for very specific applications, especially for characterizing hole-to-hole non-uniformities in mass produced fuel injection nozzles, are force impactors (Postrioti and Battistoni 2010). Using piezoelectric-force sensors traversed through the spray, the time-resolved momentum flux of a spray can be Table 7.3 Overview of non-optical methods for measuring drops and sprays. Principle

Method

Remarks

Mechanical

Collection of drops on slides Molten wax technique Drop freezing Force impactors Cascade impactors Patternators

> 3 µm

local momentum of spray used for inhalators volume/mass distributions

Charged wire technique Hot wire technique

1–600 µm

Electrical

330

Drop and Spray Diagnostics

Figure 7.3 Patternator system used in Dullenkopf et al. (1998). Reproduced with permission.

Copyright (1998) by John Wiley and Sons.

determined with high spatial and temporal resolution over the injection cycle. However, this technique is not so relevant in the context of collision phenomena and, furthermore, is commercially available only for production testing of fuel injection nozzles. The third widely used technique is spray patternation, used to determine the spatial distribution of mass flux in the spray cone (McVey et al. 1989). Generally two methods are used: systems in which the total fluid in the spray is collected, and systems using isolated probes or an array of probes in which only a fraction of the fluid in the spray is captured. Patternators are also used as a volume-flux reference for non-intrusive optical instrumentation, such as the phase Doppler technique. One such patternator is pictured in Fig. 7.3. In this implementation 50 collection tubes, each with an inner diameter of 4.8 mm, were used to achieve a high spatial resolution across the spray. After a given measurement period, the height of the fluid in each tube is measured, yielding the spatial distribution of the volume flux. The relative error in the reading at each individual tube was minimized by graduating the diameter of the tube, as sketched in the figure. Furthermore, an evacuating airflow was applied to approximate iso-kinetic sampling conditions. The average error was estimated to be around 5% (Dullenkopf et al. 1998). Most patternators in use are home-made, although some commercial instruments are available, most arising from the crop spraying industry.

7.3

Direct Imaging Advances in direct imaging over the past decade have arguably had the largest impact on the experimental investigation of collision phenomena. Not only is the adage “seeing is believing” applicable, but with the advent of ultrafast CMOS (complementary metaloxide-semiconductor) cameras providing good resolution up to 1 000 000 fps (frames per second), the physical phenomena at play during collision events can often be directly

7.3 Direct Imaging

331

Figure 7.4 Impact of a spray on a convex target.

recognized from the images. Furthermore, the laboratory setup is typically straightforward, flexible and the user obtains immediate access to results. Even modest computers provide sufficient power for advanced image processing for extraction of quantitative data. Although imaging encompasses several different measurement techniques, such as shadowgraphy, Schlieren (Settles 2012) or ballistic imaging (Linne et al. 2006), the main technique employed to investigate collision phenomena is a direct image, typically using back-lit illumination. In setting up imaging systems five issues are important: r r r r r

sensitivity/contrast spatial resolution depth of field field of view (FOV) temporal resolution/motion blur

all of which are apparent in the sample image shown in Fig. 7.4, showing the impact of a spray on a convex target, taken using a high-speed camera. Except for the depth of field, these factors are strongly technology dependent and improve with each generation of camera. The depth of field is strictly an optical measure given by the lens system used. All of these issues will be discussed below beginning with an overview of camera technology and the achievable sensitivity.

Sensitivity/Contrast The selection of CCD (charged-coupled device) and CMOS cameras is large and a recommendation for a certain camera is not possible, not only because of rapidly changing specifications, but also because each application has certain requirements on the camera. In general, imaging of collision phenomena is not as demanding as some other applications of fast cameras, such as particle image velocimetry or spectroscopic measurements, where available light intensity may necessitate an intensified camera. An intensified camera achieves higher signal strength at a much lower cost than increasing the illumination energy; however the signal-to-noise ratio (SNR) of intensified cameras is generally lower than that of cameras without intensifiers, as illustrated in Fig. 7.5 taken from Hain et al. (2007). For higher frame rates, the CMOS camera is advantageous, but the image quality of state-of-the-art CCD cameras is still better. The frame rate of cameras is usually limited by the read-out, thus if images smaller than full frame are sufficient, higher frame rates can be achieved. For instance, a CMOS

332

Drop and Spray Diagnostics

Figure 7.5 Dependence of signal-to-noise ratio on signal strength S for various CCD and CMOS cameras (Hain et al. 2007). With permission of Springer.

camera with a frame rate of 20 000 fps at full resolution of 1024 × 1024 pixels would run at 40 000 fps at a resolution of 1025 × 512 pixels or at 120 000 fps at a resolution of 512 × 264 pixels. The memory size limits the duration of recording, or number of frames that can be captured without a break for data transfer. Typically such a camera would have a sensitivity of 25 000 ISO (monochrome), a 12-bit analog-todigital converter (4096 gray levels), an electronic shutter of 1 µs and a 20 µm pixel size. Currently the fastest cameras available commercially offer frame rates up to 10 Mfps. A monochrome camera is for most applications involving collision phenomena advantageous, offering much better light sensitivity, better spatial resolution and more flexible wavelength filtering, should that be necessary. An excellent overview of developments in high-speed imaging, especially for hydrodynamic problems, can be found in Thoroddsen et al. (2008).

Spatial Resolution The spatial resolution of the optical system is important to determine the minimum size of detectable objects, or the shape resolution, for instance contact angle. The obtainable spatial resolution of an image is determined by the imaging objectives used, but the objective should be compatible with the sensor (pixel) size of the camera. Compromises are almost always necessary, since experimental setups often dictate minimum working distances (distance between object and lens). In this context it is necessary to discuss magnification and the diffraction limit of an optical system. Considering the simple imaging system depicted in Fig. 7.6, the image is in focus when the condition 1 1 1 + = so si f

(7.17)

7.3 Direct Imaging

333

Figure 7.6 Position of object and image planes assuming a thin, spherical lens.

is satisfied, where f is the focal length of the lens. This is known as the Gaussian thin lens formula (Hecht and Zajac 1974). For such a system the transversal magnification is given by MT =

yi si xi f = =− =− yo so f xo

(7.18)

and the longitudinal magnification by ML =

dxi f2 = − 2 = −MT2 . dxo xo

(7.19)

If several lenses are combined and placed close to one another, the effective focal length and magnification become 1 1 1 1 = + + ... + f f1 f2 fN

(7.20)

MT = MT 1 · MT 2 . . . · MT N .

(7.21)

However, the achievable spatial resolution is ultimately limited by the diffraction limit of the optical system. The diffraction arising from a circular aperture is the well-known Airy disk (Sir George Biddell Airy, 1801–1892) with a diameter of the main intensity peak taking the value Ds = 1.22

Rλ 2a

(7.22)

where λ is the wavelength of light. This situation is depicted in Fig. 7.7. The central intensity peak, known as the diffraction limited spot contains 84% of the light. If the distance R is approximated with the focal length f then Ds = 1.22λ f # = 1.22λ/NA, where f # = f /2a and NA is the numerical aperture. Regardless of how small the object is in the object plane , the image will always be at least Ds in diameter on the image plane. Once two objects are so close together

334

Drop and Spray Diagnostics

λ

Aperture Image plane

y

Airy disk a

Y R

z

P

Σ

Z σ

Figure 7.7 Geometric relations describing diffraction from a circular aperture.

that their Airy disks become closer than half their width, they are no longer resolvable (Rayleigh criterion), which sets the fundamental resolution limit and which is independent of the number of pixels available on the sensor. The diameter of a small particle with diameter D p will appear on the image plane with the approximate diameter 1/2  (7.23) Di ≈ D2s + MT2 D2p with Ds = 2.44(1 + MT ) f # λ

(7.24)

where MT D p is known as the geometric image diameter. Therefore, it is sensible to choose a magnification such that the Airy disk is not smaller than one pixel on the sensor, otherwise object sizes resulting in images between the diffraction spot size and one pixel cannot be resolved. Equation (7.22) indicates that an improvement of spatial resolution is possible by increasing the diameter of the focusing lens or reducing R, which is equivalent to reducing f # or increasing the NA. Reducing the wavelength (λ) significantly is not usually a viable option. Here the compromise becomes evident. If a high magnification is sought, si [R in Eq. (7.22)] must be large (Fig. 7.6), increasing the Airy disk diameter Ds and thus the diffraction-limited resolution. Albeit in many practical systems this diffraction limit is not reached. The diffraction spot size also defines the limit to resolve thin liquid fragments, free surfaces,

335

7.3 Direct Imaging

c = circle of confusion DOF 2a Dr

Df so

Object

si Lens

Image plane

Figure 7.8 Notation describing the depth of field.

contact line intersections, etc. and is therefore a fundamental limiting factor of direct imaging.

Depth of Field The depth of field (DOF) is the distance between the nearest and farthest objects that appear acceptably sharp in an image. It expresses the unsharpness when an object is out of focus and is quantified by the “circle of confusion” (c) as sketched in Fig. 7.8 and given approximately by Total DOF ≈ 2NA · c(MT + 1)/MT2 .

(7.25)

Although the sharpness of an image is dependent on many factors, including the eyesight of the observer (!), a circle of confusion of 0.01 inches (0.25 mm) is universally used to specify the Total DOF. Notice that the depth of field is primarily a function of magnification and numerical aperture. Decreasing the NA increases the depth of field; in principle a pinhole camera will have an infinite depth of field, albeit with very low light intensity. Likewise, increasing the magnification will decrease the depth of field. An approximate expression for depth of field has also been given by Adrian (1991) as   1 2 #2 f . (7.26) DOF ≈ 4λ 1 + MT A practical example is useful to illustrate limitations and conflicting demands on an imaging system. A long-distance microscope is used to observe drop impacts onto a plane target. The working distance between the objective and the impact point is 190 mm, which for this lens results in a magnification of 1.52, a numerical aperture of 0.1, a depth of field of 0.05 mm and a field of view of 4.2 mm. For a wavelength of 600 nm the diffraction spot becomes Ds = 7.32 µm, which in the object plane corresponds to 4.81 µm. Typical pixel sizes on cameras are 6–9 µm, meaning that the lens system is well matched to the camera. It is often more practical and convenient to determine the DOF empirically. This can be easily achieved by capturing an image of a tilted line pattern, as shown in Fig. 7.9a. The line pattern is placed such that the entire DOF is spanned. In the example shown in the figure the resolution of single lines is lost at about 10 mm. This can be quantified by

Drop and Spray Diagnostics

(a)

(b) 40 30 20 I – Imean

336

10 0 −10 −20 −30 −40

DOF 0

5

10

15

x (mm) Figure 7.9 (a) Line pattern image used to empirically determine the depth of field; (b) intensity

distribution along the line pattern.

examining the intensity profile along the line pattern, shown in Fig. 7.9b. The DOF can be found as the position where the amplitude of the intensity fluctuations can no longer be distinguished from the background noise level. In consideration of the above remarks some special lens systems are therefore in common use when visualizing collision phenomena. These are telecentric objectives or long distance microscopes, either lens-based or as Maksutov–Cassegrain telescopes. Telecentric lenses offer a constant magnification independent of axial object position and yield no distortion due to perspective, but also have a lower NA. Long distance microscopes offer long working distances and, by selecting addition tubes (amplifiers), the Airy disk can be matched to the pixel size.

Field of View The field of view (FOV) necessarily reduces with increasing magnification and shorter working distances. By increasing the size of the objective, large working distances at large NA values can be achieved, e.g. a 11.4 cm diameter telescope objective with a working distance of 30 cm results in an NA of 0.098, a depth of field of 56 µm, an Airy disk diameter of about 6.7 µm (λ=540 nm). With a FOV of 2 mm × 2 mm, a magnification of approximately 50 is achieved. Further details and derivations for the topics discussed above can be found in Fischer et al. (2000).

Temporal Resolution The final issue to address is the required temporal resolution. This question contains two aspects. On the one hand motion blur should be kept to a minimum. As a rule of thumb, motion blur can be considered negligible if object interfaces do not move more than one pixel during the image capture. This can be achieved either by adjusting the shutter speed or by limiting the illumination in duration, for instance using a pulsed laser as an illumination source. The magnification must of course be considered when translating object velocity into pixel movement over time. The second issue is purely

7.3 Direct Imaging

337

Figure 7.10 Example of drop detection, yielding size and velocity of individual drops.

hydrodynamical in nature and relates to the necessary frame rate to resolve the physical processes being studied, for instance the deformation of a liquid fragment or the velocity of a detected drop. Here one must also consider any movement of the frame of reference. For instance to resolve a drop impacting onto a moving target with a camera in a fixed laboratory coordinate system a much higher frame rate will be required than for a drop impacting onto a stationary target with the same relative impact velocity.

Image Processing Processing of images is a broad topic which is comprehensively covered in numerous dedicated textbooks and handbooks (Jähne 2005, Salazar 2015), which includes operations designed to enhance existing images or extract specific information from a single image or an ensemble of images. Such operations include: r r r r r r r r r

histogram sliding, stretching and equalization gray level transformations convolutions masking edge detection (Prewitt, Sobel, Robinson, Laplacian operators) frequency domain analysis high and low pass filtering image compression optical object/character recognition

most of which are available in standard programs such as MATLAB (Gonzalez et al. 2010). One such example would be the detection of single droplets in an image and the determination of their velocity, derived from tracking each droplet over two consecutive images, as is illustrated in Fig. 7.10. One particularly creative use of high camera frame rates is illustrated by the study of liquid sheet breakup, shown in Fig. 7.11 for the breakup of a droplet in a cross-flow (see Section 8.2 in Chapter 8 ). By tracking the rim propagating into the liquid sheet from one frame to the next, the rim propagation velocity (Vr ) can be determined and the

338

Drop and Spray Diagnostics

Figure 7.11 Hole evolution during bag-breakup of a drop in a cross-flow. Time between images is

91 µs. Drop is water containing 0.1 wt.-% polyether siloxane OMS-1.

liquid sheet thickness (h) can be computed as [see Eq. (2.159) in Section 2.7]: h=

2σ ρVr2

(7.27)

where ρ and σ are the fluid density and surface tension. This is particularly attractive, since there is no other viable method to measure the thickness of an unsteady free liquid sheet. One further example of image processing is presented, because of its particular application to collision phenomena involving liquid sheets and lamellae. The challenge is illustrated in Fig. 7.12a, in which the raw image of a liquid sheet issuing from a flat fan nozzle is shown. Of interest is to uniquely distinguish between the liquid sheet and the background, from which for instance the breakup behavior can be quantified. Due to the very weak absorption of light by the thin liquid sheet, the gray scale alone is insufficient to distinguish liquid from background. To overcome this problem, a texture sensitive entropy filter is used for the distinction between liquid lamellae and background (Gonzalez et al. 2010). This filter considers the 5 × 5 pixel neighborhood of each pixel and sets the center pixel intensity to e=−

NI 

P(II ) log2 (P(I j ))

(7.28)

j=1

where e is the entropy, I is the pixel intensity value, NI is the number of possible intensity values and P(I j ) is the histogram count for intensity I j . The obtained entropy e is a measure for the randomness in the observed neighborhood. This behavior is exploited by using a very homogeneous diffusor for the background illumination. The homogeneous background intensity distribution yields rather small entropy values of e. When light

7.3 Direct Imaging

339

(a) Raw image

(b) Entropy filtered image

(c) Binarized image Figure 7.12 Sample images for the entropy filtering process.

is transmitted through the liquid lamella, the intensity distribution is slightly disturbed by small capillary waves and liquid edges which are always present in the flow. The heterogeneity in intensity values is thus increased, which significantly increases the local entropy. The entropy-filtered image can then be segmented by a simple thresholding method. A binary image is then obtained, where pixels set to one are occupied by part

340

Drop and Spray Diagnostics

of the liquid sheet. Examples of the entropy-filtered image and the binarized images are shown in Figs. 7.12b and c. Finally, reference is made to the Gallery of Fluid Motion presented online by the American Physical Society Division of Fluid Mechanics, which contains a large number of high-speed video entries of hydrodynamic phenomena, illustrating many of the features discussed above (http://gfm.aps.org/).

7.4

Non-Imaging Optical Measurement Techniques Table 7.4, adapted from Tropea (2011), provides an overview of optical measurement principles and techniques available to characterize drops and sprays. Numerous of these techniques are capable of measuring size and velocity of individual drops; fewer are able to also estimate flux densities, as is often desired when investigating collision phenomena, such as spray impact onto a target. Next to direct imaging, which has already been discussed in Section 7.3 in this chapter, the phase Doppler and time-shift techniques are the most appropriate for this application. The time-shift technique is a more recent development and has been comprehensively documented in Albrecht et al. (2003) with further developments reported in Damaschke et al. (2002) and Schäfer and Tropea (2014). However, currently available instruments using the time-shift principle provide only one velocity component, which, in light of the discussion in subsection 7.1.1 in this chapter is not generally sufficient for providing flux densities relative to an arbitrarily orientated target. For this reason, the focus of this section will be on the phase Doppler technique. The phase Doppler technique is based on interferometry in which two beams of like scattering order interfere on a detector to yield the size of spherical particles/drops. The velocity is obtained using the well-known laser Doppler technique, which is an integral part of the phase Doppler instrument. The working principle is only briefly summarized here, since more complete descriptions are readily available (Albrecht et al. 2003); nevertheless some specific features and peculiarities when used in investigating collision phenomena will be addressed. The optical arrangement for the most commonly used phase Doppler instrument is depicted in Fig. 7.13. Two coherent beams are brought to intersection in a measurement volume through which drops pass. The x-velocity component (ux ) is obtained from the frequency ( fD ) of the signals received at any of the three receivers and is given by fD =

2ux sin (!st /2) λ

(7.29)

where !st is the full intersection angle of the laser beams and λ is the wavelength of the light. A second component of velocity is easily obtained by intersecting a second set of beams of different wavelength in the same measurement volume but orthogonal to the first set, and using a detector with a corresponding wavelength filter to determine the uy velocity component. This corresponds to a standard, two-velocity component configuration of a laser Doppler instrument and will be discussed further below.

