Applications of Heat, Mass and Fluid Boundary Layers 9780128179499, 1331331331, 012817949X

Table of contents :
Front Cover......Page 1
Applications of Heat, Mass and Fluid Boundary Layers......Page 4
Copyright......Page 5
Contents......Page 8
List of contributors......Page 14
Preface......Page 18
Acknowledgments......Page 20
1.1 Introduction......Page 22
1.2 The basic equations of viscous flow......Page 23
1.2.2 Derivation of the law of conservation of mass in the rectangular coordinate system......Page 24
1.2.3 Derivation of the law of conservation of mass in the cylindrical coordinate system......Page 26
1.3 Momentum equation......Page 27
1.3.1 Constitutive relations for the equation of motion for Newtonian fluids......Page 30
1.3.2 Equations of motion for Newtonian fluids: Navier-Stokes equations......Page 32
1.3.3 Constitutive equations for non-Newtonian fluids......Page 33
1.3.4 The law of conservation of energy (the first law of thermodynamics)......Page 36
1.3.5 The second law of thermodynamics: entropy production......Page 39
1.4 Velocity slip and temperature jump......Page 40
References......Page 41
2.2 Heat transfer......Page 44
2.3.1 Conduction......Page 45
2.3.2 Convection......Page 46
2.3.3 Radiation......Page 47
2.3.4 Thermodynamics of heat transfer......Page 48
2.3.5 Heat transfer and entropy production......Page 49
2.4.1 Dimensional analysis and similarity......Page 50
2.4.2 The velocity boundary layer equations......Page 52
2.4.4 The concentration boundary layer equation......Page 54
2.4.5 More on convection boundary layer flows......Page 55
2.6 External boundary layer flows......Page 61
2.7 Wake and jet boundary layers......Page 63
2.8 Hydrodynamic boundary layer stability......Page 64
2.8.2 Boundary layer separation......Page 67
2.9 Practical applications of boundary layer flow......Page 69
References......Page 73
Nomenclature......Page 76
3.1 Introduction......Page 77
3.2 Fundamental assumption......Page 78
The conservation of mass or continuity equation......Page 79
The conservation of momentum or momentum equation (Euler equation)......Page 80
3.3.1 Energy equation for adiabatic and isothermal processes......Page 82
3.3.2 Energy equation for an adiabatic process......Page 83
3.4 Entropy factors......Page 84
References......Page 86
4.1 The continuity equation......Page 88
4.2 The momentum equations......Page 89
4.3 Coutte flow......Page 91
4.4 Plane Poiseuille flow......Page 92
4.5 Hagen Poiseuille flow (pipe flow)......Page 93
4.6.1 Uniform suction on a plane......Page 95
4.7 Flow between plates with bottom injection and top suction......Page 96
4.8 Flow in a porous duct......Page 99
4.9 Approximate analytic solution (perturbation)......Page 102
4.10 Numerical solution......Page 105
4.11.1 Boundary layer governing equations......Page 106
4.12.1 Momentum thickness θ and momentum integral......Page 108
4.12.2 Displacement thickness δ*......Page 111
4.13.1 Laminar flow......Page 112
4.13.2 Turbulent flow......Page 113
References......Page 115
5.1 Introduction......Page 116
5.2 Historical perspectives of series-based methods and the Merk-Chao-Fagbenle (MCF) procedure......Page 118
5.3 Mathematical formulations of the Merk-Chao-Fagbenle method......Page 122
5.3.1 Forced convection over circular cylinder in crossflow......Page 124
5.3.2 Skin friction, displacement thickness, momentum thickness, and velocity distribution......Page 125
5.3.3 Heat transfer results......Page 130
5.3.4 Comparison with experimental heat transfer data......Page 132
5.3.6 Comparison with experimental mass transfer data......Page 135
5.4 Forced convection flow over a sphere......Page 140
5.5 Forced convection over sears' airfoils......Page 142
5.6 Consideration in forced convection (nonisothermal) boundary layer transfer......Page 145
5.8 Consideration for nanofluids and other extensions of the methodology......Page 148
References......Page 149
6.2 The spectral-homotopy analysis method (SHAM)......Page 154
6.3.1.1 SHAM solution......Page 158
6.3.1.2 Results......Page 161
6.3.2 Example 2: steady von Kármán flow......Page 162
6.3.2.1 SHAM solution......Page 163
6.3.2.2 Results......Page 166
6.4 Pros and Cons of the SHAM......Page 167