7.4 Non-Imaging Optical Measurement Techniques

341

Table 7.4 Overview of optical measurement techniques for drops and sprays, adapted from Tropea (2011). Abbreviations: PIV, Particle Image Velocimetry; PTV, Particle Tracking Velocimetry; LIF, Laser Induced Fluorescence; ILIDS, Interferometric Laser Imaging for Droplet Size; IPI, Interferometric Particle Imaging; SLIPI, Structured Laser Illumination Planar Imaging. Measurement Principle

Measured Quantities

Measurement Techniques

Imaging

size velocity shape

PIV/PTV Shadowgraphy/Schlieren Ballistic Imaging Glare-Point Separation

Intensity, Intensity Ratio SLIPI

size temperature species

Extinction/Absorption Modulation Depth Mie/LIF Ratio Two-Band/Three-Band LIF

Interferometry

size velocity refractive index temperature

Laser Doppler Velocimetry Phase Doppler Technique ILIDS/IPI Fraunhofer Diffraction Rainbow Refractometry Holography

Time Shift

size velocity (refractive index)

Time-of-Flight Pulse Displacement Time-Shift Technique

Pulse Delay

size velocity

Femtosecond Laser Methods

The phase Doppler instrument differs from the laser Doppler instrument in two fundamental respects. For one there are three receiving detectors placed at varying elevation angles ψi , as depicted in Fig. 7.13a. Furthermore, these detectors are placed at a very specific off-axis angle ϕ. The off-axis angle is chosen according to the scattering order of light which is to be used for the measurement. This is not an arbitrary choice, since one pre-requisite for the phase Doppler instrument to function properly is that one scattering order dominates the signal at the receiver. The concept of different scattering orders is illustrated in Fig. 7.14, which depicts various scattering orders (reflection, first-order refraction, second-order refraction,...) in terms of geometric optics. The question of one scattering order dominating must be resolved using appropriate programs to compute the light scattering from a spherical particle. This scattering is primarily a function of the particle size, expressed nondimensionally with the Mie parameter (xM = π D p /λ), and the relative refractive index m, which is the ratio of the particle refractive index to the refractive index of the surrounding medium. The results of such calculations, yielding conditions for which a single scattering order dominates, are summarized in the diagrams presented in Fig. 7.15. A practical example will illustrate the use of these diagrams. Water droplets in air have a relative refractive index of m = 1.33. Using parallel polarized light, an off-axis angle

342

Drop and Spray Diagnostics

(a) Est

x Θst z

(b)

U3 y U2

ϕ

ψ2

U1



(c) Phase difference ΔΦ [deg] ψr

360 deg

ΔΦ12 (Φ1−Φ2) ΔΦ13 (Φ1−Φ3)

U1 U3

ΔΦ13 (dp) ΔΦ32 (dp)

φr

y-z plane U2

ΔΦ12 (dp)

0 deg dp

dp,max

Particle diameter

Figure 7.13 (a) Optical arrangement of a standard, three-detector phase Doppler instrument. ϕ –

off-axis angle, ψi – elevation angle, !st – beam intersection angle; (b) Moiré fringe representation of interference; (c) The phase difference/diameter relations of a three-detector, standard phase Doppler system and their use to extend the size range beyond a phase difference of 2π .

Figure 7.14 Scattering orders from a spherical particle according to geometric optics.

7.4 Non-Imaging Optical Measurement Techniques

343

2.0 Perpendicular Polarization

Relative refractive index

1.8 1.6 1.4 1.2 1.0 0.8 0.6 2.0 Relative refractive index

1.8

Parallel Polarization

1.6 1.4 1.2 Dominant scattering order > 90%

1.0

Reflection (1) 1st order refraction (2)

Total reflection Rainbow angle

0.8

2nd order refraction (3)

Critical angle

0.6 0°

90°

Scattering Angle

180°

Figure 7.15 Scattering angle regions of dominant scattering order for perpendicular and parallel

polarized light as a function of scattering angle and relative refractive index. These diagrams have been processed from scattering functions of particle in the range xM =64.4. . .1290 (10 µm. . .200 µm for a wavelength of 488 nm). The contour lines correspond to the 92%, 94%, 95% and 98% regions of dominance (Albrecht et al. 2003). With permission of Springer.

of 30◦ –70◦ insures dominance of first-order refractive light at the detectors. This constraint influences strongly the design of experiments, since optical access at this angle must be available. The second differing factor between laser Doppler and phase Doppler systems is the use of three detectors. This can be explained by examining how the size of the particles is determined from the respective signals. This is most easily understood using a Moiré fringe representation of the interference taking place, illustrated in Fig. 7.13b. The two incident beams each result in a scattered field of a certain scattering order at the detectors. These two fields interfere, generating fringes in space; narrower together for large particles, farther apart for small particles. Therefore, the distance between the fringes corresponds to the particle size. The fringe spacing is not measured directly in the phase Doppler technique, but rather it is derived by measuring the frequency with which the fringes move across the detectors (directly related to the particle velocity through the measurement volume) and the phase difference between the signals registered by the two

Drop and Spray Diagnostics

Est

x Θst

Θp1



y Spherical particle (validated)

z

ϕ

EP1

Non-spherical particle (not validated)

V2

U2 V1 ψ1 ψ2

U1 Standard PD [deg]

344

Validation area

Sphericity line Phase difference planar PD [deg]

Figure 7.16 The optical configuration of a dual-mode phase Doppler instrument.

detectors. The relation between phase difference between detector 1 and 2 (12 ) and size D p are given as follows for reflected light and first-order refracted light, respectively √ √ √ 8π D p [ 1 − A + B − 1 − A − B], reflected, 12 = (7.30) √λ   8π 12 = D p [ 1 + m2 − m 2(1 + A + B) λ  (7.31) − 1 + m2 − m 2(1 + A − B)], first-order refracted, where A ≡ cos ψ cos ϕ cos(!st /2)

B ≡ sin ψ sin(!st /2).

(7.32)

Notable with these equations, as also with Eq. (7.29), is that all parameters are known from the optical arrangement; hence no calibration is required. Clearly, if the particles become larger, a 2π ambiguity will arise and this is avoided by adding a third detector, as depicted in Fig. 7.13a. The phase relation between detectors 1 and 2, compared to the phase relation between detectors 1 and 3 is given in Fig. 7.13c and from this it becomes evident that the three detectors extend the measurable size range of the phase Doppler instrument, while also resolving the 2π ambiguity. Other optical arrangements for the phase Doppler technique are possible and the most common variation is the Dual-Mode phase Doppler, shown in Fig. 7.16. This configuration combines a standard arrangement with two detectors and a planar optical arrangement with two detectors, yielding a meridional (standard) and equatorial (planar) size measurement. Only if the two size measurements are comparable, can sphericity of the particle be insured. This validation is shown in Fig. 7.16 using the relation between the two measured phase differences (standard and planar) for spherical particles. Nonspherical drops are generally not included in the processed statistic, because their size

7.4 Non-Imaging Optical Measurement Techniques

345

measurement is not reliable. The dual-mode configuration automatically extends the measurement range beyond the 2π ambiguity, since the planar slope of phase difference with size is much less than the slope of the standard system. The measurement range of size and velocity of a phase Doppler instrument will depend on numerous optical factors and features of the signal processing used. Normally a lower size limit of 1 µm is assumed, arising primarily from the ability to detect the phase difference within noise to a sufficient degree. The size range is typically 60:1. In principle there is no upper limit to measurable sizes; however non-sphericity usually becomes a problem when dealing with large drops. For measurements involving collision phenomena the phase Doppler technique is used primarily to measure the size and flux density of drops in a spray impinging on a target and of drops ejected from the impingement events. As outlined in subsection 7.1.1 in this chapter, measurement of flux density requires the velocity of each drop and also a reference area Aγ i (D p,i , v p,i ). Herein lies a major source of error with the phase Doppler technique, since the cross-sectional area of the detection volume lying perpendicular to the desired flux vector must be determined and this area will depend not only on the sensitivity of the detectors, but also on the drop size and, if an aperture is used, on the drop trajectory. The measurement volume is formed by laser beams with a Gaussian intensity profile; hence the fringes of the volume are much more weakly illuminated than the central portions. Small particles at the edge of the volume may no longer be detected as well as large particles, thus the effective detection volume becomes smaller for small particles. This effect is well known and documented, and various in-situ calibration procedures have been developed; nevertheless, the achievable accuracy remains poor, especially for sprays and drop streams in which large variations of velocity direction occur. Reference is made to Albrecht et al. (2003) for a more complete discussion and a detailed derivation of the effective area can be found in Roisman and Tropea (2001). A further consideration, especially relevant for measurement of spray impingement onto walls and liquid films, where the important parameters to be quantified include the deposition rate of the liquid onto the wall and characteristics of the splashed fraction − velocity, size and flux density of the drops in the secondary spray. When applying the phase Doppler instrument to characterize spray/wall impingement, the measurement volume is usually placed at a finite distance above the wall surface. The in-going (primary) droplets are then distinguished from out-going (secondary) droplets through the sign of the velocity component normal to the wall. It can be demonstrated that the portion and size of the wall surface associated with the impinging drops or the secondary drops must be determined by analyzing the drop trajectories between the measurement position and the wall. If the tangential velocity of the drops is significant or if the wall surface is curved, then the flux densities referenced to particular wall surface areas can be significantly different when neglecting these trajectories. These two effects are best explained using Figs. 7.17 and 7.18. In Fig. 7.17 the impingement of a hollow cone spray onto a hemispherical target is schematically depicted, whereby a nominal impingement point is marked on the target surface. To investigate the outcome of the spray impingement, the measurement volume of a phase Doppler instrument is traversed around the nominal impingement point at a

346

Drop and Spray Diagnostics

Figure 7.17 Tangentially moving secondary spray with erroneous declaration as primary drops.

constant distance above the surface. Two such measurement points on either side of the nominal impingement point are marked with a cross. The phase Doppler instrument is so aligned, that two velocity components U2 and V2 are acquired. The U2 component is aligned normal to the target at the mean impingement point. If now primary drops are differentiated from secondary drops using the sign of the U2 velocity component (negative is primary, positive is secondary), then errors will arise at the two shown measurement points. Secondary drops coming from the mean impingement point will be detected as primary drops if they are moving too close tangentially along the surface. This error can be alleviated if a transformation of the measured velocity into components aligned with the local target surface is performed. A second error arises due to the uncertainty of the wall contact point. This effect is illustrated in Fig. 7.18. Every wall contact point bears a spatial uncertainty due to the finite size of the detection volume of the phase Doppler instrument. The detection area Aγ i (D p,i , v p,i ) represents the projection of the detection volume in the direction of the moving drop. It can be considered as the spatial uncertainty of the drop position upon its detection, since the position of the drop within the detection volume can no longer be resolved. For spray/wall interaction studies the spatial uncertainty is not dt

dt

dw

x dw

Θcon Θcon

Figure 7.18 Spatial uncertainties of the wall contact points of two exemplary drops. The diameter

of the detection volume is denoted dt and the projected length of the diameter onto the wall is denoted dw .

7.5 Measurement Techniques for Liquid Films

347

required at the measurement position but on the target surface. Hence, Aγ i (D p,i , v p,i ) must be transferred onto the target surface for every drop, which is done analogously to the determination of the wall contact point. For large surface curvatures the values of effective impingement area can depend strongly on the direction of the velocity vector and the position of the measurement point. A complete discussion of these and further aspects of applying the phase Doppler technique to spray/wall interaction studies can be found in Mühlbauer et al. (2011).

7.5

Measurement Techniques for Liquid Films In the context of collision phenomena, the measurement of film thickness is mainly of interest when liquid drops impact onto rigid surfaces and leave a residual liquid film long enough such that following drop impacts will interact with the film. Information about film thickness, resolved in time and space, is of great interest since the residual liquid film after an impact event strongly influences the outcomes of the impact. Indeed, it is only due to a residual liquid film that the impact of a spray cannot be treated as the superposition of individual drop impacts (see Chapters 6 and 9). Moreover, the heat flux through a heated wall cooled using a spray will be strongly dependent on the liquid film thickness residing on the surface. Nevertheless, the measurement of liquid film thickness is not straightforward. The drop/film interaction is highly unsteady and may exhibit very short time scales. Furthermore, liquid films arising from such events can exhibit very irregular surfaces and may span a wide range of thickness, from micro to millimeter scale. Finally, intrusive measurement techniques are undesirable, since the films involved may also be highly unstable to disturbances. Given the demands on the measurement technique for this application only a subset of the numerous techniques available can be considered and some of these, like ultra-fast X-ray tomography (Zboray et al. 2011) or cold neutron imaging (Zboray and Prasser 2013) are somewhat exotic and require very special laboratory equipment and conditions. Alekseenko et al. (1994) outlined the main techniques used in liquid film thickness measurement, including mechanical, electrical and optical. Another more recent review of techniques is given in Lel et al. (2005), where also two optical techniques were introduced and applied. Of all the electrical methods available, few are non-intrusive with the exception of a high-speed liquid film sensor developed by Damsohn and Prasser (2009) and based on electrical conductance. This sensor has a time resolution of 10 kHz, 64 × 16 measuring points with a spatial resolution of 3.12 mm2 , and a maximum film thickness range of 0.8 mm. Optical methods are preferable because of their non-intrusiveness and can be classified according to imaging, light deflection, absorption, fluorescence, and diffraction/interferometry, all of which provide sufficient spatial and temporal resolution to describe dynamic liquid films. Nevertheless, many can only be used under very specific experimental conditions, e.g. transparent target, or require elaborate calibration procedures. Very few of the techniques documented in the literature are available commercially. Of the above methods only imaging and fluorescence will be discussed in

348

Drop and Spray Diagnostics

Spray

Dry Target

hfilm Contour line detection

Figure 7.19 Example application of shadowgraphy to measure liquid film thickness.

the subsequent subsections, since these are the most widely used methods presently in use. Shedd and Newell (1998) discuss a light deflection method exploiting total internal reflection of light entering the film through a transparent substrate. Calibration of the instrument is necessary and a precision of approximately 10 µm is achievable. Light absorption methods have been employed for many years; however, major drawbacks came from the poor quality of light sources and detectors. More recent studies demonstrate significant improvements using laser diode sources (Mouza et al. 2000) and Greszik et al. (2011) have compared absorption methods with a laser-induced fluorescence (LIF) method and a method based on spontaneous Raman scattering of liquid water.

7.5.1

Liquid Film Thickness Measurement: Imaging Two main imaging approaches are used for liquid film thickness measurements. The first is shadowgraphy, which is straightforward and widely used. The principle is illustrated in Fig. 7.19. For obvious reasons, it is advantageous when employing this technique if the target surface is convex and indeed, this is the main motivating factor for using such target geometries. An example of how this technique can be used is given in Section 4.6 in Chapter 4, Figs. 4.19 and 4.20. The accuracy of this method depends heavily on the resolving features of the imaging optics, as discussed in depth in Section 7.3 in the present chapter; however a resolution of several microns is in many instances possible. The second imaging technique for liquid film thickness measurements is chromatic confocal imaging, a technique which is normally used to investigate the surface structure of solid bodies. This technique exploits chromatic aberration of a lens when a white light source is used for illumination and has been known for many years (Tiziani and Uhde 1994), but has only more recently been adapted for liquid films and offered as a commercial instrument (Kunkel and Schulze 2004). The working principle is sketched in Fig. 7.20. A polychromatic light source is focused through a lens to the surface and, due to chromatic aberrations, the focal point varies with wavelength. The reflected light,

7.5 Measurement Techniques for Liquid Films

349

Figure 7.20 Measurement principle of confocal chromatic imaging.

passed to a spectrometer, carries information about the surface position in the wavelength with the highest intensity. Position errors of approximately 1 µm can be achieved. Chromatic confocal line sensors are now available, which combine a large number (≈200) of individual focal points to deliver the film thickness along a line. Typical focal point diameter is 3–10 µm with a pitch between spots of 7–20 µm, yielding a line length of 1.5–4 mm and a depth of field of 0.1–2.6 mm. Such instruments can operate with surface slope angles of 20◦ to 40◦ at working distances of 5–50 mm and obtaining an accuracy of 0.1–5 µm (Precitec 2016, Stil 2016). With steep wavefronts the amount of reflected light collected by the spectrometer is too low to be evaluated accurately. Repetition rates of several kHz are possible. If the film is transparent and the depth of field is sufficient, both the wall and the film surface produce a signal with two peaks at two different wavelengths, allowing a direct measurement of film thickness. Otherwise the wall position must be determined beforehand in a calibration. Several more recent studies have demonstrated the use of such an instrument for liquid film thickness measurements, albeit mostly for falling film situations (Lel et al. 2005, Zhou et al. 2009). The instrument can be positioned above the film, or in the case of spray/drop impact, behind a transparent target.

7.5.2

Liquid Film Thickness Measurement: Laser-Induced Fluorescence The measurement of film thickness using laser-induced fluorescence (LIF) relies on the presence of a fluorescent dye tracer (e.g. Comarin 152a, Rhodamin B, Rhodamin 101) excited with light-emitting diodes (LED), allowing the fluorescent light to be separated from normally scattered light using suitable optical filters. The scattering properties of fluorescent tracers are uniform for all scattering angles, which is beneficial for optical configurations operating in backscatter. The principle of the technique is given by the Beer–Lambert law, which relates the decrease in light intensity passing through a fluid

350

Drop and Spray Diagnostics

to the absorption strength (Eckbreth 1988), i.e. the transmitted light intensity It to the incident light intensity I0 . ILIF = It = I0 · (−cdye · h · #λ ) (7.33) where cdye (in mol/l) is the dye concentration in solution, h is the liquid film thickness (in m) and #λ is the molar absorption of the fluorescent dye (l/mol m) at the given wavelength λ.  is a factor which accounts for the mode of operation, but is absorbed into the calibration. If working in backscatter, for instance the light source and detector are both above the film, the factor  expresses the backscatter intensity in terms of the transmitted intensity. If working in forward scatter, for example the light from a source enters from below the film through a transparent substrate, and the detector is placed above the film, then  will take on a different value. It is obvious that this technique is suitable only for non-evaporating fluids, since otherwise the concentration cdye would change in time. The LIF method requires both a geometric and an intensity calibration. The geometric calibration addresses the optical properties of the imaging system, such as magnification, DOF, etc., but is straightforward by simply placing a target pattern in the system, yielding a pixel per mm factor. The intensity calibration requires test cells of liquid to relate the gray levels of the camera chip to the film thickness. These calibration steps must be carried out with great care and furthermore, changes of the incident intensity I0 in time, for example through heating of the LED, must be accounted for. The incident intensity and dye concentration must be adjusted to the detector sensitivity and the film thickness. Whereas Greszik et al. (2011) used very low fluorescent dye concentrations to avoid absorption and measure the fluorescent intensity directly, most applications work on the principle described above and the dye concentration is adjusted to match the absorption to the film thickness (Hagemeier et al. 2012, Lel et al. 2005).

7.6

References Adrian, R. J. (1991). Particle-imaging techniques for experimental fluid mechanics, Annu. Rev. Fluid Mech. 23: 261–304. Albrecht, H., Borys, M., Damaschke, N. and Tropea, C. (2003). Laser Doppler and Phase Doppler Measurement Techniques, Springer, New York. Alekseenko, S., Nakoryakov, V. and Pokusaev, B. (1994). Wave Flow of Liquid Films, Begell House Publishers, Danbury. ASAE-S-572 (1999). Spray nozzle classification by droplet spectra, Am. Soc. Agricult. Eng., ASAE S 572. ASTM-E-799 (1981). Standard E799-81: Practice for determining data criteria and processing for liquid drop size analysis, Annual Book of ASTM Standards. ASTM-E-1260 (2003). Standard E1260: Standard test method for determining liquid drop size characteristics in a spray using optical nonimaging light-scattering instruments, Annual Book of ASTM Standards. ASTM-E-1620 (2004). Standard E1620: Standard terminology relating to liquid particles and atomization, Annual Book of ASTM Standards.

7.6 References

351

ASTM-E-2798 (2011). Standard E2798: Characterization of performance of spray drift reduction adjuvants for ground application, Annual Book of ASTM Standards. Bachalo, W. (1994). Experimental methods in multiphase flows, Int. J. Multiph. Flow 20: 261– 295. Bachalo, W. (2000). Spray diagnostics for the twenty-first century, Atom. Sprays 10: 439–474. Barndorff-Nielsen, O. (1977). Exponentially decreasing distributions for the logarithm of particle size, Proc. R. Soc. London Ser. A-Math. 353: 401–419. Bhatia, J. C. and Durst, F. (1989). Comparative study of some probability distributions applied to liquid sprays, Part. Part. Syst. Charact. 6: 151–162. Chigier, N. (1983). Drop size and velocity instrumentation, Progr. Energy Comb. Sci. 9: 155–177. Crowe, C. T. (2005). Multiphase Flow Handbook, CRC Press, Boca Raton. Damaschke, N., Nobach, H., Semidetnov, N. and Tropea, C. (2002). Optical particle sizing in backscatter, Appl. Optics 41: 5713–5727. Damsohn, M. and Prasser, H.-M. (2009). High-speed liquid film sensor for two-phase flows with high spatial resolution based on electrical conductance, Flow Meas. Instr. 20: 1–14. DIN-EN-299 (2009). Oil pressure atomizing nozzles – determination of the angle and spray characteristics, Deutsches Institut für Normung. DIN-SPEC-91325 (2015). Characterization of sprays and spraying processes by measuring the size and velocity of non-transparent droplets, Deutsches Institut für Normung. Dullenkopf, K., Willmann, M., Wittig, S., Schöne, F., Stieglmeier, M., Tropea, C. and Mundo, C. (1998). Comparative mass flux measurements in sprays using a patternator and the phasedoppler technique, Part. Part. Syst. Charact. 15: 81–89. Eckbreth, A. (1988). Laser Diagnostics for Combustion Species and Temperature, Abacus, Cambridge. Fansler, T. D. and Parrish, S. E. (2015). Spray measurement technology: a review, Meas. Sci. Technol. 26: 012002. Fischer, R. E., Tadic-Galeb, B., Yoder, P. R. and Galeb, R. (2000). Optical System Design, Penn State University, Citeseer. Gonzalez, R. C., Eddins, S. L. and Woods, R. E. (2010). Digital Image Processing using MATLAB, Tata McGraw Hill, Noida, UP, India. Greszik, D., Yang, H., Dreier, T. and Schulz, C. (2011). Laser-based diagnostics for the measurement of liquid water film thickness, Appl. Optics 50: A60–A67. Hagemeier, T., Hartmann, M., Kühle, M., Thévenin, D. and Zähringer, K. (2012). Experimental characterization of thin films, droplets and rivulets using LED fluorescence, Exp. Fluids 52: 361–374. Hain, R., Kähler, C. J. and Tropea, C. (2007). Comparison of CCD, CMOS and intensified cameras, Exp. Fluids 42: 403–411. Hecht, E. and Zajac, A. (1974). Optics, Addison-Wesley, New York. ISO-11545 (2009). Agricultural Irrigation Equipment-Centre-Pivot and Moving Lateral Irrigation Machines with Sprayer or Sprinkler Nozzles-Determination of Uniformity of Water Distribution, Beuth Verlag GmbH, Berlin. ISO-22856 (2008). Equipment for crop protection methods for the laboratory measurement of spray drift wind tunnels, Beuth Verlag GmbH, Berlin. ISO-9276 (2004). Darstellung der Ergebnisse von Partikelgrößenanalysen, Beuth Verlag GmbH, Berlin. Jähne, B. (2005). Digital Image Processing, Springer, Berlin.

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Jones, A. (1977). A review of drop size measurement – the application of techniques to dense fuel sprays, Progr. Energy Comb. Sci. 3: 225–234. Kunkel, M. and Schulze, J. (2004). Mittendicke von Linsen – berührungslos Messen, Photonik 6: E505T007. Lefebvre, A. (1989). Atomization and Sprays, Hemisphere Publishing Corporation, New York. Lel, V., Al-Sibai, F., Leefken, A. and Renz, U. (2005). Local thickness and wave velocity measurement of wavy films with a chromatic confocal imaging method and a fluorescence intensity technique, Exp. Fluids 39: 856–864. Linne, M., Paciaroni, M., Hall, T. and Parker, T. (2006). Ballistic imaging of the near field in a diesel spray, Exp. Fluids 40: 836–846. Liu, H. (1999). Science and Engineering of Droplets: Fundamentals and Applications, William Andrew, Norwich, New York. McVey, J. B., Kennedy, J. B. and Russell, S. (1989). Application of advanced diagnostics to airblast injector flows, J. Eng. Gas Turbines Power 111: 53–62. Mouza, A., Vlachos, N., Paras, S. and Karabelas, A. (2000). Measurement of liquid film thickness using a laser light absorption method, Exp. Fluids 28: 355–359. Mugele, R. and Evans, H. (1951). Droplet size distribution in sprays, Ind. Eng. Chem. 43: 1317– 1324. Mühlbauer, M., Roisman, I. V. and Tropea, C. (2011). Evaluation of spray/wall interaction data, Meas. Sci. Technol. 22: 065402. Nukiyama, S. and Tanasawa, Y. (1939). Experiments on the atomization of liquids in an air stream, Trans. Soc. Mech. Eng. Jpn. 5: 62–67. Postrioti, L. and Battistoni, M. (2010). Evaluation of diesel spray momentum flux in transient flow conditions, Technical report, SAE Technical Paper. Precitec (2016). Product information, Precitec GmbH & Co. KG, Schleussnerstrasse 54, 63263 Neu-Isenburg, Germany. Roisman, I. V. and Tropea, C. (2001). Flux measurements in sprays using phase doppler techniques, Atom. Sprays 11: 673–705. Rosin, P. and Rammler, E. (1933). Gesetzmassigkeiten in der Kornzusammensetzung des Zementes, Zement 31: 427–433. Salazar, N. (2015). Digital Image Processing Handbook, Clanrye International, New York. Schäfer, W. and Tropea, C. (2014). Time-shift technique for simultaneous measurement of size, velocity, and relative refractive index of transparent droplets or particles in a flow, Appl. Optics 53: 588–597. Settles, G. S. (2012). Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transparent Media, Springer, Heidelberg. Shedd, T. A. and Newell, T. (1998). Automated optical liquid film thickness measurement method, Rev. Sci. Instr. 69: 4205–4213. Stil (2016). Product information, Stil SA, 595 rue Pierre Berthier, Domaine de Saint Hilaire, 13255 Aix en Provence, Cedex 3, FRANCE. Thoroddsen, S., Etoh, T. and Takehara, K. (2008). High-speed imaging of drops and bubbles, Annu. Rev. Fluid Mech. 40: 257–285. Tiziani, H. J. and Uhde, H.-M. (1994). Three-dimensional image sensing by chromatic confocal microscopy, Appl. Optics 33: 1838–1843. Tropea, C. (2011). Optical particle characterization in flows, Annu. Rev. Fluid Mech. 43: 399–426. Tropea, C. and Roisman, I. V. (2000). Modeling of spray impact on solid surfaces, Atom. Sprays 10: 387–408.

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353

Villermaux, E. (2007). Fragmentation, Annu. Rev. Fluid Mech. 39: 419–446. Zboray, R., Guetg, M., Kickhofel, J., Barthel, F., Sprewitz, U., Hampel, U. and Prasser, H. (2011). Investigating annular flows and the effect of functional spacers in an adiabatic doublesubchannel model of a bwr fuel bundle by ultra-fast x-ray tomography, Proc. 14th Int. Topical Meeting on Nuclear Thermal-hydraulics (NURETH-14), Toronto, Canada. Zboray, R. and Prasser, H.-M. (2013). Measuring liquid film thickness in annular two-phase flows by cold neutron imaging, Exp. Fluids 54: 1–15. Zhou, D., Gambaryan-Roisman, T. and Stephan, P. (2009). Measurement of water falling film thickness to flat plate using confocal chromatic sensoring technique, Exp. Thermal Fluid Sci. 33: 273–283.

8

Atomization and Spray Formation

The process of atomization involves the generation of drops from bulk fluid, achieved using a wide variety of atomization concepts, depending on the desired local drop number, size and velocity flux densities, as well as on the bulk fluid and its properties, e.g. pure liquids, dispersions, suspensions, emulsions, etc. In the context of collision phenomena, atomization plays a key role in applications such as spray cooling, touchless cleaning and spray coating, whereby the latter can be understood in a very broad sense, encompassing applications such as spray painting, crop spraying, spray based encapsulation, domestic sprays (e.g. hair sprays, polishes) or even inhalators. Indeed, a majority of liquid collision phenomena involve atomization for the generation of individual drops and this fact motivates the present examination of the atomization process in more detail, with the aim of establishing an understanding between the atomization conditions and the resulting properties of the spray. This chapter divides the atomization process into primary atomization (Section 8.1), i.e. overcoming the consolidating influence of surface tension by the action of internal and external forces (Lefebvre 1989), secondary atomization, and binary drop collisions in a spray, whereby several special modes of secondary atomization are treated in the final four sections. The causes of secondary atomization are manifold and can significantly alter the size distribution in a spray and are therefore important to consider. Typical causes of secondary atomization include aerodynamic forces whenever a drop is exposed to a relative air flow; covered in Section 8.2. Binary drop collisions can also lead to secondary atomization, as they occur in dense sprays, interacting sprays or when spray drops impinging onto a surface interact with drops ejected from the surface. Binary drop collisions are discussed in Section 8.3. Another cause of secondary atomization is when a drop impinges onto or is forced off a filament, for instance in a filter. This atomization scenario is the topic of Section 8.4. Finally, secondary atomization can also be electrically driven. In this case, evaporation in flight of electrified drops issued from electrostatic atomizers diminishes the drop surface area, while the electric charges they carry remain the same. As a result, the shrinking drop size can reach the so-called Rayleigh limit. Then, an electrically driven instability sets in and the drop atomizes into smaller fragments, as described in Section 8.5. There are many other specific processes leading to primary or secondary atomization which will not be discussed in detail. These include the deformation and breakup involving non-Newtonian drops (Ha and Leal 2001, Joseph et al. 2002), acoustic modulation (Yarin et al. 2002) or high levels of turbulence (Sevik and Park 1973).

8.1 Primary Atomization

355

Table 8.1 Number and surface area of drops arising from one liter of bulk fluid after an idealized mono-dispersed atomization. Total Volume [l] 1 1 1 1 1 1

Diameter of Drops [mm]

Number of Drops [-]

Total Surface Area [m2 ]

124 10 1 0.1 0.01 0.001

1 1910 1.91 x 106 1.91 x 109 1.91 x 1012 1.91 x 1015

0.0048 0.6 6.0 60.0 600.0 6000.0

However, before proceeding to these specific modes of atomization, some remarks about atomization processes in general should be made. To begin, we examine the effect of breaking up bulk liquid into smaller drops. Table 8.1 shows for one liter of liquid the number of drops and their total surface area if the liquid is ideally atomized into equally large drops of various sizes. The huge number of drops involved if the liquid is atomized into drops of small diameter, and the remarkable increase in surface area underline the advantages of atomization if processes like drying, combustion or coating are targeted. At the same time, it is instructive to examine the energy required to create the new liquid surface. Using the example from Table 8.1 in which the bulk liquid is atomized into droplets of 1 µm diameter, the total surface energy, using the surface tension of water (72.75 ×10−3 N/m), amounts to 436 J. An atomization nozzle with a volume flow rate of 1 l/s (60 l/min), which is moderately high, would therefore require ≈436 W for the creation of the new liquid surface. In terms of required pressure, this translates into p = surface energy / volume = 4.36 bar. Considering that such a nozzle producing such small droplets would typically require >1000 bar, one recognizes that almost all of this energy is expended to accelerate the droplets (kinetic energy) and only fractions of a percent are used for the actual atomization, i.e. the creation of new surface area. In this sense the process of atomization is energetically extremely inefficient!

8.1

Primary Atomization The purpose of this section is not to comprehensively cover the fundamentals of atomization, since that topic is the subject of numerous textbooks, monographs and handbooks (Ashgriz 2011, Lefebvre 1989, Lin 2003, Liu 1999, Nasr et al. 2013, Taylor 1959b, Wozniak 2013, Yarin 1993). Rather the focus is placed on the extent to which properties of sprays can be selected to be suitable for particular collision applications, such as coating, cleaning, cooling etc. The basic processes involved in primary atomization are manifold, depending on the type of atomizer used. However frequently, if not always, some form of hydrodynamic instability is involved, as has been discussed in detail in Section 1.10 in Chapter 1. Indeed, with many atomizers a cascade of different

356

Atomization and Spray Formation

instabilities can be recognized over the different stages of primary atomization. Examples will be given in the following discussion of atomizers. There exists a large number of different atomizers and atomization processes, the most common being: r r r r r r r r

pressure atomization rotary atomization two-fluid atomization, air assist two-fluid atomization, air blast effervescent atomization electrostatic (ESTA) ultrasonic atomization flash-boiling atomization

whereby the first three make up the largest contingent for industrial purposes. However, in volume, domestic atomizers (e.g. hair spray, cleansing sprays, etc.) are by far the most frequent atomizer and these are based for the most part on flash-boiling atomization, using a binary mixture of the liquid to be dispersed (solvent) and a suitable propellant. The atomizing nozzle (sudden expansion) has the task of bringing the mixture to a pressure sufficiently below its saturation pressure, such that a rapid boiling process is initiated. The advantages, drawbacks and applications of each atomizer concept have been summarized in tabular form in the handbook literature cited above and will not be repeated here. However, two basic processes will be discussed in more detail, because of their fundamental importance in relation to collision phenomena. The first is the disintegration of a liquid jet and the second is the breakup of a liquid sheet.

8.1.1

Disintegration of a Liquid Jet A first discussion of the capillary instability of liquid jets and different factors that affect it was given in subsection 1.10.1 in Chapter 1, with reference to Fig. 1.8. Beyond the point D in this figure the Rayleigh capillary instability no longer prevails and Kelvin– Helmholtz instabilities become more important for low-viscosity liquids. At higher velocities of the jet, aerodynamic effects increase and also the turbulence level in the liquid jet starts influencing the atomization. These effects were first characterized by Ohnesorge (1936) and cast in a form resembling the diagram presented in Fig. 8.1. Further contributions complementing the work of Ohnesorge came from Castleman (1931), Sauter (1926) and Miesse (1955). The current interpretation of this diagram is due largely to Reitz (1978) and is summarized in Table 8.2, adapted from Lefebvre (1989). Alternative categorization schemes for disruption modes of round jets have been proposed (Chigier and Farago 1992, Lin and Reitz 1998), taking into account both the gas Weber number (Weg ) and the jet Reynolds number (see Section 1.2 in Chapter 1 having in mind that in the present chapter these numbers are based on the nozzle

357

8.1 Primary Atomization

Table 8.2 Summary of jet breakup regimes shown in the Ohnesorge diagram (Fig. 8.1). Wel is the Weber number using liquid properties, Weg is the Weber number using gas properties.

Regime Rayleigh regime First wind-induced regime Second wind-induced regime Atomization regime

Criteria for transition to next regime Wel >8, Weg 40 is good. These modified Ohnesorge number Oh = 0.32. The agreement for We results are also compared with the theoretical predictions for Oh = 1, corresponding to a much higher liquid viscosity. The change of the predicted radius R1 with the modified Ohnesorge number is rather small. This result indicates that, at the initial stage of drop deformation the effect of viscosity is small, at least for low-viscosity liquids.

8.3.4

Computational Algorithm The computational algorithm is based on the hypothesis that, at the initial stage of the drop collision, the rim is only beginning to form. Its mass, and thus the left-hand side of Eq. (8.26), are assumed to be small and can be neglected. Therefore, the momentum balance equation can be simplified and reduced to its quasi-stationary form )

* * 6ηR2Ri 6 RRi (ti + τ )2 + ˙ . + exp RRi = − ˜ i + τ) ˜ (ti + τ )2 ti + τ Re(t ηWe

(8.30)

388

Atomization and Spray Formation

We = 826 We = 480 We = 102

We = 1165 We = 536 We = 255

Figure 8.27 Collisions of two drops of the same liquid with the modified Ohnesorge numbers

Oh = 0.27–0.3. Predictions of the evolution of the dimensionless drop diameter in comparison with the experimental data from Willis and Orme (2000, 2003) (Roisman et al. 2012). Reproduced with permission.

The initial radius RRi is then adjusted to meet the condition that the predicted value of the maximum spreading diameter is “saturated,” i.e. does not depend on RRi . In this study the time instant ti = 0.5 is used to start the numerical integration of Eq. (8.26).

8.3.5

Model Validation In order to validate the model, comparison is made between the theoretical predictions and existing experimental data for high-speed binary collisions of drops of the same liquid. The evolution of the dimensionless drop diameter is shown in Fig. 8.27 in comparison with the experiments from Willis and Orme (2000, 2003). The modified Ohnesorge number varies in a relatively small range Oh = 0.27–0.33. The agreement is rather good for all the experimental conditions. The results for Oh = 0.1 are shown in Fig. 8.28. Some underprediction of the maximum spreading diameter can be explained by the significant deformation of the rim cross-section due to its deceleration. As we have shown above, there is now the possibility to “tune” the initial conditions in order to fit the theoretical results with the experiments. The total duration of the drop spreading is then predicted accurately, also for the case of Oh = 0.1. In Fig. 8.29, the results of the theoretical predictions for the maximum drop diameter are shown in comparison with the available experimental data from Willis and Orme (2003). Consider now the theoretical results for binary collisions of drops of different immiscible liquids. Figure 8.30 presents the results of comparisons of the theoretical predictions of the maximum drop diameter with the experimental data from Roisman et al. (2012). In this figure, the data on collisions of 200 µm oil drops (mixture of oils SO M5+M10, mixture of oils SO M10+M20, or pure oil SO M20) with drops of 50% glycerol solution are shown. The corresponding comparison of the measured and theoretically predicted values of the time instant tmax at which the drop spreading diameter is maximum is shown in Fig. 8.31.

8.3 Drop–Drop Binary Collisions in Sprays

389

We = 543 We = 284 We = 89

Figure 8.28 Collisions of two drops of the same liquid with the modified Ohnesorge number

Oh = 0.1. Predictions of the evolution of the dimensionless drop diameter in comparison with the experimental data from Willis and Orme (2000, 2003) (Roisman et al. 2012). Reproduced with permission.

The theory predicts that, in the range 0 < Oh < 1, the effect of the viscosity on the maximum drop diameter is minor. The maximum drop diameter can thus be described solely by the modified Weber number. Exemplary results of the theoretical predictions for the evolution of the drop diameter obtained by numerical integration of Eq. (8.26) are shown in Fig. 8.32 in comparison with the experimental data from Roisman et al. (2012). These cases correspond to the

We Figure 8.29 Collisions of two equal viscous drops. Theoretically predicted maximum drop

diameter in comparison with the experimental data from Willis and Orme (2003) (Roisman et al. 2012). Reproduced with permission.

390

Atomization and Spray Formation

We Figure 8.30 Collisions of two immiscible viscous drops: oil drops (SO M20, or mixture of SO

M5+M10 or of SO M10+M20) and 50% glycerol solution drops. Theoretically predicted maximum drop diameter in comparison with experimental data from Roisman et al. (2012) (Roisman et al. 2012). Reproduced with permission.

head-on collision of pairs of 200 µm drops: SO M20 oil drop with 50% glycerol solution drop in Fig. 8.32a, and SO M10 oil drop with a 50% glycerol solution drop in Fig. 8.32b. The agreement between the theoretical predictions and the experiments is good. The drop diameter first increases, reaches its maximum value, and then recedes.