References......Page 168
Nomenclature......Page 170
7.1 Introduction......Page 171
7.2 Mathematical formulation......Page 173
7.3 Similarity analysis......Page 174
7.3.1 Boundary conditions......Page 175
7.4.1.1 Overview......Page 176
7.4.1.2 Some important details......Page 177
7.4.1.3 Application......Page 178
7.5 Results and discussion......Page 180
7.6 Conclusion......Page 193
References......Page 196
8.1 Introduction......Page 198
8.2 Bivariate interpolated spectral quasilinearization method......Page 200
8.3 Results and discussion......Page 204
8.4 Conclusion......Page 209
References......Page 210
9.1 Introduction......Page 212
9.1.1 Some historical background and literature review......Page 213
9.1.2 Objectives of the study......Page 214
9.2 Governing equations for horizontal elliptic duct rotating in parallel mode......Page 215
9.2.1 Equations for the working fluid......Page 216
9.2.2 Normalization parameters......Page 218
9.3 Governing equations for vertical elliptic duct rotating in parallel mode......Page 220
9.4 Parameter perturbation analysis for horizontal elliptic ducts in parallel mode rotation......Page 222
9.4.1 Boundary conditions......Page 223
9.4.2 Power series......Page 224
9.4.3 Zeroth-order solutions......Page 226
9.4.4 First-order solutions......Page 228
9.4.5 Second-order solutions......Page 232
9.4.6 Solutions of power series approximations......Page 238
9.4.7 Peripheral Nusselt number......Page 240
9.5 Parameter perturbation analysis for vertical elliptic ducts in parallel mode rotation......Page 242
9.5.2 Zeroth-order solutions......Page 243
9.5.4 Second-order solutions......Page 244
9.5.5 Peripheral local Nusselt number......Page 245
9.6 Discussion and conclusions of the effects of the variables on fluid flow and heat transfer......Page 246
References......Page 251
10.1 Introduction - laminar boundary layer equations......Page 254
10.1.2 One-dimensional unsteady heat equation......Page 255
10.2 Numerical methods - general background and the most important techniques in the context of the laminar boundary layer ODEs......Page 256
10.2.1 General overview - Euler scheme example......Page 257
10.2.2.1 Runge-Kutta methods......Page 260
10.2.2.2 Multistep methods - Adams-Bashforth......Page 261
10.3.1 Blasius equation as a numerical boundary value problem......Page 262
10.3.2.1 Change of variables and first steps......Page 265
10.3.2.2 Secant method for the iteration towards finding the accurate . F |η= 0......Page 266
10.3.2.3 Bisection method for the iteration towards finding the accurate . F | η=0......Page 268
10.4.1 Program description......Page 270
10.4.2 Results, interpretation, and sensitivity study......Page 271
10.5 Application of the finite-element method for one-dimensional unsteady heat equation......Page 274
10.6 Practical implementation of the finite-element method for one-dimensional unsteady heat equation......Page 275
10.7 Summary and outlook......Page 277
References......Page 279
11.2 Viscous dissipation and magnetic field effects on convection flow over a vertical plate......Page 280
11.3 Free convection mhd flow past a semiinfinite flat plate......Page 284
11.4 Convection of non-darcy flow, Dufour and Soret effects past a porous medium......Page 286
11.5 Unsteady mhd convective flow with thermophoresis of particles past a vertical surface......Page 289
11.6 Thermal conductivity effects on compressible boundary layer flow over a vertical plate......Page 295
References......Page 299
12.1 Introduction......Page 302
12.1.1 Some background and history of nanofluids......Page 304
12.2 Current understanding of nanofluids......Page 306
12.2.1 Production of nanofluids......Page 308
12.2.2 Some effective medium theories of nanofluids......Page 309
12.3.1 Simple nanofluids......Page 310
12.3.2 Hybrid nanofluids......Page 337
12.3.3 Carbon nanotube nanofluids......Page 344
12.4 Thermophysical properties of nanofluids......Page 345
12.4.1 Thermal conductivity (knf)......Page 347
12.4.2 Dynamic viscosity (μnf)......Page 349
12.4.3 Density (ρnf ), specific heat ( Cp ) nf , and thermal expansion ( β)nf coefficient......Page 351
12.4.4 Other factors......Page 352
12.5 Convective heat transfer of nanofluids......Page 353