We Figure 8.31 Collisions of two immiscible viscous drops: oil drops (SO M20, or mixture of SO

M5+M10 or of SO M10+M20) and 50% glycerol solution drops. Theoretically predicted dimensionless time instant tmax in comparison with the experimental data from Roisman et al. (2012) (Roisman et al. 2012). Reproduced with permission.

8.4 Secondary Drop Detachment from a Filament

We = 40 We = 17 We = 7.6

391

We = 18 We = 9 We = 6.5

Figure 8.32 Collisions of two immiscible viscous drops: (a) SO M20 drops or (b) SO M10 drops

and 50% glycerol solution drops. Predictions of the evolution of the dimensionless drop diameter in comparison with the experimental data from Roisman et al. (2012) (Roisman et al. 2012). Reproduced with permission.

The calculations are terminated when the rim merges, i.e. when the radius of the rim centerline is equal to the radius of the rim cross-section, RR = a. It is not surprising that the agreement is better for higher modified Weber numbers, since the theory is valid for long times after impact and for inertia-dominated drop collisions. In fact it is more curious that the theory is applicable for cases with the relatively small Weber numbers shown in Fig. 8.32. To summarize, note that the presented theoretical model for drop spreading in binary collisions is based on the consideration of the dynamics of the rim bounding a radially spreading lamella. The inertial, viscous and capillary effects are accounted for. On the basis of the theoretical model, appropriate expressions for the modified impact Weber and Reynolds numbers are proposed. The theory describes fairly accurately the evolution of the spreading diameter of the merged drop. The theory is valid for long times after collision and for high values of the modified Weber and Reynolds numbers. It has been shown that the agreement between the theory and the experimental data is rather good, even for relatively small modified Weber numbers.

8.4

Secondary Drop Detachment from a Filament Blowing of drops off filaments or wires by a cross-flow is kindred to some extent to disintegration of free drops falling in air at rest or in the air flow in a wind tunnel or a shock tube considered in Section 8.2 in this chapter (see Fig. 8.12). This type of secondary aerodynamic atomization is important, for example, in relation to plant disease spreading after rain, when wind can blow off drops containing various fungi and viruses from wet branches and leaf stems of infected plants and transmit them to the still healthy ones. The breakup regimes and disintegration patterns of drops on filaments or wires in cross-flow of air were studied in Sahu et al. (2013) in the air velocity range from

392

Atomization and Spray Formation

Table 8.6 Oscillation frequencies of silicone oil and water drops in the direction of blowing. The experimental frequencies are an average taken over several trials (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry. Kinematic viscosity Experimental (Hz) Theoretical (Hz)

10 cst 132.2 184

20 cst 153.6 184

50 cst 97 184

100 cst No oscillations 184

Water 208 339

7.23 m/s to 22.7 m/s. Accordingly, the Weber number varied from 2 to 40 and the Ohnesorge number from 0.07 to 0.8. In this section the gas Weber number We is defined as We = ρaV 2 2Re /σ , and the Ohnesorge number Oh = μ/(ρσ 2Re )1/2 , where ρa is the air density, ρ is the liquid density (in the range 0.93–0.96 g/cm3 for the oils used), σ is the surface tension, μ is the liquid viscosity, 2Re is the volume-equivalent diameter of the primary drop and V is the relative air velocity. Sahu et al. (2013) introduced the lower and upper critical Weber numbers to distinguish between the following two cases: one in which the drop starts breaking off the filament and the other one in which the bag– stamen breakup begins. The essential details of that study are discussed in this section below. The experiments were conducted with silicone oil drops of different kinematic viscosities (10 cst, 20 cst, 50 cst and 100 cst) on filaments aligned perpendicularly to the air flow. Also water drops were used for comparison (in Table 8.6 discussed below). The critical velocity of oil drops blowing off the filament can be evaluated from the condition that the Weber number We=1, which means that the dynamic pressure of air is sufficient to overcome the resistance of surface tension and significantly deform the drop. This corresponds to the critical blowing velocity of the order of 5 m/s. The air jet was issued from a nozzle (Fig. 8.33) at different pressures (5 psi to 30 psi in steps of

Di

re cti

on

of

Bl

ow in

g

A B B

A

Figure 8.33 Sketch of the experimental setup used for blowing off a drop from a filament.

Blowing was perpendicular to the filament (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

8.4 Secondary Drop Detachment from a Filament

393

Figure 8.34 Position of the drop with respect to the filament and the nozzle exit. Panel (a)

corresponds to the viewing direction A–A in Fig. 8.33, and panels (b) and (c) correspond to the viewing direction B–B in Fig. 8.33. The position and volume of the drop on the filament before and after blowing is illustrated in panels (b) and (c), respectively. The nozzle diameter is d =2.2 mm (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

5 psi). The corresponding exit velocity range was from 7.23 m/s to 22.7 m/s, i.e. the velocities were in the supercritical range. The volumetric flow rate through the nozzle was calibrated for different pressure readings of the regulator using a digital flow meter (Omega FMA-5610). Drop evolution was recorded using a high-speed CCD camera (Phantom Miro 4). A 30G needle was used to deposit silicone oil or water drops onto the filament at any desired position. The distance between the filament and the nozzle exit issuing the jet was kept constant at 1 cm. Silicone oil drops were placed on the filament and care was taken to keep them approximately near the nozzle center. Figure 8.34 shows the drop view with respect to the nozzle and filament from two different directions. The breakup modes of drops located on a filament in air cross-flow depend on the maximum air velocity at the nozzle exit. Different breakup modes were observed at different air velocities. At low air velocity a vibrational type breakup was observed. The vibrational type breakup refers to drop disintegration into large fragments comparable to the initial drop size. Three different types of the vibrational breakup were observed and referred to as V1, V2 and V3. Figure 8.35 shows the vibrational type breakup of type V1 of a silicone oil drop at We = 2.8 and Oh = 0.07. The initiation of the breakup process in the case of a drop on a filament is different from that of a freely falling (see Fig. 8.12 in Section 8.2 in this chapter) or a levitating one. The drop on a filament is sometimes asymmetric depending on its volume and the contact angle. The friction forces acting at the contact surface between the drop and filament tend to counteract the drop distortions due to the aerodynamic forces. Moreover, the drop on the filament is free to reshape itself while still being attached to the surface. The initial configuration of the drop in Fig. 8.35 corresponds to t = 0 ms. As air blowing begins, the drop reshapes itself and becomes aligned along the direction of blowing with the shape resembling that of an inflated balloon as is seen in Fig. 8.35 at t = 3 ms and t = 7.8 ms. At low blowing velocities the drop does not immediately break up and exhibits oscillations at its lowest eigenfrequency, presumably associated with the inertia and surface tension. The theoretical eigenfrequency of drop oscillations is thus evaluated as f = (1/2π )(8σ /ρR3e )1/2 (Landau and Lifshitz 1987) and is compared to the measured

394

Atomization and Spray Formation

t = 0 ms

t = 3 ms

t = 7.8 ms

t = 11.4 ms

t = 25.2 ms

Figure 8.35 Several snapshots of a drop blowing off in the vibrational type breakup of type V1.

Silicone oil drop at We = 2.8 and Oh = 0.07 at different time moments. The direction of blowing is from left to right. The drop volume-equivalent diameter 2Re = 0.9 mm. Scale bars, 1 mm (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

one in Table 8.6. The comparison shows that at low blowing velocity of 7.23 m/s the low-viscosity oil drops on the fiber oscillated not very far from the eigenfrequencies of the free drop oscillations. Moreover, no filament vibrations were visible, since it was firmly taut and clamped at the two end supports. Then, practically the entire drop is blown off the filament at t = 11.4 ms. The effect of the surface tension is predominant in the low-viscosity cases, whereas the viscosity becomes the dominant factor for liquids of higher viscosity. After being blown off, the drop is swept by the air flow with practically no further change in its shape, since the capillary waves are damped very rapidly (t = 25.2 ms in Fig. 8.35). With an increase in the blowing velocity to 11.14 m/s (We = 6.19), the drop experiences a different mode of the vibrational breakup of type V2. In this case the drop initially aligns itself in the blowing direction and then is stretched to some extent as is shown in Fig. 8.36. A major portion of the drop is still attached to the filament with a liquid stem. The aerodynamic force is the origin of the stem stretching. At a later time the stem breaks off the filament, presumably due to the capillary instability (see subsection 1.10.1 in Chapter 1). Fragments of smaller radius originating from the stem breakup attain higher velocities than the larger fragments, as could be seen from the positions of two fragments of different sizes at t = 13.2 ms and t = 18.3 ms. The aerodynamic form drag F scales with the fragment radius r as F ∝ r2 . On the other hand, the fragment mass m scales as m ∝ r3 . As a result, the acceleration increases as r decreases, and thus a higher velocity is attained. There is always a residual amount of liquid left on the filament after drop blowing off as is evident in Fig. 8.36 (t = 18.3 ms). This residue left on the filament is close to the filament size and sheds vortices in its wake. Such vortex shedding is a source of filament vibrations, which were observed only after the primary drop has been blown off. The maximum amplitude of the filament

8.4 Secondary Drop Detachment from a Filament

t = 0 ms

t = 4.5 ms

t = 7.5 ms

t = 9.6 ms

395

t = 13.2 ms

t = 18.3 ms Figure 8.36 Several snapshots of drop blowing off corresponding to the vibrational breakup of

type V2. Silicone oil drop at We = 6.19 and Oh = 0.07 at different time moments. The direction of blowing is from left to right. The volume-equivalent initial drop diameter 2Re = 0.88 mm. Scale bars, 1 mm (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

vibrations was measured to be in the 20 − 40 µm range about its mean position, which is insignificant compared to the drop size. At relatively low values of the Weber number the drop does not detach from the filament, but keeps oscillating and remains suspended on the stem. The threshold value of the Weber number for this pattern is the lower critical Weber number. It depends on the Ohnesorge number and the contact angle of the drop with the filament surface. At the same We, a liquid drop with a higher viscosity could keep oscillating without breaking off the filament, whereas a drop of a low-viscosity liquid breaks off the filament. Another mode of the vibrational breakup of type V3 was observed at We = 7.39, as shown in Fig. 8.37. In this case the drop formed a long double-stem structure (t = 9 ms) which was then stretched in the blowing direction. The stems are subjected to the capillary instability and break up into smaller fragments. The fragment size resulting from the secondary stem (still attached to the filament at t = 9 ms in Fig. 8.37) is smaller than that from the primary thicker stem. As the We number is further increased, the bag–stamen type breakup is observed as the upper critical Weber number is surpassed. Figure 8.38 shows the bag–stamen type breakup of a liquid drop at We = 11.67. The stamen is highlighted in Fig. 8.38 in the panel corresponding to t = 5.1 ms. It is emphasized that the stamen is issued from the bag in the direction opposite to the direction of air blowing.

t = 0 ms

t = 4.2 ms

t = 9 ms

t = 11.4 ms

t = 14.4 ms Figure 8.37 Several snapshots corresponding to the vibrational breakup of type V3. Silicone oil

drop at We = 7.39 and Oh = 0.07 at different time moments. The direction of blowing is from left to right and the volume-equivalent initial drop diameter 2Re = 1.1 mm. Scale bars, 1 mm (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

Bag

Stamen

t = 0 ms

t = 2.7 ms

t = 4.5 ms

t = 5.1 ms

t = 6.6 ms Figure 8.38 Snapshots of the bag–stamen type breakup of a silicone oil drop at We = 11.67 and Oh = 0.07 at different time moments. The blowing direction is from left to right and the volume-equivalent diameter of the initial drop 2Re = 0.94 mm. Scale bars, 1 mm (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

8.4 Secondary Drop Detachment from a Filament

397

Bag

t = 0 ms

t = 2.4 ms

t = 3.6 ms

t = 4.5 ms

t = 5.4 ms Figure 8.39 Snapshots of the bag type breakup of a silicone oil drop at We = 17.17 and Oh = 0.07 at different time moments. The direction of blowing is from left to right and the volume-equivalent diameter of the initial drop 2Re = 1 mm. No stamen is visible here. Scale bars, 1 mm (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

With an increase in the Weber number We, the aerodynamic forces acting on the drop also increase in comparison to the capillary pressure, and the central part of the drop is thinned into a sheet surrounded by a rim at the periphery (Fig. 8.38, t = 4.5 ms). At t = 5.1 ms the central part of the bag–stamen structure is further increased in size. The rim of the bag has developed surface waves (see Fig. 8.38 at t = 5.1 ms). Both the bag and stamen then disintegrate into very small fragments. Figure 8.39 shows the bag type breakup at We = 17.17. In distinction from the bag– stamen breakup in Fig. 8.38, there is no stamen present in this case. The instability of the bag is more evident in this case at t = 4.5 ms compared to the bag–stamen breakup in Fig. 8.38. The instability of the bag and its rim results in disintegration into smaller fragments than in the case of the vibrational breakup of type V3 in Fig. 8.37. The total time to breakup decreases as the Weber number We increases at a fixed Ohnesorge number, as is evident in Figs. 8.35–8.39. The bag and bag–stamen types of breakup could switch with a slight difference in the blowing velocity. It should be emphasized that the bag–stamen breakup of a drop blown off a filament shown in Fig. 8.38 resembles to some extent the morphologies seen in the bag-like aerodynamic breakup of a free drop shown in Fig. 8.12 in Section 8.2 in this chapter. As the Ohnesorge number Oh increases with the increase in viscosity, the energy dissipation also increases, which then reduces the drop deformation. Figure 8.40 shows the vibrational breakup of type V2 at We = 17.73 and Oh = 0.13. At practically the same value of the Weber number, the drop of a lower viscosity liquid (of a lower Ohnesorge

398

Atomization and Spray Formation

t = 0 ms

t = 2.8 ms

t = 6.5 ms

t = 11.7 ms

t = 18.8 ms

Figure 8.40 The vibrational breakup of type V2 of a silicone oil drop at We = 17.73 and Oh = 0.13 at different time moments. The direction of blowing is from left to right and the volume-equivalent diameter of the initial drop 2Re = 1 mm. Scale bars, 1 mm (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

number Oh = 0.07) undergoes the bag type breakup (Fig. 8.39) in distinction from the scenario in Fig. 8.40 for a higher liquid viscosity (a higher Oh = 0.13). A non-Newtonian viscoelastic polymer [poly(ethylene oxide), PEO] solution was also tested. Figure 8.41 shows the breakup pattern of a PEO drop at a relatively low We = 7.5. The elastic nature of the polymer solution (see Section 1.9 in Chapter 1) allows it to be significantly stretched under the action of the aerodynamic drag, as is evident from Fig. 8.41 (t = 37 ms). The stem spanning the drop and filament continues stretching as its radius shrinks. Moreover, the part of the drop which remains attached to the filament is depleted in time, which is similar to the tubeless viscoelastic siphon (Bird et al. 1987). Finally, the stem breaks off the filament, leaving no PEO solution on the filament, in distinction from the Newtonian liquids where a portion of the initial drop is always left behind on the filament. The We–Oh plane can be delineated into domains corresponding to different breakup modes of the Newtonian liquids (oils), as shown in Fig. 8.42. It is clear that at higher values of Oh the bag type breakup is delayed and the domain of the vibrational types of breakup increases, as the viscous dissipation diminishes deformation caused by the aerodynamic forces. The domain I corresponding to the vibrational type V1 possesses almost the same thickness at different values of Oh, which expresses the fact that surface tension is dominant at lower values of We. The breakup time tb is defined as the time between the onset of the blowing and the detachment of a drop from the filament. The breakup time was measured using the experimental data for different blowing velocities (different We) and liquid viscosities. It could be seen in Fig. 8.43 that at low values of the Weber number the breakup time significantly increases as the viscosity increases. At higher We the increase in the breakup time with viscosity is insignificant and tb becomes practically independent of viscosity. In the case of Newtonian silicone oils a part of a liquid drop always remains on the filament at the end of the blowing process after detachment of the main body. The volume-equivalent radius of the residual part left on the filament was calculated and normalized with the initial drop radius. It is denoted as a. Such data from multiple

8.4 Secondary Drop Detachment from a Filament

t = 0 ms

t = 13.5 ms

t = 74.3 ms

399

t = 37 ms

t = 202 ms

t = 240 ms

Figure 8.41 The breakup of 1 wt% aqueous solution of PEO (molecular weight Mw = 2000 kDa) at We = 7.5 and Oh = 0.25 at different time moments. The direction of blowing is from left to right and the volume-equivalent diameter of the initial drop 2Re = 1 mm. Scale bars, 1 mm (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

experiments was used to evaluate the probability density function of the size of the residual part under different conditions. It is shown in Fig. 8.44 for four different air blowing velocities and four different kinematic viscosities of silicone oil. It was found that the normal distribution best fits the corresponding probability density functions. Figure 8.44d clearly shows that the expected volume-equivalent radius of the residual drop left on the filament diminishes as the blowing velocity increases. Two physical parameters of liquid could determine the size of the residual droplet left on the filament: the surface tension and/or the viscosity. The radius of the droplet which can withstand the aerodynamic drag (without being blown off) solely due to the surface tension is found as σ (8.31) a∼ CD ρaV 2 where CD is the drag coefficient (see Section 2.6 in Chapter 2). On the other hand, the radius of the droplet which can withstand the aerodynamic drag (without being blown off) solely due to the viscous dissipation is found as  1/2 μa f (8.32) a∼ CD ρaV where a f is the cross-sectional filament radius.

Atomization and Spray Formation

Figure 8.42 The We–Oh plane with the domains corresponding to different types of breakup of

Newtonian liquid drops on a filament in cross-flow: I– vibrational breakup of type V1; II– vibrational breakup of type V2; III– vibrational breakup of type V3; IV– bag–stamen type breakup; V– bag type breakup (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry. 0.040 ν = 10cst

0.035

ν = 20cst ν = 50cst

0.030

ν = 100cst

0.025 tb (s)

400

0.020 0.015 0.010 0.005 0.000 0

5

10

15

20

25

30

35

40

We Figure 8.43 Breakup time versus the Weber number for different kinematic viscosities of silicone

oil (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

401

8.5 Secondary Electrically Driven Drop Breakup

(a) 6

Count

5

(b) ν = 10 cst ν = 20 cst ν = 50 cst ν = 100 cst

6 5

4 3

3 2

2

1

1 0 0.0

0.1

0.2

0.3 0.4 R/R0

0.5

0.6

(d) ν = 10 cst ν = 20 cst ν = 50 cst ν = 100 cst

10 8 Count

Count

8

0 0.0

0.7

(c) 10

6

4

2

2

0.1

0.2

0.3 0.4 R/R0

0.5

0.6

0.7

0.1

0.2

0.3 0.4 R/R0

0.5

0.6

0.7

0.3 0.4 R/R0

0.5

0.6

0.7

ν = 10 cst ν = 20 cst ν = 50 cst ν = 100 cst

6

4

0 0.0

ν = 10 cst ν = 20 cst ν = 50 cst ν = 100 cst

4 Count

7

0 0.0

0.1

0.2

Figure 8.44 Probability density function of the size of the residual droplet left on the filament. (a)

The blowing velocity V = 11.14 m/s, (b) V = 14.43 m/s, (c) V = 17.46 m/s, (d) V = 20.13 m/s. The lines spanning the bar charts are the corresponding normal distributions. In the notation near the horizontal axes R = a and R0 = Re (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

Equations (8.31) and (8.32) predict a ∼ V −2 and a ∼ V −1/2 , respectively. These predictions are compared to the experimental data in Fig. 8.45. The scaling predicted by Eq. (8.32) is closer to the data, albeit a good numerical agreement in the exponent value is found only in the case of ν = 20 cst in Fig. 8.45b.

8.5

Secondary Electrically Driven Drop Breakup: The Rayleigh Limit Liquids can be electrified, which is widely used in such processes as electrospraying and electrospinning (Shrimpton 2009, Yarin et al. 2014). This can be achieved by various means but most frequently, by submerging an electrode attached to a high-voltage

Atomization and Spray Formation

(b) –0.6 –0.8

–1.0

–1.0

–1.2

–1.2

–1.4

In (a¯ )

In (a¯)

(a) –0.8

y = –1.44x –0.124

–1.4

–1.6

–1.6

–1.8

–1.8

–2.0

y = –0.168x –0.56

–2.0 2.0

2.5 – In (V )

2.4

3.0

2.6

2.8 – In (V )

3.0

3.2

3.0

3.2

(c) –0.2 –0.4 (d) –0.0

–0.6 –0.8

–0.5

–1.0 In (a¯ )

In (a¯)

402

–1.2 y=

–0.928x –0.3

–1.4

–1.0 y = –1.21x –0.21 –1.5

–1.6 –2.0

–1.8 2.4

2.6

2.8 – In (V )

3.0

3.2

2.4

2.6

2.8 – In (V )

Figure 8.45 Normalized volume-equivalent radius of the residual droplet left on the filament

versus the normalized air blowing velocity. Here the blowing velocity is rendered dimensionless by V∗ = 1 m/s. Symbols show the experimental data (the average taken over 20 trials for each point) spanned by a power-law fit written in each panel. (a) the kinematic viscosity of oil ν = 10 cst, (b) ν = 20 cst, (c) ν = 50 cst, (d) ν = 100 cst (Sahu et al. 2013). Reproduced with permission from The Royal Society of Chemistry.

DC-source into a liquid before it is issued from a nozzle. As a result, free uncompensated ions are created via different possible mechanisms (Yarin et al. 2014), and such a liquid acquires an excessive electric charge, i.e. loses its original electroneutrality. Liquids used for spray formation are typically very poor ionic conductors, which results from ion mobility in the electric field. Due to their low electrical conductivity they are called leaky dielectrics (Melcher and Taylor 1969, Russel et al. 1992, Saville 1997, Castellanos and Pérez 2007, Chang and Yeo 2010, Yarin et al. 2014). In spite of the name, the main hydrodynamic effects attracting attention to them are due to their electrical conductivity, rather than their dielectric polarization. Their electrical conductivity matters in the processes where the characteristic charge relaxation time τC = ε/(4π σe )

8.5 Secondary Electrically Driven Drop Breakup

403

of a liquid is much shorter than the characteristic hydrodynamic time τH . Here and hereinafter Gaussian (CGS) units are used, which is convenient in situations where both electric and hydrodynamic processes are involved; the electrical conductivity is denoted σe , and the relative permittivity is denoted ε. In most of the cases of interest in electrospraying τC ≤ 1 ms, whereas τH (which is understood, for example, as the jet or drop formation or existence time) is of the order of 1 s. Due to the inequality τC  τH , ions always have enough time to escape to the free surface, where ultimately all the free charges accumulate. Then, such liquid bodies very rapidly become equipotential, with all the electric charges located at the free surface and completely eliminating any electric field inside. Therefore, in this limit they become indistinguishable from perfect conductors. It should be emphasized that in spite of the presence of the electric charges, atomization of liquid jets is normally still driven purely hydrodynamically (e.g. due to the Rayleigh capillary instability, or the Kelvin–Helmholtz instability due to the interaction with the surrounding gas, see Section 1.10 in Chapter 1), while the charges for a while are relatively passively redistributed between the diminishing liquid bodies. However, at a certain stage, after the individual electrically charged drops have already been formed in the course of the primary atomization, the electrical forces can become dominant and trigger the secondary atomization resulting in a further drop division and formation of much tinier droplets. Such an outcome leads to a phenomenon first demonstrated by Lord Rayleigh (1882), that charged drops become unstable purely electrodynamically when they become sufficiently small and reach the so-called Rayleigh limit. The mechanical action of the electric field is intrinsically associated with the emergence of the additional stresses associated with the electric field strength E (Panofsky and Phillips 2005). In the absence of significant magnetic effects, which is usually true in the atomization processes, the Maxwell stress tensor σ M reads σM =

  1 εs E ⊗ E− (E · E) I . 4π 2

(8.33)

This is the second-rank tensor, with E ⊗ E being the dyadic product of the electric field strength E, E · E being the scalar product, I being the tensor unit and εs being the relative permittivity of the surrounding gas. In the case of a single free spherical drop of radius a, with an overall electric charge Q at its surface, the electric potential  in the space surrounding it is =−

Q εs r

(8.34)

where the radial coordinate r ≥ a is reckoned from the drop center. Note that an additive constant in the potential is omitted as immaterial. Then, the electric field strength defined as E = − ∇ has the only non-zero component normal to the drop surface, the radial component Er = Q/(εs r2 ). The fact that the tangential components are zero is expected for an equipotential body with all the free charges being at the surface. Accordingly, the only non-zero component of the Maxwell

404

Atomization and Spray Formation

stress at the drop surface at r = a found from Eq. (8.33) reads  σrrM r=a =

1 Q2 . 8π εs a4

(8.35)

This stress pulls the drop surface outward due to the attraction of the free charges at the surface to a grounded electrode at infinity. The Maxwell stress, Eq. (8.35), effectively diminishes the capillary pressure, 2σ /a, with σ being the surface tension, which means that in the presence of the Maxwell stress the effective capillary pressure becomes 2σ /a − Q2 /(8π εs a4 ). Only a positive capillary pressure is capable of keeping liquid surface curved, which means that when the drop radius diminishes to that extent that the electric term fully compensates the surface-tension-related one, the drop cannot exist anymore. This happens at the Rayleigh limit when a = aR , which is given by the following expression  aR =

Q2 16π σ εs

1/3 .

(8.36)

Equation (8.36) corresponds to the Rayleigh limit when the electric charge is fixed, while the drop size has diminished, for example due to drop evaporation. On the other hand, one can consider a given drop of size a, with the electric charge being continuously added to the surface. Then, the Rayleigh limit can be formulated in the terms of charge, as (Pruppacher and Klett 2012) 1/2  . (8.37) QR = 16π σ εs a3 At a fixed electric charge, as the destabilizing Maxwell stresses increase much faster than the stabilizing capillary pressure, as the radius a decreases below aR , one can expect that below the Rayleigh limit, a transient process will occur where the droplet becomes unstable and breaks up. Alternatively, at a > aR , the capillary pressure becomes dominant and should stabilize a perturbed drop. Indeed, as it was shown by Lord Rayleigh (1882) in the case of inviscid drops, small perturbations of the drop surface of the general form vary in time in the linear approximation as ζ (t, θ , ϕ) = ζ0 exp(−γ t )Pnm (cos θ ) exp(−imϕ)

(8.38)

with the increment γ being given by the following characteristic equation  1/2  σ 4a3 γ = ± − 3 n(n − 1) n + 2 − 3R . ρa0 a0

(8.39)

Here the small perturbation of the drop surface ζ is defined as ζ (t, θ , ϕ) = [a(t, θ , ϕ) − a0 ]/a0  1, with a being the current local radial position of the drop surface, and a0 being an unperturbed radius of the spherical drop, θ and ϕ being the zenith and angular spherical coordinates and t being time. Also, Pnm (•) denote the Legendre polynomials, and m is the angular wavenumber, ρ is liquid density and i is the imaginary unit. It should be emphasized that in Eqs. (8.38) and (8.39) n > 2.

405

8.5 Secondary Electrically Driven Drop Breakup

Figure 8.46 Breakup of a charged ethylene glycol drop below the Rayleigh limit (Duft et al.

2003). Reprinted by permission from Macmillan Publishers Ltd. Copyright (2003) by Nature Publishing Group.

Equation (8.39) shows that the electrically driven drop instability sets in only when n+2
aR , cannot increase since both solutions of Eq. (8.39) are imaginary, thus determining capillary waves at the drop surface according to Eq. (8.38). It should be emphasized that perturbations of all values of n are present simultaneously in reality. Thus, highly charged drops will have such large Rayleigh limits aR , that the inequality in Eq. (8.40) will be fulfilled not only at n = 2 but at some n > 2. Then, several multilobal perturbations will exponentially grow simultaneously and real jetting can start from the drop surface. The jetting modes of free individual drop breakup below the Rayleigh limit were experimentally demonstrated in Duft et al. (2003), Grimm and Beauchamp (2010) and others (see Fig. 8.46). Miloh et al. (2009) studied the effect of viscosity μ on the electrically driven instability of free drops. They showed that the Rayleigh limit always determines the onset of the instability of viscous drops. In particular, in the case of highly viscous drops, for which inertial effects are negligibly small, perturbations of the drop surface of the type given by Eq. (8.38) possess the increment γ given by the following expression   σ n(2n + 1) n + 2 − 4a3R /a30 . (8.41) γ = 2μa0 2n2 + 4n + 3

406

Atomization and Spray Formation

The value of γ determined by Eq. (8.41) becomes negative for drops smaller than the Rayleigh limit, i.e. at a0 < aR , and thus, according to Eq. (8.38) perturbations will exponentially grow at the drop surface and such a drop will break up. Note that charged liquid cylinders also demonstrate an electrically driven instability resulting in multiple undulations and jetting from their surface, the phenomenon known and observed mostly with some polymer solution jets in electrospinning and referred to as branching (Yarin et al. 2005, Yarin et al. 2014).

8.6

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8.6 References

411

Villermaux, E. and Bossa, B. (2009). Single-drop fragmentation determines size distribution of raindrops, Nat. Phys. 5: 697–702. Weber, C. (1931). Zum Zerfall eines Flüssigkeitsstrahles, ZAMM-J. of Appl. Math. Mech. 11: 136– 154. Wierzba, A. (1990). Deformation and breakup of liquid drops in a gas stream at nearly critical Weber numbers, Exp. Fluids 9: 59–64. Willis, K. and Orme, M. (2000). Viscous oil droplet collisions in a vacuum, Exp. Fluids 29: 347– 358. Willis, K. and Orme, M. (2003). Binary droplet collisions in a vacuum environment: an experimental investigation of the role of viscosity, Exp. Fluids 34: 28–41. Wozniak, G. (2013). Zerstäubungstechnik: Prinzipien, Verfahren, Geräte, Springer, Heidelberg. Wu, P.-K. and Faeth, G. M. (1993). Aerodynamic effects on primary breakup of turbulent liquids, Atom. Sprays 3: 265–289. Wu, P.-K., Miranda, R. F. and Faeth, G. M. (1995). Effects of initial flow conditions on primary breakup of nonturbulent and turbulent round liquid jets, Atom. Sprays 5: 175–196. Xu, L., Zhang, W. W. and Nagel, S. R. (2005). Drop splashing on a dry smooth surface, Phys. Rev. Lett. 94: 184505. Yarin, A. L. (1993). Free Liquid Jets and Films: Hydrodynamics and Rheology, Longman and John Wiley & Sons, Harlow and New York. Yarin, A. L., Kataphinan, W. and Reneker, D. H. (2005). Branching in electrospinning of nanofibers, J. Appl. Phys. 98: 064501. Yarin, A. L., Pourdeyhimi, B. and Ramakrishna, S. (2014). Fundamentals and Applications of Micro- and Nanofibers, Cambridge University Press. Yarin, A. L. and Weiss, D. A. (1995). Impact of drops on solid surfaces: self-similar capillary waves, and splashing as a new type of kinematic discontinuity, J. Fluid Mech. 283: 141–173. Yarin, A. L., Weiss, D. A., Brenn, G. and Rensink, D. (2002). Acoustically levitated drops: drop oscillation and break-up driven by ultrasound modulation, Int. J. Multiph. Flow 28: 887–910. York, J. L., Stubbs, H. E. and Tek, M. R. (1953). The mechanism of disintegration of liquid sheets, Trans. ASME 75: 1279–1286.

9

Spray Impact

The impact of sprays onto walls is of great industrial importance and for this reason has attracted the attention of researchers in an effort to predict the outcome. While some applications expressly avoid splashing, e.g. coating or spray painting, many result in a secondary spray. In fact, spray impact may even be used to intentionally change the size distribution of droplets in a spray, such as with inhalation nebulizers or in direct injection fuel systems. While Chapters 4, 5 and 6 dealt with the impact of single drops onto surfaces or liquid layers, the present chapter addresses the impact of sprays onto such surfaces. Fundamentally, similar questions are asked: how many secondary droplets of what size and velocity are generated and what part of the impacting liquid remains on the surface? If heat transfer is involved then interest lies with the heat flux density at the surface or the effective Nusselt number, which, as with single drops, will depend strongly on the temperature of the surface; hence on which regime of the Nukiyama curve describing heat transfer at the surface is applicable (Nukiyama 1934, Kutateladze 1963, Carey 1992). This chapter encompasses spray impact onto liquid films (Section 9.1), discusses the secondary spray formation in Section 9.2 and outlines useful empirical correlations in Section 9.3 Two main approaches are commonly used to predict the outcome of a spray impact with a rigid wall or with a wall covered by a liquid film. The first approach is formulated in the framework of an Euler/Lagrange numerical simulation and describes the spray as the superposition of a large number of isolated, non-interacting drops (Cossali et al. 2005). Numerous models for single drop impact have been proposed (Bai and Gosman 1995, Stanton and Rutland 1996, Mundo et al. 1998, Lee and Bergman 2002), all having empirical origins. Roisman et al. (1999), Moreira et al. (2010) and Park and Watkins (1996) have provided overviews of many existing models and presented also direct comparisons of their predictive capabilities. It is a characteristic of all models that they provide reliable predictions at most over the narrow range of impact parameters from which they were derived. Indeed, several models in use for spray impact are even based on the impact of single drops onto a dry surface, which may be completely inappropriate. The second approach is to formulate purely empirical correlations based on experimental data obtained for specific conditions, accepting the corresponding lack of universality. Better approaches are not yet completely clear; however, one can assume that the situation should improve if physical modeling can be properly introduced into the second approach, which is attempted in Section 9.1.

Spray Impact

Multiple Drop Interactions

413

Liquid Film Influence

hf Crown interactions

Average film thickness and velocity

df Drop–drop interactions

Oscillations of the film

Figure 9.1 Factors distinguishing spray impact from individual drop impact.

The main difference between single drop impacts and spray impact is that the impact of individual drops of a spray occurs onto a thin liquid film, accumulated from the impact of previous drops. This immediately introduces two basic mechanisms through which the impact of a single drop can be influenced by the impact of neighboring and/or previous drops onto the surface: (i) interactions of multiple neighboring drops can occur, and (ii) a liquid film may be present, which can influence the impact outcome and secondary atomization. Indeed, these two mechanisms, pictorially sketched in Fig. 9.1, are identified by Moreira et al. (2010) in a comprehensive review article as being the main unresolved issues in closing the gap between understanding the impact of individual drops and modeling an impacting spray. These two mechanisms are addressed respectively in the present chapter, following which empirical correlations for spray impact phenomena are reviewed in Section 9.3. A modest step in formulating a more physical model of the spray impact would be to consider simple drop–drop interactions occurring with multiple drop impacts onto substrates. Experiments investigating the impact of two drops simultaneously or in quick succession onto a dry, solid surface show that such interactions can enhance splash (Roisman et al. 2002a, Fujimoto et al. 2002, Kalb et al. 2000). The spreading lamellae of individual impacting drops interact to create strong uprising liquid jets and sheets. However, investigations to date have been restricted to very specific impact conditions. The situation for two neighboring impacting drops will be addressed in subsection 9.1.1 of this chapter. The next degree of complication is if a drop impacts onto an existing film, as discussed in Sections 6.2, 6.3 and 6.7 in Chapter 6 or depicted in Fig. 8.5 in subsection 8.1.2 in Chapter 8. If the film is fluctuating in thickness or flows along the wall due to previous impact events, additional uprising liquid jets and sheets, and subsequently secondary droplets, can be generated. Such phenomena were examined by Roisman and Tropea (2005) and will be elaborated in the subsection 9.1.3. Four different liquid film types can be identified to characterize the liquid film thickness. Table 9.1, adapted from Kalantari and Tropea (2007), summarizes these types and provides a threshold Weber number (Weth ) required for the onset of splashing. The

414

Spray Impact

Table 9.1 Different types of liquid films on the wall relevant for drop impact phenomena. Dimensionless film thickness, h∗

Wall film condition

Variation of Weth for the onset of splashing

Correlation for Weth

h∗ ≤ 0.1 0.1 < h∗ ≤ 1 1 < h∗ ≤ 2 h∗ ≥ 2

Wetted wall Thin liquid film Shallow liquid film Deep liquid layer

Constant Increasing Decreasing Constant (the asymptotic value)

∼480–500 1366 h∗ +354 −0.54 ∼1657 h∗ ∼1100

value of the Weber number required for the onset of splashing is constant for the case of very thin films with h∗ ≤ 0.1 (wetted wall) as proposed by Schmehl et al. (1999), where the mean film thickness is rendered dimensionless using the impacting drop diameter, i.e. h∗ = h/D. The threshold Weber number then increases monotonically with an increase in the dimensionless film thickness up to h∗ = 1 (thin liquid film) and then decreases until ∼ h∗ = 2 (shallow liquid film) and finally achieves an asymptotic value corresponding to a deep liquid layer (deep pool condition). The borderline value h∗ = 2 was proposed by Macklin and Metaxas (1976). This classification accounts for the data obtained by Wang et al. (2002) for a 70% glycerol–water solution. For the deep film condition (i.e. h∗ > 2) inertia dominates and an impacting drop creates a crater in the liquid film, leading to bubble entrainment inside the film and formation of an uprising central jet. This phenomenon is well known and described for the case of a single drop impact onto a stationary deep liquid layer in Section 6.5 in Chapter 6 and by Oˇguz and Prosperetti (1989) or Fedorchenko and Wang (2004). Before discussing modeling efforts for spray impact further, it is instructive to roughly estimate conditions under which the superposition of single drop impact events is no longer adequate to describe spray impact. It is customary to describe spray impact in terms of mass flux density m, ˙ with the units of kg/m2 s, impinging on the surface. This can be converted into a frequency of single drop impact events only after the drop size and the drop velocity are specified. Figure 9.2 illustrates the number flux density as a function of mass flux density for various mono-dispersed sprays in the range of mass flux densities typical for spray cooling applications in the metallurgical industry (0.1–100 kg/m2 s). The huge number of drops involved for even modest mass flux densities indicates that a statistical approach to describe or model spray impact is more viable than tracking single drop impacts (note the logarithmic scale on the ordinate of Fig. 9.2) The mass flux density at which the impacts of individual drops begin to directly interact with one another on the target surface is estimated next. Similarly to Chapter 4, the expressions derived for the maximum spreading diameter of a single drop impacting onto a smooth surface and the time at which this maximum is achieved for Re We are used (Roisman et al. 2002b) Dmax = D[0.87Re1/5 − 0.48Re2/5 We−1/2 ]

(9.1)

tmax = 5D/V0

(9.2)

where D and V0 are the drop diameter and velocity upon impact, respectively. Assuming that the drop spreads to Dmax and does not recede, and that an appreciable

Spray Impact

415

Figure 9.2 Number flux density as a function of mass flux density and drop size. Assumed

mono-dispersed spray and a density of 1000 kg/m3 .

interaction will only occur if another drop impacts within one Dmax diameter and within double the tmax time, and furthermore, assuming the drops follow a Poisson spatial distribution before impact and that all drops impact normal to the surface with the same velocity, the probability of interaction can be estimated as follows. Given that a drop impacts at some time t0 , the probability that one or more further drops exists in a volume Vint of area D2max and length 2tmaxV0 , given a drop concentration λconc within the volume is sought. The approach is illustrated in Fig. 9.3, where the volume Vint over the impacting drops is depicted (assuming drops of the same size and same impact instant). The dimensionless drop concentration λconc can be obtained simply as the product of the number concentration, c (number of drops/m3 ), times the volume, i.e. λconc = cVint . The number concentration is derived from the mass flux density m˙ and the assumed drop size and velocity as: c = 6m/(ρπ ˙ D3V0 ) where ρ is the drop density of the drop.