12.5.1 Convective heat transfer of nanofluids in porous media......Page 356
12.5.2 Convective mass transfer of nanofluids......Page 358
12.5.3 Convective heat transfer of nanofluids in magnetic and electric fields......Page 359
12.5.4 Nanofluid flow and heat transfer in turbulent flow......Page 361
12.5.5 Boundary slip considerations in nanofluids......Page 363
12.5.6 The lattice-Boltzmann method for convective nanofluid research......Page 364
12.6 Global nanofluid research - developments, policy perspectives, and patents......Page 365
12.7 Applications of nanofluids......Page 366
12.7.1 Application in conventional and renewable energy......Page 369
12.7.2 Applications of nanofluids in automotive systems......Page 371
12.7.3 Applications of nanofluids in electronic cooling systems......Page 372
12.7.5 Other notable areas of application of nanofluids......Page 373
12.8 Research gaps and outlook......Page 375
12.9 Conclusion......Page 376
References......Page 378
13.2 Internal combustion engines - Otto and Diesel cycles......Page 404
13.3 Electrical power generation - ideal basic Rankine cycle......Page 409
13.4 Refrigeration systems - ideal vapor compression refrigeration cycle......Page 412
13.5 Gas turbine systems - ideal air-standard Brayton cycle......Page 415
13.6 Desiccant and subcooling dehumidification......Page 419
13.7 Evaporative cooling......Page 423
13.8 Entropy generation in boundary layer flow and heat transfer......Page 427
References......Page 431
14.2 Background......Page 434
14.2.1.1 Shear-thinning, or pseudo-plastic, fluids......Page 436
14.2.1.2 Shear-thickening, or dilatant, fluids......Page 439
14.2.2.1 Thixotropic and rheopectic fluids......Page 440
14.2.3 Non-Newtonian laminar boundary layer flows......Page 441
14.2.3.1 Laminar boundary layer flows through pipes......Page 442
14.2.3.2 Laminar boundary layer flow over a flat plate......Page 443
14.2.3.4 Laminar boundary layer flow over a porous surface......Page 444
14.2.4 Heat transfer in non-Newtonian laminar boundary layers......Page 445
14.2.4.1 Thermodynamic-entropy generation in non-Newtonian boundary layers......Page 446
14.2.5 Non-Newtonian nanofluid boundary layer transfer......Page 447
14.2.5.1 Extensions of the Merk-Chao-Fagbenle method to non-Newtonian fluids......Page 448
14.3.1 Biological/biomedical systems: vascular fluid dynamics......Page 449
14.3.2 Chemical systems: pharmaceutical products......Page 450
14.3.4 Geosciences: drilling muds......Page 451
14.3.5 Transportation systems: transport of crude oil emulsions......Page 452
References......Page 453
15.1 Introduction......Page 458
15.2 Climate change......Page 459
15.2.1 Global warming......Page 460
15.3.1 Industrial activity......Page 461
15.3.2 Deforestation......Page 462
15.4.1 Major greenhouse gases......Page 463
15.4.2 The greenhouse effect......Page 464
15.6 Some climate change trends......Page 465
15.6.1 Modeling of climate change......Page 466
15.7.2 Marine transport......Page 467
15.9 Climate change mitigation and adaptation......Page 468
15.11 Climate change and agriculture......Page 470
15.12 The role and integration of renewable energy technologies......Page 472
15.13 Climate changes in China......Page 474
15.14 Climate changes in Malaysia......Page 475
15.15 Climate changes in Nigeria......Page 478
15.16 Climate changes in Brazil......Page 480
15.17 Climate changes in India......Page 481
15.18 Climate changes in South Africa......Page 483
15.19 Climate changes in Ecuador......Page 485
15.20 Conclusion......Page 486
References......Page 487
A Some mathematical background of fluid mechanics......Page 494
Transport theorem......Page 497
Conservation of mass (continuity equation)......Page 498
Alternative forms of the continuity equation......Page 500
Sum of forces......Page 501
Further analysis of FS......Page 503
Construction of the matrix T......Page 505
The Navier-Stokes equations......Page 506
References......Page 509
B.1 Types of fluid flow......Page 512
Compressible and incompressible flows......Page 513
B.2.1 Pathlines......Page 514
Properties of a stream tube......Page 515
References......Page 516
Index......Page 518
Back Cover......Page 532

Citation preview

WOODHEAD PUBLISHING SERIES IN ENERGY

APPLICATIONS OF HEAT, MASS AND FLUID BOUNDARY LAYERS Edited by

R.O. FAGBENLE, O.M. AMOO, S. ALIU, A. FALANA

Applications of Heat, Mass and Fluid Boundary Layers

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Woodhead Publishing Series in Energy

Applications of Heat, Mass and Fluid Boundary Layers Edited by

R.O. Fagbenle Center for Petroleum, Energy Economics and Law, CPEEL (A John D. and Catherine T. McArthur Center of Excellence) University of Ibadan Ibadan, Oyo State, Nigeria

O.M. Amoo Department of Mechanical Engineering University of Ibadan Ibadan, Oyo State, Nigeria

S. Aliu Department of Mechanical Engineering University of Benin Benin City, Nigeria

A. Falana Department of Mechanical Engineering University of Ibadan Ibadan, Oyo State, Nigeria

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2020 Elsevier Ltd. All rights reserved. MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-817949-9 For information on all Woodhead Publishing publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Brian Romer Acquisition Editor: Maria Convey Editorial Project Manager: Joanna Collett Production Project Manager: Joy Christel Neumarin Honest Thangiah Designer: Victoria Pearson Typeset by VTeX

This seminal volume is dedicated to the blessed and loving memory of Professor Bei Tse Chao of the University of Illinois Urbana-Champaign (1918–2011) and to the recent passing in June 2019 of Professor Richard Olayiwola Fagbenle (1943–2019), during the course of this work, and also to all those looking to examine things from a different perspective, to see things with not just the eye of analysis, but also curiosity. Prof. Fagbenle was a doctoral student of Prof. Chao, while the three other editors of this volume were all doctoral students of Prof. Fagbenle, in the area of fluid boundary layer transfer.

Professor Bei Tse Chao

Professor Richard Olayiwola Fagbenle

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Contents

List of contributors Preface Acknowledgments 1

2

3

Physics of fluid motion O.C. Okoye, B.O. Bolaji 1.1 Introduction 1.2 The basic equations of viscous flow 1.3 Momentum equation 1.4 Velocity slip and temperature jump References Mechanisms of heat transfer and boundary layers Sufianu Aliu, O.M. Amoo, Felix Ilesanmi Alao, S.O. Ajadi 2.1 Introduction 2.2 Heat transfer 2.3 Modes of heat transfer 2.4 The boundary layer equations 2.5 Internal boundary layer flows 2.6 External boundary layer flows 2.7 Wake and jet boundary layers 2.8 Hydrodynamic boundary layer stability 2.9 Practical applications of boundary layer flow 2.10 Conclusions References On some basics of compressible fluid flows Felix Ilesanmi Alao, Samson Babatunde Nomenclature 3.1 Introduction 3.2 Fundamental assumption 3.3 Basic equations of compressible fluid flow 3.4 Entropy factors 3.5 A note on applications of compressible fluid flow References

xiii xvii xix 1 1 2 6 19 20 23 23 23 24 29 40 40 42 43 48 52 52 55 55 56 57 58 63 65 65