Figure 9.3 The volume over which the drop concentration λconc is determined.

(9.3)

Spray Impact

100 Probability of drop interaction

416

(a)

(b)

10–2

10–4 m˙ m˙ m˙ m˙

10–6 1

10

= 1 kg/m2s = 5 kg/m2s = 20 kg/m2s = 50 kg/m2s

100

1000

D [μm] Figure 9.4 (a) Probability of drop interaction on a target surface as a function of drop diameter

and mass flux density (V0 =25 m/s); (b) Drop impact pattern on the target surface for a drop size of 30 µm and a mass flux density of m=50 ˙ kg/m2 s.

The probability of more than one further drop appearing in the specified volume is p(≥ 2, λ) = 1 − p(0, λ) − p(1, λ)

(9.4)

which for a Poisson distribution (Feller 1968) is p(≥ 2, λ) = 1 − e−λ (1 + λ).

(9.5)

This drop interaction probability is plotted in Fig. 9.4a as a function of drop size and mass flux density for an impact velocity of 25 m/s. This estimate of direct drop impact interaction was made using idealized conditions (monodispersed drops, uniform velocity, etc.). A more refined estimate of interaction probability was conducted using a Monte Carlo simulation, in which the time evolution of the drop splat was taken into account when simulating the drops impinging onto the surface. For instance, in Fig. 9.4b such a simulation is used to create a picture of the instantaneous drop impact pattern for a drop size of 30 µm and a mass flux density of m˙ = 50 kg/m2 s, whereby the growth of the drop splat according to Eq. (9.1) has been invoked in the simulation. One observation is that even for very high loadings (e.g. 50 kg/m2 s), the probability of direct drop interaction on the surface remains quite low and that this probability is a strong function of drop size. At small drop diameters and low velocities these estimates are no longer too reliable, since the underlying correlations are generally considered valid only for Reynolds numbers above 1000 and Weber numbers above 10. However, practically, the question of drop impact interaction is closely related to the history of the drop splat after impact. On a very hot surface the liquid may evaporate and the above estimations may be applicable. Similarly, for hydrophobic surfaces the drop may rebound after some interaction time (possibly even less than tmax ) and again these estimations may be quite reasonable. In many cases the liquid remains on the surface and accumulates into a film, in which case a drop/film interaction arises, which can influence the outcome of the impact immensely. This is the subject of the following sections.

9.1 Spray Impact onto Liquid Films

9.1

417

Spray Impact onto Liquid Films Spray impact is very often accompanied by the presence of a residual liquid film on the target surface and, as indicated in Fig. 9.1, this may strongly influence the overall outcome in terms of splash threshold and generation of secondary droplets. How thick the liquid film is depends on numerous factors, including the local volume flux density of the spray, the evaporation (or boiling) rate of the liquid in the case of heated surfaces, the wettability of the surface and, often disregarded, the far-field boundary conditions on the film flow. If for instance the film cannot flow off the surface, a deep pool eventually forms and the impact conditions change dramatically. Fundamentally, the impact of drops onto a liquid film differs significantly from impact onto a dry surface, as highlighted by the discussions in Chapters 4 and 5. A liquid film gives rise to the kinematic discontinuity responsible for the emergence of a crown and the subsequent secondary droplets (Yarin and Weiss 1995). For very thick films, approaching deep liquid pools, the impact phenomena have been discussed in Section 6.5 in Chapter 6 and are described in detail, theoretically and numerically in Roisman et al. (2008), van Hinsberg et al. (2010) and Berberovi´c et al. (2009), and will not be discussed further in this section. Given that a liquid film exists on the surface, its influence on the spray impact phenomena is twofold; on the one hand the film can exhibit fluctuating flow, which directly affects the momentum balance in the kinematic discontinuity responsible for the crown formation. This means that even if the drop velocity is normal to the wall, the impact is in fact not axisymmetric but locally oblique. Also the wavy surface of the liquid film may have some influence on the impact. On the other hand, the fluctuating flow in the liquid layer may lead directly to interacting crowns, lamella and finger-like jet formation and subsequently to the creation of secondary droplets. The complexity of the situation is graphically illustrated in Fig. 9.5, in which the variation of the liquid film thickness is evident in the irregularity of the crown development and collapse into secondary droplets. Evidently liquid jets and secondary droplets also appear due to the fluctuating liquid layer. The crown development can be highly asymmetric because of non-zero local film velocities at the instant of impact or because of a varying film thickness. This difference of crown development due to the fluctuating liquid layer on the surface was investigated experimentally by Sivakumar and Tropea (2002). Figure 9.6 summarizes their results, showing the evolution of the crown radius in time for impact Weber numbers in the range 414–1096 and Reynolds numbers in the range 2016–4954. There is no observable systematic change in crown radius growth with Weber number; however all experimental data lie close to a growth rate proportional to t 0.2 which is considerably different from the t 0.5 growth rate predicted by Yarin and Weiss (1995). The situation of spray impact onto a liquid film is shown pictorially in Fig. 9.7 taken from Roisman et al. (2006). Additional mechanisms through which secondary droplets can be formed are: r ejection after pinch-off from individual cusps of the crown (see Section 6.7 in Chapter 6)

Spray Impact

Liquid film (a)

(i)

(b)

(j)

(c)

(k)

(d)

(l)

(e)

(m)

(f)

(n)

(g)

(o)

(h)

(p)

3 mm Figure 9.5 Sequential images showing the variation in liquid film thickness over the impact

surface during spray impact. The diameter of the impact surface is 10 mm. The number concentration of impinging spray is 2.78 drops/mm3 . The images were recorded with a frame rate of 1825 fps (Sivakumar and Tropea 2002). Reprinted with the permission of AIP Publishing. 5

We = 695, Yarin and Weiss (1995) We = 1232, Yarin and Weiss (1995) Variation line with τ 0.2

4

3 rc /d

418

2 414 638 685 812

1 0 0

5

525 639 746 833

10

546 676 790 905

15

598 681 811 1096

20

25

τ Figure 9.6 Instantaneous variation of crown radius, rc , with dimensionless time, τ = tV0 /d, for different impinging drop Weber numbers We (Sivakumar and Tropea 2002). Reprinted with the permission of AIP Publishing.

9.1 Spray Impact onto Liquid Films

419

Figure 9.7 Sketch of drops in a spray impacting onto a liquid film (Roisman et al. 2006).

Reprinted with the permission of AIP Publishing.

r capillary breakup of ligaments of the collapsing crown (see Fig. 1.6c in Section 1.10 in Chapter 1) r jetting from the surface with droplet separation from the tip r jetting and droplet separation when crowns collide In all cases, the situation must be considered statistically, because of the random occurrence of impact events and impact parameters (size and velocity of drops). In considering the formation of secondary droplets from a fluctuating liquid layer upon which a spray is impacting it is instructive to recall that very successful models for describing phenomena, such as crowns, jetting, splash, etc., have all been achieved by assuming that the flow is inertial-dominated, i.e. that the Kelvin–Helmholtz instability associated with the gas flow, capillary waves, short compressible shock wave, etc. are secondary and can be neglected.

9.1.1

Multiple Drop Impact onto a Dry Substrate In the present subsection the experimental observation of the interaction of two drops impacting a solid substrate and the theoretical modeling of this phenomenon is examined. This can be considered a first step to the extrapolation of results from single-drop impacts to a dense spray. The theoretical analysis builds on the results of Chapter 6, where the formation of the crown-like shape during a single drop impact onto a liquid layer is described as the propagation of a “kinematic discontinuity” in the liquid layer (Yarin and Weiss 1995), which was generalized to the arbitrary case, including oblique impact and interaction of two crowns in Roisman and Tropea (2002) [cf. Section 6.2 in Chapter 6].

420

Spray Impact

Drop generators

Mirror Top-looking video camera

Illumination source

System controller

Dx

Light barrier

Side-looking video camera Substrate

Figure 9.8 Experimental setup to investigate multiple drop impacts (Roisman et al. 2002a).

Reprinted with permission from Elsevier.

Experimental Investigation The experimental setup is shown in Fig. 9.8. Two pendant drops detach from two parallel capillaries under the action of a mechanical disturbance and fall to the dry, horizontal steel surface by gravity. The distance between the drops and the delay between their detachment can be precisely controlled. Both a side-viewing and top-viewing video camera allows multiple exposures at three different times (two in flight and one at a certain pre-programmed instant after impact). This allows the initial drop diameters and impact velocities to be determined. An example of two drops of pure water falling from a height of about 60 cm is shown in Fig. 9.9a. The impact velocity of both drops is V01 = V02 = 3.36 m/s, the diameter of the drops is D1 = D2 = 2.5 ± 0.05 mm, the time interval between impacts is t = 0.9 ± 0.01 ms, and the distance between the impact centers is x = 8.4 mm. The third exposure is taken at the instant t = 2.81 ± 0.005 ms after the impact of the first drop. The second camera captures the top view at two different instants of time t after impact of the first drop (Fig. 9.9b,c). The two cameras are synchronized so that the image of the top view corresponds to the third exposure made by the side-viewing camera. Collision of the drops generates an uprising sheet along their intersection line (see Fig. 9.9a), whereas the spreading on the dry part of the substrate is not modified by the presence of the other drop. When there is a time lag between the impacts, the sheet is inclined from the vertical towards the earlier drop. The interaction of the drops produces several cylindrical jets formed in close proximity to the substrate. These jets break up and create a number of secondary droplets, whereas for the single-drop impact under the same conditions, deposition without breakup is observed. The generation of jetting and splashes is shown in Fig. 9.10, obtained with a smaller inter-drop distance of x = 4.75 mm. For smaller inter-drop distances, the uprising sheet is more stable and breaks up into jets at some distance from the substrate.

(a)

(a) 1 mm

1 mm

Sheet

(b) (b)

(c) (c)

Figure 9.9 Impact of two distant water drops

Figure 9.10 Impact of two water drops onto a

weakly interacting on a steel substrate. D1 = D2 = 2.5 mm, V01 = V02 = 3.36 m/s, t = 0.9 ms, x = 8.4 mm. (a) Side views at three different times, third exposure at t = 2.81 ms; (b) top view, t = 2.81 ms; (c) top view, t = 3.81 ms (Roisman et al. 2002a). Reprinted with permission from Elsevier.

steel substrate showing the formation of a crown-like instability toward the first drop, whereas a festoon-like instability is displayed on the dry side. D1 = 2.66 mm, D2 = 2.75 mm, V01 = V02 = 3.45 m/s, t = 1.31 ms, x = 4.75 mm. (a) Side views at three different times, third exposure at t = 1.96 ms; (b) top view, t = 1.96 ms; (c) top view, t = 2.96 ms (Roisman et al. 2002a). Reprinted with permission from Elsevier.

422

Spray Impact

y (a)

Δx

x Rr1

Rr2

y

(b)

Δx

Xi XB0 XB

Rr2

x

Collision line

Rr1

Figure 9.11 Sketch of two lamellae produced by drop impacts (a) before interaction, t < t0 ; (b) after interaction, t > t0 (Roisman et al. 2002a). Reprinted with permission from Elsevier.

Theory Two drops of the same liquid of initial diameters D1 and D2 impacting onto a dry substrate with impact velocities V01 and V02 are considered. A Cartesian coordinate system {x, y, z} with base vectors (ex , ey , ez ) is used (see Fig. 9.11a). The z axis is directed normal to the substrate. The impact of the first drop onto the substrate at the origin of the coordinate system (0, 0, 0) is taken as the instant of time t = 0. The second drop impacts onto the substrate at the point (x, 0, 0) at the time instant t. The velocity fields Vl1 and Vl2 in the two spreading lamellae, and their thickness, hl1 and hl2 , can be obtained with the help of Eq. (6.12) (in Section 6.2 in Chapter 6) and are given in Eqs. (6.39)–(6.42) and (6.45)–(6.47). The condition for intersection of the two crowns is given by Eq. (6.49), yielding for the coordinates of the crowns’ intersection contour the Eqs. (6.53) and (6.54). Note that the parabolic dependence of the radius of the rim on time assumes an initial increase of Rr , followed by a receding phase. Therefore, there are two analytical solutions for the time instant t0 . One is in the advancing phase; the second is in the receding phase. Only the smaller positive value corresponding to the first interaction of the two rims is considered.

423

9.1 Spray Impact onto Liquid Films

VSR

(a)

hSR

Rim of the second drop

z VS

Sheet rim XR

Rim of the first drop

Sheet

x

hS

XS

hB

XB

VB

Vn

y VW

Substrate

VB (b)

e'z

V1s

V2s

hB e'B

V1n

V2n

e'τ

hl1

hl2 e'n

ez

XB

ey ex Figure 9.12 Sketch of the uprising sheet. (a) General view; (b) zoom for the region of the collision

line (Roisman et al. 2002a). Reprinted with permission from Elsevier.

The theoretical analysis of the uprising sheet formed at the collision line is given in Roisman and Tropea (2002). This theory, as described in subsection 2.2.1 in Chapter 2, is now used to describe the sheet appearing due to the interaction of two lamellae. The main factor influencing the velocity of the uprising sheet and its shape is the inertia of the fluid. The viscosity and the surface tension are therefore neglected in the solution for the velocity VB of the uprising sheet at the collision line and its shape defined as x = XS (see Fig. 9.12).

424

Spray Impact

The uprising sheet is depicted in Fig. 9.12a. The median surface of the sheet is defined in the parametric form x = XS (ti , tB , t ), where t is the time and ti and tB are independent parameters. Therefore, the equation of motion of a material element belonging to the sheet can be written in the (Lagrangian) form ∂ 2 XS (ti , tB , t ) =0 ∂t 2

(9.6)

which must be solved subject to the initial conditions XS (ti , tB , t ) = XB (ti , t ) at t = tB ∂XS (ti , tB , t ) = VB (ti , t ) at t = tB . ∂t

(9.7) (9.8)

This means that the material element ejected from the collision line XB (ti , t ) at the time instant t = tB with the velocity VB (ti , t ), reaches at the time instant t the location XS (ti , tB , t ). The solution of Eq. (9.6) is therefore XS (ti , tB , t ) = XB (ti , tB ) + XB (ti , tB )(t − tB ).

(9.9)

Equation (9.9) is the analytic expression for the shape of the uprising sheet. The thickness of the sheet can be obtained from the mass conservation of the considered material element and is given in Eq. (2.81). The shape of the collision line predicted by the model is compared with the experimental results in Fig. 9.13. Note however, that at the instant when these two drops intersect, the dimensionless time after the impact of the second drop is less than unity, and the theoretical expression for the velocity in its lamella is not yet valid. Therefore, the drop diameter at these initial, very short times after impact (t < 0.1) can be approximated simply by a truncated sphere. Overall, the agreement between theory and experiment can be considered good. In Fig. 9.14 the shape of the collision line (white curves) is shown at various instants of time for the parameters of impact given in the caption. The material parameters of the liquid correspond to water. The gray circles correspond to the expanding lamellae. The instant shown in Fig. 9.14a corresponds to t = t0 . In Fig. 9.14b the spreading phase is shown. In Figs. 9.14c and d the lamellae begin to recede. The shape of the uprising sheet in the plane y = 0 is shown in Fig. 9.15. In fact, this sheet is a three-dimensional surface bounded by a free rim. The calculation of the motion of the sheet rim is a complicated numerical task; however the motion of the sheet rim does not change the shape of the sheet; it only cuts off some upper part of the sheet. The theoretical predictions of the height of the sheet rim evolving in time are shown in Fig. 9.16 for the symmetric case. The vertical uprising sheet appears at the time instant t0 when the two drops intersect. The maximum height of the sheet (at the plane y = 0) first ascends and then descends due to the motion of the free rim. At some instant tend , the rim falls back to the substrate. The results of the prediction indicate that the height of the sheet is very sensitive to the interdrop distance x. Overall, the model presented in this subsection is valid for the inertia dominated impact, when the dimensionless Reynolds and Weber numbers are much higher than

9.1 Spray Impact onto Liquid Films

425

(a)

(b) 3 2 1 0 −1 −2 −2

0

2

4

6

Figure 9.13 Interaction of two water drops on a steel substrate, top view. (a) experiment; (b)

theoretical prediction. D1 = D2 = 2.5 mm, V01 = V02 = 3.36 m/s, t = 0.9 ms, x = 8.4 mm (Roisman et al. 2002a). Reprinted with permission from Elsevier.

unity and when the drops are weakly interacting, which correspond to sparse spray conditions. Nevertheless, this analysis assumes a dry wall around the intersecting drops, while typically, spray impact on a wall results in a liquid film accumulating on the wall. Therefore, the next stage is to consider the hydrodynamics of the liquid wall film and its effect on the impact process.

9.1.2

Sparse Spray Impact A necessary element in the understanding and modeling of spray impact is the hydrodynamics of the flow in a liquid film on a solid substrate. The motion of this film is initiated by drop impacts. Consider the motion in the film during the time period when it is not disturbed by other droplets. The theory of the flow in the film is given in Yarin and Weiss (1995) for the axisymmetric case and is generalized in Roisman and Tropea (2002) for the case of a two-dimensional plane film with the resulting film thickness given by Eqs. (2.7) and (2.9) in subsection 2.1.1 in Chapter 2. It can be easily shown that if det(∇ζ F ) is positive, then the denominator on the righthand side in Eq. (2.9) in subsection 2.1.1 in Chapter 2 vanishes at some positive instant

Spray Impact

4

(a)

t U1/D1 = 2.6

3

3

2

2

1

1 y/D1

y/D1

4

0

−1

−2

−2

−3

−3

−4

−4

4

1

2 x/D1

3

4

5

t U1/D1 = 10

t U1/D1 = 5

−3 −2 −1 0

6

4

(c)

(b)

0

−1

−3 −2 −1 0

3

3

2

2

1

2 3 x/D1

4

5

t U1/D1 = 13

6

(d)

1 y/D1

1 y/D1

426

0

0

−1

−1

−2

−2

−3

−3

−4

−4 −3 −2 −1 0

1

2 x/D1

3

4

5

6

−3 −2 −1 0

1

2 x/D1

3

4

5

6

Figure 9.14 The shapes of the two lamellae (gray circles) and the collision line (white curve).

D1 = 2.5 mm, D2 = 1.25 mm, V01 = 3.36 m/s, V02 = 2.38 m/s, t = 0.9 ms, x = 8.4 mm (Roisman et al. 2002a). Reprinted with permission from Elsevier.

in time. In this case the solution produces a “kinematic discontinuity” (Yarin and Weiss 1995, Roisman and Tropea 2002) leading to the formation of an uprising sheet. At this kinematic discontinuity (or at the “base of the sheet”) the velocity in the film jumps from V1 (x, t ) to V2 (x, t ) and the film thickness jumps from h1 (x, t ) to h2 (x, t ) (see Fig. 2.5 in Section 2.2 in Chapter 2). The position of the kinematic discontinuity is XB and its velocity of propagation is found in Section 2.2 in Chapter 2 and in subsections 6.2.1 and 6.2.2 in Chapter 6. Free liquid sheets, including uprising sheets, are bounded by a rim. This free rim is formed due to capillary forces and its velocity of propagation over the sheet V is given by Eq. (2.159) in Section 2.7 in Chapter 2. When the upward sheet velocity is larger than V , it reaches a maximum height; then it collapses (when the sheet velocity becomes smaller than V ) and falls back onto the wall. In some cases (when the impact velocity

9.1 Spray Impact onto Liquid Films

427

Jet, t- = 13

Jet, t- = 10

Jet, t- = 7 Drop D2, t = 0

Jet, t- = 5

Drop D1, t = 0

Wall Figure 9.15 Predicted shapes of the uprising sheet at various time instants, as well as the position

of two impacting drops at t = 0. The parameters of the impact are the same as in Fig. 9.14 (Roisman et al. 2002a). Reprinted with permission from Elsevier.

exceeds the splashing threshold) the rim centerline deforms, giving rise to several fingerlike jets. These jets then break up, forming a number of secondary droplets. The droplets are formed at the end of each finger (see Fig. 1.6c in Section 1.10 in Chapter 1.) The velocity of such droplets is smaller than the velocity in the finger-like jet. The finger-like jet (ligament) is approximated by a cylindrical jet of diameter dF . Then the momentum 0.40

Dx = 9.6 mm

0.35

Dx = 10.6 mm

0.30

Dx = 11.2 mm

– Zr

0.25 0.20 0.15 0.10 0.05 0.00 0

1

2

3 – t

4

5

6

tend

Figure 9.16 Symmetric impact of two drops. Theoretical prediction of the height of the rim as a

function of the time after impact. D1 = D2 = 2.5 mm, V01 = V02 = 3.36 m/s (Roisman et al. 2002a). Reprinted with permission from Elsevier.

428

Spray Impact

balance of the droplet described as the balance of the inertia of the liquid entering the drop, capillary forces and the capillary pressure pσ in the jet π dF2 π dF2 ρUD2 + π dF σ − pσ = 0, 4 4

(9.10)

where UD is the droplet velocity relative to the jet, and the capillary pressure is pσ = 2σ /dF . The solution of Eq. (9.10) for the relative velocity of the droplet is  2σ UD = . (9.11) ρ dF The expression (9.11) determines the initial velocity of secondary droplets produced by the splash if the velocity of the finger-like jets (ligaments) is known. If the drop impacts with high Reynolds and low Froude numbers, the acceleration of material elements in the ejected sheet is negligibly small, and the velocity of the fingers is thus known (Peregrine 1981, Roisman and Tropea 2002). In Section 6.6 in Chapter 6 the mechanism of cusp formation and fingering is discussed based on the assumption that the rim propagates with a constant velocity. In Roisman and Tropea (2002) the instant of cusp formation is estimated. In order to validate this model, measurements of the drop diameter and two components of the velocity were performed in a water spray using the phase Doppler technique. The detection volume was located 1 mm above a metal polished target. The sign of the u-component of the drop velocity was used to distinguish drops before impact (u > 0) from secondary droplets (u < 0), with u being the velocity normal to the wall. The spray was produced using a commercial pressure swirl atomizer. The parameters of the spray were varied, changing the distance of the nozzle from the impact target, atomization pressure, volumetric flux, type of the nozzle and impingement angle. In Fig. 9.17 examples of the velocity distributions of drops are shown: the measured velocity vector on the left-hand side, and the theoretical predictions on the right-hand side. The gray level in the contour plot is proportional to the logarithm of the probability density function f (u, v), where v is the transverse (parallel to the wall) component of the velocity vector. The direction of motion of the secondary droplets coincides with the direction of the sheet propagation, although the magnitude is smaller by the value of the relative drop velocity, determined by Eq. (9.11). This relative velocity is estimated assuming the diameter of the finger-like jets (ligaments) to be of the same order as the diameter of the secondary droplets (∼40 µm, known from the measurements). For the given conditions this relative velocity is approximately 2 m/s. In the three cases shown in Fig. 9.17 the impact velocity for the simulations was chosen from the measurement data as an average velocity of drops before impact exceeding the splashing threshold We4/5 Re2/5 ≥ 2800 (Tropea and Roisman 2000). In all the cases the predicted velocity distribution is of the same order as the measured velocity distribution. The wider and smoother “cloud” depicting the measured velocities can be associated with the interaction between droplets, which is not accounted for in the simulations. Note, that the characteristic velocity of film fluctuations is much smaller than the drop velocity in

429

9.1 Spray Impact onto Liquid Films

25

25

Primary drops, u > 0

(a)

20

15

15

10

10

5

5

u, m/s

u, m/s

20

0 −5

0 −5

−10

−10

−15

−15

−20 −25 −25 −20 −15 −10 −5

−20

Secondary drops, u < 0 0

5

10 15

20

−25 −25 −20 −15 −10 −5

25

v, m/s 25 20

25

5

10 15

20

25

5

10 15

20

25

10

5

u, m/s

u, m/s

20

15 Primary drops

0 −5

5 0 −5

−10

−10

−15

−15

−20 −25 −25 −20 −15 −10 −5

−20

Secondary drops 0

5

10 15

20

−25 −25 −20 −15 −10 −5

25

v, m/s

0

v, m/s 25

(c)

Primary drops

20

15

15

10

10 5

5

u, m/s

u, m/s

10 15

20

10

20

5

25

(b)

15

25

0

v, m/s

0 −5

0 −5

−10

−10

−15

−15

−20 −25 −25 −20 −15 −10 −5

−20

Secondary drops 0

v, m/s

5

10 15

20

25

−25 −25 −20 −15 −10 −5

0

v, m/s

Figure 9.17 Normal impact of a water spray. Comparison of experimental data for the velocity

distribution (left of the vertical centerline) with the theoretical prediction for the secondary spray (right from the vertical centerline). The impact velocity used in the simulations is: (a) V0 = 10 m/s; (b) V0 = 15 m/s; (c) V0 =20 m/s (Roisman and Tropea 2005). Reprinted with permission from Elsevier.

Spray Impact

25 20

u, m/s

15 10

25

(a) : Experimental data

20

Simul. (d) Simul. (c) Simul. (b) Primary drops

5 0 −5

5 0

−10

−15

−15 −20

−25 −10

Secondary drops −5

0

5 10 v, m/s

15

20

25

Primary drop

−5

−10 −20

(b) : Simulations

15 10 u, m/s

430

−25 −10

Secondary drops −5

0

5 10 v, m/s

15

20

25

Figure 9.18 Oblique impact of a water spray. Comparison of experimental data (a) for the velocity

distribution with the simulations for the secondary spray (b). The impact velocities used for the simulations are: [Simul. (b)] u = 7.5 m/s, v = 6.3 m/s; [Simul. (c)] u = 9.7 m/s, v = 8 m/s; [Simul. (d)] u = 12.45 m/s, v = 10 m/s. The parameters of primary drops used for the simulations are also shown in (a) as white circles (Roisman and Tropea 2005). Reprinted with permission from Elsevier.

such relatively sparse sprays. Their influence on the velocity of the secondary droplets is neglected. The case of oblique impact of water spray is shown in Fig. 9.18. The velocity distribution in this case is very different from that in the normal spray impact. The velocities of the secondary droplets after normal impact are directed in all directions, whereas the velocities after an oblique impact are directed mainly in one direction. This can be explained as follows. There is no known splashing threshold for the case of oblique drop impacts. In some existing models of spray impact the normal velocity component is used to describe such a threshold, which is correct only in the case of impact onto a dry surface when the drag force associated with the parallel to the wall motion is negligibly small (Šikalo et al. 2005). However, the hydrodynamics of the oblique drop impact onto a liquid film depends significantly on the impact angle (Roisman and Tropea 2002) and the splash threshold as a function of impact angle is not known at present. Therefore, the magnitude of the impact velocity V0 used in the simulations was adjusted to fit the results with the data, whereas the impact angle used was 45◦ , i.e. the same as in the experiments.

9.1.3

Typical Scales of Film Fluctuations on the Wall The hydrodynamics of impact of relatively dense sprays, influenced considerably by the film fluctuations and interactions of drops, has not yet been studied in detail. The impact of a Diesel spray onto a wall is depicted in Fig. 9.19 from which it is obvious that the film must be modeled with special attention to its character. Particularly, several important problems are not yet solved or even considered in the literature, namely, prediction of:

9.1 Spray Impact onto Liquid Films

431

Figure 9.19 Image of a Diesel spray impacting onto an inclined target. The uprising finger-like

jets can be clearly seen. The faster, incoming drops appear as vertical streaks due to their higher velocity than the ejected jets (Roisman and Tropea 2005). Reprinted with permission from Elsevier.

r magnitudes for typical film velocities or frequencies of the film fluctuations produced by drop impacts; r characteristic length scales of these fluctuations; r effect of the inertia of the fluctuations on the average film motion and the average film thickness; r influence of these fluctuations on a single drop impact, and of splash conditions on the final outcome. In the following, an attempt is made to characterize the motion in the liquid film associated with the fluctuations of the dynamic pressure. This cannot be considered as an exact model, but more as an estimation for the characteristic velocity, length and time scales. Consider the impact of a dense spray onto a flat, horizontal, rigid substrate. Assume that the thickness of a liquid film created on the substrate is much smaller than both the characteristic size of the spray and the characteristic size of the target. The first assumption allows one to substitute the parameters of the spray at the fluctuating film surface by the corresponding parameters at the steady wall surface. The second assumption implies that the main flow is directed parallel to the wall. Consider also the long-time behavior of spray impact using the assumption of the steady-state stochastic process. The flow in the film produced by spray impact is influenced by the inertia associated with the velocity fluctuations. These fluctuations appear due to individual drop impacts. It is possible to determine a number of characteristic scales describing a single drop impact. The usual scales are the initial diameter, D, as a length scale, the normal impact velocity, V0 , as a velocity scale, and D/V0 as a time scale. In the case of the train of drops produced by the drop generator (Yarin and Weiss 1995) (see Section 6.7 in Chapter 6) an additional time scale, 1/ f , can be introduced, where

432

Spray Impact

f is the frequency of drop impact. The length scale characterizing the spray transport is c−1/3 , where c is the number concentration of the drops in the spray. This characteristic length scale can be associated with the average distance between the drops in the spray. However, it is impossible to determine either the length scale or the time scale characterizing the polydisperse spray impact directly from the given spray parameters. Below a model is considered allowing one to determine such scales from the balance of the inertial forces in the spray and in the fluctuating liquid layer. Consider the impact of a very dense, uniform spray such that the viscous and capillary effects can be neglected in comparison to the inertia of the liquid. The velocity field u in the liquid can be subdivided into two parts: a time-averaged part u and a fluctuating part u , as u = u + u . Note, that u is associated with the fluctuations produced by individual drops impacting onto the liquid layer, whereas u is time-averaged over the time comparable with the characteristic time of the spray. This means that u is constant for the stochastically steady-state spray, and is not constant for intermittent sprays. In the analysis below only a stochastically steady-state spray is considered. Assume for simplicity that the time-averaged velocity u in the film vanishes and the time-averaged parameters of the spray are uniform. This situation arises near the center of symmetry of the target, r → 0. Assume also axial symmetry around the spray impact axis, such that u s = u s (r)er . The subscript s means that the velocity is associated with ˙ which is assumed to a single drop impact. The time-averaged pressure in the film is P, ˙ be uniform, i.e. ∂ P/∂r = 0. However, the temporal and spatial fluctuations of pressure, p s (r, t ), play a role of a force driving flow fluctuations in the film. In this case the continuity equation and the radial component of the momentum balance equation for the fluctuating part of the flow over the wall can be written in the form ∂h 1 ∂ (r h u s ) + = 0, ∂t r ∂r r h ∂ p s ∂ (r h u s ) ∂ (r h u 2s ) + =− . ∂t ∂r ρ ∂r

(9.12) (9.13)

The average number of drops impacting onto a circle of radius r during the time interval t after impact is λ = π r2 t n, ˙ where the area number flux density of the spray onto the film n˙ is assumed to be constant. The question is, what is the distribution of drops around an individual drop under consideration? When a single drop is removed from the ensemble of the uniformly distributed drops, a “hole” is formed. It is obvious that if λ 1 the average number of drops around the drop under consideration is λ1 ≈ λ − 1. Assuming that the drops of the spray are distributed randomly in space and in time, the probability P (k, λ) that exactly k drops impact onto the considered circle can be described by the Poisson distribution (Feller 1968) P (k, λ) =

e−λ λk . k!

(9.14)

The latter assumption is valid even in relative close proximity to the nozzle of water spray, as was shown experimentally using the phase Doppler instrument (Roisman and Tropea 2001).

9.1 Spray Impact onto Liquid Films

433

In order to determine the distribution P1 of drops around an individual drop, the case k = 0 with the probability P (0, λ) = e−λ should be excluded from the set of possible events. The minimum possible k value is 1 (because the individual drop under consideration has already hypothetically impacted onto the circle). Thus, the distribution P1 can be found with the help of Eq. (9.14) as P1 (k1 , λ) =

e−λ λk1 +1 (1 − e−λ )(k1 + 1)!

(9.15)

where k1 = k − 1 is the number of drops impacted onto the circle, excluding the one under consideration. The term (1 − e−λ ) expresses the probability that at least one drop (including the one under consideration) will impact onto the circle. It appears in the denominator of the expression for P1 as a normalization parameter. The average number of drops impacting onto the circle around the one associated with the center of impact of the individual drop under consideration is λ1 (r, t ) =

∞ 

k1 P1 = λ eλ ξ −1 − 1

(9.16)

k1 =0

where ξ = eλ − 1. This value depends on the radius of the area around the individual drop under consideration and time. On the other hand, the number of drops λ1 can be determined using the statistically averaged number flux density n˙ 1 (r, t ) of the drops around the individual drop under consideration t r λ1 = 2π rn˙ 1 (r, t )drdt. (9.17) 0

0

The expressions (9.16) and (9.17) are used to determine n˙ 1 (r, t ) in the form n˙ 1 (r, t ) = G(r, t ) n˙ where # $ 1 ∂ 2 λ1 = eλ ξ −1 − 3λξ −2 + (2 + ξ )λ2 ξ −3 . (9.18) 2π r n˙ ∂t∂r The function G expresses the ratio of the statistically averaged number flux n˙ 1 of drops around the individual drop under consideration to the given constant number flux n. ˙ This function approaches unity in the limit λ → ∞. This means that the average distribution of the drops far from the one under consideration is unaffected by the drop impact. Variation of G in the radial direction means that the statistically averaged pressure p s = G P˙ applied to the liquid film by the drops surrounding the one under consideration also varies in r, causing flow fluctuations in the film. This statistically averaged flow will be analyzed below. Using the length scale $, the time scale T and the velocity scale ϒ in the form   1/6  ρ 1/3 P˙ $ , T= , ϒ= (9.19) $= 2 2 n˙ π ρ T P˙ nπ ˙ G(r, t ) =

Equations (9.12) and (9.13) are rendered dimensionless and the equations for the statistically averaged mass balance and the momentum in the radial direction can be obtained

434

Spray Impact

using Eqs. (9.18) in the following dimensionless form ∂ h˜ 1 ∂ (˜r h˜ u˜ s ) + = 0, ∂ t˜ r˜ ∂ r˜ ∂G ∂ u˜ s ∂ u˜ s + u˜ s =− ∂ t˜ ∂ r˜ ∂ r˜

(9.20) (9.21)

where the variables with tilde are dimensionless. The parameter λ can be written as λ = r˜2 t˜. Consider first the approximate solution of Eqs. (9.20) and (9.21) as r → 0. The linearized right-hand side of Eq. (9.21) obtained using Eq. (9.18) is −

2 ∂G ≈ − r˜t˜ ∂ r˜ 3

(9.22)

for the initial conditions h˜ = h˜ 0 = const.,

u˜ s = 0

at t˜ = 0.

(9.23)

Then the solution is u˜ s = −˜r t˜2 /3

  ; 5/3; −2t˜3 /27 ,  3 0 F1 ; 2/3; −2t˜ /27 0 F1

h˜ = h˜ 0

0 F1



−2 ; 2/3; −2t˜3 /27

(9.24)

where 0 F1 (; a; b) are the confluent hypergeometric limit function. It can be shown that this solution becomes singular near the axis r˜ = 0 at the time instant t˜∗ ≈ 2.274. This instant corresponds to the kinematic discontinuity in the velocity gradient (a “shock”) causing formation of an uprising central jet. This phenomenon was observed in the experiments with the Diesel spray impact (see Fig. 9.19). Also, such jets not associated with any crown produced by an individual drop impact were observed even during not so “aggressive” water spray impact (Tropea and Roisman 2000, Sivakumar and Tropea 2002). The nature of this kinematic discontinuity is very similar to the process leading to the formation of the uprising crown-like sheets produced by single drop impacts (Yarin and Weiss 1995), inclined sheets produced by the oblique drop impacts (Roisman et al. 2002b) or drop interactions on a substrate discussed in Chapter 2 and 6 and in the previous subsection (Roisman et al. 2002a). Consider again the system of nonlinear differential Eqs. (9.20) and (9.21). The numerical solution of this system, valid for all the radii, obtained using the expression (9.18) for the function G is shown in Figs. 9.20 and 9.21. The velocity distribution u˜ s in the film is shown in Fig. 9.20. As time approaches the critical moment t˜∗ = 2.274, the velocity gradient at the center (˜r = 0) tends to −∞, whereas the film thickness grows infinitely (see Fig. 9.21). Note however, that the model for the flow fluctuations implies the long-wave approximation in the thin film, and is not valid for times t˜ > t˜∗ . Therefore, it cannot describe formation of the central jet. The flow in this jet is mostly longitudinal, normal to the wall, leading to the jet stretching. Moreover, at the leading end of the jet, an almost spherical droplet is formed due to the capillary forces. The motion of this droplet determines the length of the jet, which is in reality finite.