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4

5

6

Contents

Boundary layer equations in fluid dynamics Hafeez Y. Hafeez, Chifu E. Ndikilar 4.1 The continuity equation 4.2 The momentum equations 4.3 Coutte flow 4.4 Plane Poiseuille flow 4.5 Hagen Poiseuille flow (pipe flow) 4.6 Flow over porous wall 4.7 Flow between plates with bottom injection and top suction 4.8 Flow in a porous duct 4.9 Approximate analytic solution (perturbation) 4.10 Numerical solution 4.11 The boundary-layer equations 4.12 Influence of boundary layer on external flow 4.13 The flat-plate boundary layer References The Merk–Chao–Fagbenle method for laminar boundary layer analysis R. Layi Fagbenle, Leye M. Amoo, S. Aliu, A. Falana 5.1 Introduction 5.2 Historical perspectives of series-based methods and the Merk–Chao–Fagbenle (MCF) procedure 5.3 Mathematical formulations of the Merk–Chao–Fagbenle method 5.4 Forced convection flow over a sphere 5.5 Forced convection over sears’ airfoils 5.6 Consideration in forced convection (nonisothermal) boundary layer transfer 5.7 Consideration in mixed convection boundary layer transfer 5.8 Consideration for nanofluids and other extensions of the methodology 5.9 Conclusion References The spectral-homotopy analysis method (SHAM) for solutions of boundary layer problems S.S. Motsa, Z.G. Makukula 6.1 Introduction 6.2 The spectral-homotopy analysis method (SHAM) 6.3 Examples 6.4 Pros and Cons of the SHAM 6.5 Conclusion References

67 67 68 70 71 72 74 75 78 81 84 85 87 91 94

95 95 97 101 119 121 124 127 127 128 128

133 133 133 137 146 147 147

Contents

7

8

9

10

On a new numerical approach of MHD mixed convection flow with heat and mass transfer of a micropolar fluid over an unsteady stretching sheet in the presence of viscous dissipation and thermal radiation S. Shateyi, G.T. Marewo Nomenclature 7.1 Introduction 7.2 Mathematical formulation 7.3 Similarity analysis 7.4 Methods of solution 7.5 Results and discussion 7.6 Conclusion Acknowledgment References On the bivariate spectral quasilinearization method for nonlinear boundary layer partial differential equations Vusi M. Magagula, Sandile S. Motsa, Precious Sibanda 8.1 Introduction 8.2 Bivariate interpolated spectral quasilinearization method 8.3 Results and discussion 8.4 Conclusion Acknowledgments References Mixed convection heat transfer in rotating elliptic coolant channels Olumuyiwa Ajani Lasode 9.1 Introduction 9.2 Governing equations for horizontal elliptic duct rotating in parallel mode 9.3 Governing equations for vertical elliptic duct rotating in parallel mode 9.4 Parameter perturbation analysis for horizontal elliptic ducts in parallel mode rotation 9.5 Parameter perturbation analysis for vertical elliptic ducts in parallel mode rotation 9.6 Discussion and conclusions of the effects of the variables on fluid flow and heat transfer References Numerical techniques for the solution of the laminar boundary layer equations O.M. Amoo, A. Falana 10.1 Introduction – laminar boundary layer equations

ix

149 149 150 152 153 155 159 172 175 175

177 177 179 183 188 189 189 191 191 194 199 201 221 225 230

233 233

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10.2

Numerical methods – general background and the most important techniques in the context of the laminar boundary layer ODEs 10.3 Application of different numerical methods for the solution of the Blasius equation 10.4 Implementation of the shooting method for the solution of the Blasius equation 10.5 Application of the finite-element method for one-dimensional unsteady heat equation 10.6 Practical implementation of the finite-element method for one-dimensional unsteady heat equation 10.7 Summary and outlook References 11

12

13

On a selection of convective boundary layer transfer problems Anselm Oyem 11.1 Introduction 11.2 Viscous dissipation and magnetic field effects on convection flow over a vertical plate 11.3 Free convection mhd flow past a semiinfinite flat plate 11.4 Convection of non-darcy flow, Dufour and Soret effects past a porous medium 11.5 Unsteady mhd convective flow with thermophoresis of particles past a vertical surface 11.6 Thermal conductivity effects on compressible boundary layer flow over a vertical plate 11.7 Conclusion References Advanced fluids – a review of nanofluid transport and its applications Leye M. Amoo, R. Layi Fagbenle 12.1 Introduction 12.2 Current understanding of nanofluids 12.3 Classes of nanofluids 12.4 Thermophysical properties of nanofluids 12.5 Convective heat transfer of nanofluids 12.6 Global nanofluid research – developments, policy perspectives, and patents 12.7 Applications of nanofluids 12.8 Research gaps and outlook 12.9 Conclusion References On a selection of the applications of thermodynamics L.M. Amoo