435

9.1 Spray Impact onto Liquid Films

0.0 −0.2 −0.4 ∼ t = 0.5 ∼ t = 1.0 ∼ t = 1.5 ∼ t = 2.0 ∼ t = 2.26

∼ ur

−0.6 −0.8 −1.0 −1.2 0

1

2

∼ r

3

4

5

Figure 9.20 The dimensionless, statistically averaged radial velocity u˜ s = u˜r in the film (Roisman

and Tropea 2005). Reprinted with permission from Elsevier.

The parameters characterizing the spray can be rewritten in the more convenient form as vz  =



P˙ , ρ q˙

D =

6q˙ π n˙

1/3 .

(9.25)

Equation (9.25) defines the averaged normal drop velocity vz  and diameter D in the spray. Therefore, the expressions for the scales given by Eqs. (9.19) transform to   D vz  1/6 , T= # ϒ = [q˙ vz ]1/2 . (9.26) $ = D $1/3 , 2 36 q˙ 6 q˙ vz  10 ∼ t = 0.5 ∼ t = 1.0 ∼ t = 1.5 ∼ t = 2.0 ∼ t = 2.26

9 8 7

∼∼ h/h0

6 5 4 3 2 1 0 0

1

2

∼ r

3

4

5

˜ (Roisman and Tropea Figure 9.21 The dimensionless, statistically averaged film thickness h 2005). Reprinted with permission from Elsevier.

Spray Impact

80

60

40 u, m/s

436

20

0

−20 −50

−40

−30

−20

−10

0

10

20

30

40

50

v, m/s Figure 9.22 Normal impact of a dense Diesel spray. Experimental data for the distribution of the

drop velocity (Roisman and Tropea 2005). Reprinted with permission from Elsevier.

An example of an oblique impact of a very dense Diesel spray is shown in Fig. 9.19. A similar experiment with a normal impact was conducted and the experimental data for the drop velocity distribution in the spray, immediately above the target surface, are shown in Fig. 9.22. These data, obtained using a phase Doppler instrument, reveal impact velocities up to 70 m/s, whereas the velocity of the secondary droplets is smaller than 10 m/s. The small secondary droplets can be formed only by the breakup of fingerlike jets (ligaments). In the case of low-velocity, sparse sprays, or impact of a single drop, these jets appear after the breakup of crown-like uprising sheets. In the case of dense Diesel spray impact, crowns are not observed, but rather single jets appearing directly from the film. The hypothesis is now checked that these jets can be associated with the fluctuations of the dynamic pressure in the film. The diameter of the secondary drops is of the order of the characteristic size of film fluctuations, whereas their velocity magnitude is of the order of the characteristic velocity of such fluctuations. The value of the average local volume flux density q˙ is estimated by simply measuring the volume of the injected and collected fuel during a defined time. In the experiments of Roisman and Tropea (2005) this value is of the order of 0.2 m/s at the injection pressure of 150 bar, and 0.3 m/s at the injection pressure of 300 bar. The values of q˙ are used in Eqs. (9.26) to determine the scales of the film fluctuations. In Figs. 9.23 and 9.24 the average droplet diameter, Da , and the magnitude of the average normal velocity, Ua , of the secondary spray produced by the Diesel spray impact are shown as functions of $ and ϒ, determined for various spray parameters. It should be emphasized that both the predicted length scale of the film fluctuations and the predicted velocity scale are of the

9.1 Spray Impact onto Liquid Films

437

30

25

Da, μm

20

15

10

5

0 5

0

10 Λ, μm

15

20

Figure 9.23 Normal impact of a dense Diesel spray. Average diameter of the secondary droplets as a function of the length scale for the film fluctuations $ (Roisman and Tropea 2005). Reprinted with permission from Elsevier.

1.0 0.9 0.8 0.7 Ua, m/s

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5 ϒ, m/s

2.0

2.5

Figure 9.24 Normal impact of a dense Diesel spray. Average normal velocity of the secondary

droplets as a function of the velocity scale for the film fluctuations ϒ (Roisman and Tropea 2005). Reprinted with permission from Elsevier.

438

Spray Impact

same order as the Da and Ua . Moreover, Da and Ua are well-correlated with $ and ϒ, respectively. This correlation provides a qualitative validation of the model presented above.

9.1.4

Film Produced by a Single Drop Impact In this subsection the characteristic film thickness associated with the drop impact is considered, because of its fundamental importance for the analysis of splashing, and diameters and velocities of the secondary droplets. The thickness of the lamella produced by drop impact, and the flow in this lamella was analyzed in Roisman et al. (2002b). In this study a kinematically admissible velocity field was assumed in the liquid layer on the wall. This flow satisfies the continuity equation and the boundary conditions at the wall and at the upper surface of the drop moving with the impact velocity V0 . Since the volume of this thin wall layer is much smaller than the drop volume, the inertial effects are also negligibly small in comparison to the inertia of the entire drop, and thus are neglected. Surface tension and wettability effects are taken into account. The total axial momentum balance of the impacting drop yields an equation for the characteristic thickness of the lamella hL at the time instant D/V0 , where D is the drop initial diameter, and V0 is the impact velocity. A very similar approach successfully predicts the films produced by binary drop collisions (Roisman 2004), cf. subsection 8.3.2 in Chapter 8. Even though some relatively simple expressions for the lamella thickness can be obtained from the experimental data, it is still important that these results are related to the basic physics. The relation between hL and the characteristic diameter DL of the lamella formed by drop impact (usually reported in the literature) can be estimated from the volume balance of the drop in the form π D2L hL /4 = π D3 /6 resulting in hL /D = (2D2 )/(3D2L ). In Fig. 9.25 the experimental data (Roisman et al. 2002a, Roisman et al. 2002b, Šikalo 2003, Fukai et al. 1995) for hL estimated at the time instant D/V0 are shown as a function of the Reynolds number, Re = DV0 /ν, where ν is the kinematic viscosity. None of the cases shown in Fig. 9.25 leads to splash. Two ranges of the Reynolds numbers can be clearly distinguished. At high Reynolds numbers, Re > 2000, the lamella thickness is scaled as hL ∼ DRe−1/2 . At smaller Reynolds numbers, Re < 1000, the film thickness is scaled in the form hL ∼ DRe−1/3 . At very high Reynolds numbers the thickness hL of the lamella produced by drop impact onto a dry wall is defined by the thickness of the viscous boundary layer corresponding to Stokes’ first (or Rayleigh’s) problem as  (9.27) hL ∼ νD/V0 = DRe−1/2 . The idea of using this value of the viscous boundary layer as a scale for the film thickness produced by a train of periodic drop impacts was first introduced by Yarin and Weiss (1995) (see Section 6.7 in Chapter 6). In the study of Tropea and Roisman (2000) the expression (9.27) has been used to scale the thickness of the crown produced by a single drop impact onto a liquid film. A similar approach was subsequently used to estimate the values of velocity gradients and shear stresses in the boundary layer during the first deformation stage of a single drop impact onto a uniform liquid film

9.1 Spray Impact onto Liquid Films

439

Figure 9.25 Single drop impact onto a dry wall. Experimental data [◦ – Roisman and Tropea (2002), Roisman et al. (2002b),  – Šikalo (2003),  – Fukai et al. (1995), + – Roisman et al. (2002a)] for the dimensionless thickness of the lamella at the time instant D/V0 as a function of the Reynolds number (Roisman et al. 2006). Reprinted with the permission of AIP Publishing.

(Roisman and Tropea 2002). These assumptions about the scaling of the crown thickness have been confirmed numerically in Josserand and Zaleski (2003). At smaller Reynolds numbers the pressure gradient in the radial direction becomes significant and the assumption leading to Eq. (9.27) is no longer valid. The main idea of the model discussed below can be explained considering the orders of magnitude of the main parameters involved in the description of drop impact. In a thin layer of thickness hL the velocity changes from uz ∼ V0 in the normal-to-the-wall (z) direction to ur in the radial r direction. The continuity equation requires that ur ∼ V0 r/hL in the time interval 0 < t < D/V0 . Therefore, the pressure gradient in this layer ∂ p/∂r ∼ μ∂ 2 ur /∂z2 yields p ∼ μV0 r2 /h3L . The net force applied to the drop at the wall in the normal direction, which can be obtained by the integration of the pressure p over the wetted spot, is approximately F ∼ μV0 D4 /h3L . During the initial stage of drop deformation, 0 < t < D/V0 , the total initial momentum of the drop is balanced by the impulse of the force F ρπ D3 D V0 ∼ F . 6 V0

(9.28)

Condition (9.28) yields an expression for the thickness of the lamella at the instant t = D/V0 in the form hL ∼ D0 Re−1/3 valid at relatively small Reynolds numbers (see Fig. 9.25).

(9.29)

440

Spray Impact

It is interesting that for the impact parameters corresponding to Reynolds numbers around 2000, the jump of the film thickness from one branch (hL ∼ Re−1/3 ) to the other (hL ∼ Re−1/2 ) takes place. The low Reynolds impacts are affected also by surface tension. In Roisman et al. (2002b) the value of hL is estimated as a root of the following cubic equation 3

3We + 5(1 − cos θ )RehL = 10ReWe hL ,

hL = hL /D0 ,

(9.30)

accounting for the surface tension, where θ is the average value of the dynamic contact angle. Two asymptotic solutions of Eq. (9.30) can be easily found: hL ∼ Re−1/3 if Re  We3/2 (which means that the terms associated with the wettability are negligibly small), or hL ∼ We−1/2 if Re We3/2 . Note, that Eq. (9.30) is obtained not accounting for the time-dependent viscous boundary layer and is not valid for drop impacts characterized by very high Reynolds numbers. In the following, the description for high Reynolds number impacts leading to splash is of interest, therefore the relevant scale defined in Eq. (9.27) will be used. The focus is on the experimental description of a quasi-steady, normal impact of a polydispersed spray onto a smooth, rigid wall. The aim is the prediction of the distribution of droplets in the secondary spray produced by this impact. The intensity of the primary impacting spray can be changed varying the injection pressure, the distance between the atomizer and the substrate and the type of the nozzle. The intensity is affected by local flux density distribution in the spray, distribution of the drop diameters and the velocity field. It is virtually impossible to vary only one parameter in such sprays, keeping all other parameters constant. A precise description of drop distributions in polydispersed sprays requires a large number of parameters. Therefore, even a straightforward parametric study of spray impact requires an enormous number of experiments. The situation with the modeling of spray impact is complicated by the fact that even the outcome of a single drop impact is hard to describe. There are no reliable models for the number and diameter of the secondary droplets produced by a splash of a single drop onto a stationary uniform liquid film of a given thickness, as outlined in Chapter 6. The approach is, therefore, to develop an experimental method of spray characterization (particularly to determine the local spray flux density vectors and local momentum flux density tensors), and to describe the main parameters of the secondary spray using the scaling laws based on the fundamental hydrodynamics of splash.

9.2

Description of the Secondary Spray In this section the instabilities initiating fingering and splashing will be addressed, leading to a description of the secondary spray in terms of droplet size and velocity. All inertially dominated flows associated with drop and spray impact onto a cold stationary rigid target can be subdivided into three main stages:

9.2 Description of the Secondary Spray

441

Cusps

(a)

(b) Jets

(c)

(d)

Figure 9.26 Stages of the splash: (a) growth of the transverse instabilities of the rim centerline, (b)

cusp formation, (c) finger-like jets ejection from the cusps, and (d) breakup of the jets leading to the appearance of secondary droplets (Yarin 1993, Roisman et al. 2006). Reprinted with the permission of AIP Publishing.

r transformation of a nearly spherical drop into a film r transformation of a film into a jet, formation of a rim r breakup of a jet and creation of secondary droplets. These phenomena are influenced by secondary effects typical for spray impacts: interaction of liquid sheets, hole formation in the films, etc. The important elements of splash modeling are a definition of the commensurate scale for the film thickness, Eq. (9.27), the splash threshold given by Eq. (6.134) in Section 6.7 in Chapter 6, and the description of the dynamics of a rim formed at the edge of the sheet due to the capillary forces, Eq. (2.159) in Section 2.7 in Chapter 2. From the modeling point of view the splash process can be subdivided into several stages shown schematically in Fig. 9.26: bending of the rim centerline and development of its instability, described in Section 6.6 in Chapter 6, cusp formation (Yarin and Weiss 1995), and the ejection of the finger-like jets at the cusp locations (analyzed in Sections 6.6 and 6.7 in Chapter 6). The last stage of the splash – capillary breakup of a jet leading to formation of secondary droplets – is discussed in subsection 1.10.1 in Chapter 1.

9.2.1

Characterization of Spray Impact Spray Measurements using the Phase Doppler Instrument The procedure of spray impact measurements using the phase Doppler instrument is explained in detail in Tropea and Roisman (2000). The experimental setup and the

Spray Impact

Z

Receiving optics

Impacting spray

Detection volume

dt

Detection volume

Y Y

dt Ls

1mm

Spherical target

Transmitting optics

X

Figure 9.27 Sketch of the experimental setup: (a) front view, (b) top view (Roisman et al. 2006).

Reprinted with the permission of AIP Publishing.

v

geometry of the detection volume are shown schematically in Fig. 9.27. The laboratory coordinate system {X , Y, Z} is related to the optical configuration of the phase Doppler instrument. The Z axis coincides with the direction of the transmitting optics, whereas the vertical X axis is directed along the spray axis. Two components of drop velocity, u and v corresponding to the X and Y axes, and the diameter D are measured 1 mm above the north-pole of a metal, spherical (93 mm in diameter) target. The sign of the normal-to-the-wall u velocity allows primary drops before wall interaction (u > 0) to be distinguished from secondary droplets (u < 0). A sample correlation of the drop size and the u and v components of drop velocity for the spray directed normal to the sphere is shown in Fig. 9.28 for one experimental condition. All the points with negative u velocity correspond to secondary droplets. The impacting drops move mostly normal to the target, whereas the secondary droplets move at various angles. The average velocity of the primary drops increases with the drop diameter. This behavior is typical for decelerating sprays. One can recognize also

u

442

(a)

(b) u

Figure 9.28 Normal impact onto a spherical target. Scatter diagram (a) – [D, u] and (b) – [u, v].

Each point corresponds to the detected and validated drop (Roisman et al. 2006). Reprinted with the permission of AIP Publishing.

9.2 Description of the Secondary Spray

443

a “cloud” of relatively small droplets (D < 40 µm) moving towards the wall with relatively high velocities, comparable with the velocity of the primary spray. This phenomenon can be explained by the fact that some small droplets are accelerated by an airflow induced by the fast and large neighboring primary drops. The geometry of the detection volume must be determined accurately in order to calculate the correct drop distributions and local flux densities in spray (Roisman and Tropea 2001). The length of the detection volume is determined by the scattering angle φ and the projected thickness of the slit Ls of the receiving optics. The effective diameter of the detection volume, dt , depends on the laser beam intensity, system configuration and the drop diameter. The diameter dt (D) is evaluated analyzing the statistics of the burst lengths of the detected droplets. The third velocity component, w, is not measured in this configuration of the phase Doppler instrument. Therefore, the general three-dimensional case cannot be correctly characterized using this system. Nevertheless, in the considered normal, axisymmetric spray impact, the problem can be solved noting that the w velocity component has the same probability density function as the transverse component v. The azimuthal component of drop velocity vanishes at the spray axis and the transverse velocity is expressed in the form v = ur cos ϕ, where ur is the unknown radial component of the drop velocity and ϕ is the unknown azimuthal angle. In the axisymmetric spray considered here the value of the radial velocity is independent of the azimuthal angle. Thus, the averaged square values of the velocity components are related by  2π 1 1 2 2 v = ur cos2 ϕ dϕ = u2r . (9.31) 2π 0 2 Therefore, statistically correct values for the magnitude of the velocity vector in the axisymmetric case can be obtained if it is assumed that  (9.32) uabs = u2 + 2v 2 . The flux density Q˙ X in the axial, normal-to-the-wall direction of a property q is determined using  Nsv  ηv q uX 1 ˙ QX = √ T i=1 A u2 + 2v 2

(9.33)

where T is the observation time, Nsv is the number of validated signals, ηv is the correction factor accounting for the contribution of the non-validated droplets to the flux and also taking into account the probability of the detection of two or more droplets simultaneously [see subsection 7.1.1 and Roisman and Tropea (2001)]. The projected area A depends on the drop diameter and the direction of the velocity vector. It is determined from the analysis of the distribution of burst lengths of the detected drops. In Eq. (9.33) q = 1 is used ˙ q = π D3 /6 for the volume  2for the2 number flux density n, 3 flux density q, ˙ q = ρπ D u + 2v /12 for the kinetic energy flux density e˙K , q =  ρπ D3 u2 + 2v 2 /12 + π D2 σ for the total mechanical energy e˙T , q = ρπ D3 u/6 for the axial momentum flux density p, ˙ etc.

444

Spray Impact

Spray impact is characterized by the number of drops impacting onto a unit area of the substrate. This is actually the “plane” distribution of a spray moving in the X -direction. The algorithm for calculation of such a distribution is explained in Roisman and Tropea (2001). The average normal velocity U10 , the average diameter D30 and the average velocity magnitude Vabs20 of the spray are calculated as   1/3  p˙ 6q˙ 2e˙K 1/2 , D30 = U10 = , Vabs20 = . (9.34) ρ q˙ π n˙ ρ q˙ The corresponding average parameters are calculated separately for the primary drops before impact: Ub , Db and Vb , and for the secondary droplets after wall impact: Ua , Da and Va . Then, the dimensionless parameters of the impacting spray are defined by Reb =

Ub Db , ν

Web =

ρUb2 Db , σ

4/5 Kb = Re2/5 b Web .

(9.35)

The parameters of the impacting spray can be varied changing the injection pressure in the nozzle, using nozzles of different inner diameter or by varying the distance from the nozzle to the target. In the experiments with water (ρ = 103 kg/m3 , ν = 10−6 m2 /s, σ = 73 × 10−3 N/m) the average normal velocity of the primary spray has been varied [4, 21] m/s, the average diameter Db ∈ [30, 135] µm, the volume in the range Ub ∈ $ # flux density q˙ ∈ 2.5 × 10−4 , 1.1 × 10−2 m/s. Additionally, similar experiments with a Diesel injection spray (ρ = 817 kg/m3 , ν = 3.5 × 10−6 m2 /s, σ = 26.3 × 10−3 N/m) were performed. The Diesel spray injection is not continuous. However, during the injection time (8 ms) the average velocity and diameter of the spray are approximately stationary. The average velocity of the primary Diesel spray during the injection time has been varied in the range Ub ∈ [10, 26] m/s and the average diameter Db ∈ [16, 21] µm. It is important to note here that the proposed method of characterization of normal spray impact is based on the assumption that the spray is symmetric relative to its axis and that the gradients of the spray parameters along the wall vanish near the symmetry axis. In the case of an inclined spray impact this condition is no longer valid and the distribution of the droplets detected by the instrument is influenced by the gradients of spray parameters. The secondary droplets arrive at the detection volume from various locations at the wall surface with different impact conditions. The analysis of such spray impact measurements requires analysis of the