235 241 249 253 254 256 258 259 259 259 263 265 268 274 278 278 281 281 285 289 324 332 344 345 354 355 357 383

Contents

13.1 13.2 13.3 13.4

Introduction Internal combustion engines – Otto and Diesel cycles Electrical power generation – ideal basic Rankine cycle Refrigeration systems – ideal vapor compression refrigeration cycle 13.5 Gas turbine systems – ideal air-standard Brayton cycle 13.6 Desiccant and subcooling dehumidification 13.7 Evaporative cooling 13.8 Entropy generation in boundary layer flow and heat transfer 13.9 Conclusions References 14

15

Overview of non-Newtonian boundary layer flows and heat transfer Leye M. Amoo, R. Layi Fagbenle 14.1 Introduction 14.2 Background 14.3 A note on current research status and applications of non-Newtonian fluids 14.4 Future directions References Climate change in developing nations of the world Leye M. Amoo, R. Layi Fagbenle 15.1 Introduction 15.2 Climate change 15.3 Anthropogenic influences on climate change 15.4 Greenhouse gases (GHGs) 15.5 Earth’s energy budget 15.6 Some climate change trends 15.7 Climate change and the transport sector 15.8 Climate change and the industrial sector 15.9 Climate change mitigation and adaptation 15.10 Climate change and conflict 15.11 Climate change and agriculture 15.12 The role and integration of renewable energy technologies 15.13 Climate changes in China 15.14 Climate changes in Malaysia 15.15 Climate changes in Nigeria 15.16 Climate changes in Brazil 15.17 Climate changes in India 15.18 Climate changes in South Africa 15.19 Climate changes in Ecuador 15.20 Conclusion References

xi

383 383 388 391 394 398 402 406 410 410 413 413 413 428 432 432 437 437 438 440 442 444 444 446 447 447 449 449 451 453 454 457 459 460 462 464 465 466

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A

B

Contents

Some mathematical background of fluid mechanics Felix Ilesanmi Alao, Samson Babatunde, Zounaki Ongodiebi References

473

Some fundamentals of fluid mechanics Zounaki Ongodiebi B.1 Types of fluid flow B.2 Flow visualization References

491

Index

488

491 493 495 497

List of contributors

S.O. Ajadi Faculty of Science, Department of Mathematics, O.A.U, Ile Ife, Nigeria Felix Ilesanmi Alao Department of Mathematical Sciences, University of Texas, Richardson, TX, United States Federal University of Technology, Akure, Nigeria Sufianu Aliu University of Benin, Department of Mechanical Engineering, Benin City, Nigeria Leye M. Amoo Stevens Institute of Technology, Hoboken, NJ, United States University of California, Los Angeles, Los Angeles, CA, United States Olaleye M. Amoo Department of Mechanical Engineering, University of Ibadan, Oyo State, Ibadan, Nigeria Samson Babatunde Department of Mathematics, University of Texas, Dallas, TX, United States B.O. Bolaji Department of Mechanical Engineering, Federal University, Oye Ekiti, Ikole Ekiti Campus, Ekiti State, Nigeria Ayodeji Falana Department of Mechanical Engineering, University of Ibadan, Oyo State, Ibadan, Nigeria Hafeez Y. Hafeez Federal University Dutse, Dutse, Jigawa State, Nigeria Olumuyiwa Ajani Lasode Department of Mechanical Engineering, University of Ilorin, Ilorin, Nigeria

xiv

List of contributors

R. Layi Fagbenle Mechanical Engineering Department and Center for Petroleum, Energy Economics Law, CPEEL (A John D. and Catherine T. McArthur Center of Excellence), University of Ibadan, Ibadan, Oyo State, Nigeria Vusi M. Magagula School of Mathematics, Statistics and Computer Science, University of KwaZuluNatal, Pietermaritzburg, South Africa Faculty of Science and Engineering, Department of Mathematics, University of Swaziland, Kwaluseni, Swaziland DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoEMaSS), Pretoria, South Africa Z.G. Makukula Department of Mathematics, University of Swaziland, Kwaluseni, Swaziland G.T. Marewo University of Limpopo, Department of Mathematics & Applied Mathematics, Sovenga, South Africa Sandile S. Motsa School of Mathematics, Statistics and Computer Science, University of KwaZuluNatal, Pietermaritzburg, South Africa Faculty of Science and Engineering, Department of Mathematics, University of Swaziland, Kwaluseni, Swaziland Chifu E. Ndikilar Federal University Dutse, Dutse, Jigawa State, Nigeria O.C. Okoye Department of Mechanical Engineering, Federal University, Oye Ekiti, Ikole Ekiti Campus, Ekiti State, Nigeria Zounaki Ongodiebi Department of Mathematics and Computer Science, Niger Delta University, Bayelsa, Nigeria Anselm Oyem Department of Mathematical Sciences, Federal University Lokoja, Lokoja, Kogi State, Nigeria

List of contributors

xv

S. Shateyi Department of Mathematics & Applied Mathematics, University of Venda, Thohoyandou, South Africa Precious Sibanda School of Mathematics, Statistics and Computer Science, University of KwaZuluNatal, Pietermaritzburg, South Africa

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Preface

Fluid flow remains a fascinating subject in both physiology and nature. It sees attention for its prodigious assortment of industrial applications that range from the processing of chemicals to aircraft performance. Fluid motion, along with the inherent mathematical difficulty of the nonlinear differential equations necessary to represent it, has fascinated numerous great applied mathematicians and physicists of the past. These include Isaac Newton, Leonardo da Vinci, Leonard Euler, the Bernoulli family, and Joseph-Louis Lagrange, to name a few. When examining the situation involving heat transfer issues, the need arises to understand internal and external flows because although real differences exist, the major ideas are the same. In recent past, and as can be observed from the literature, the research field of fluid boundary layers is mostly saturated by theoreticians with many studies placing less emphasis on practical implications. As such, opportunities for application of boundary layer results in real-world problems abound. Notably, in the application of boundary layer results, however, care must be taken to ensure the conditions governing boundary layer analysis do pertain, in particular to a distinguishing free-stream within the flow under consideration. The boundary layer is inherently a mathematically dense area of fluid mechanics. In recent past, there has been remarkable progress in boundary layer transfer analysis, especially towards the study of advanced heat transfer fluids known as nanofluids, with the unifying theme being energy (cooling/heating and efficiency) and sustainability. Suffices to say, there is a sense of heightened and vigorous activity in the thermal-fluid-mass research area. Every text provides roughly the same information and differs mainly in emphasis, philosophy, depth, applications, mode of presentation, and organization. The main justification for another monograph on fluid boundary layer transfer is the level of distinctiveness and potential exploration of new perspectives on boundary layers considering many advances on the subject, the study of nanofluids, contrasted with some specific mathematical methods or procedures thereof. The monograph explores what has been achieved and contemplates what the future has in store in the field. This multiauthored monograph is an exposition of the state-of-the-art overviews with new research and development activities of the ever-evolving dynamics, and applications of boundary layer research. This volume is largely concerned with viscous incompressible boundary layer transfer and may serve as a reference text on a broad range of fluid boundary layer transfer problems. The chapters have been contributed by selected researchers in the field of thermal-fluid-mass considering boundary layers, heat transfer, thermodynamics, entropy generation, nanofluids, energy, and climate change with special focus on areas of applications. The theoretical concepts and analyses are presented to serve as a more reliable guide for generalization, correlation, and prediction versus any quantity of empiricism. The monograph provides a unique overview of the fundamentals and

xviii

Preface

insights into advanced concepts. Future research opportunities are also highlighted. Thermal-fluid-mass problems today are predominantly solved numerically using several numerical techniques and tools of computational fluid dynamics, as opposed to the experimental approach. These numerical solutions are sometimes validated by experimental data, other times with benchmark numerical work. This volume is aimed at providing scholars, researchers, the thermal-fluid-mass industry, and students both an introductory and advanced overview of boundary layers and related subjects and towards areas of application such as energy and climate change which are the most vivid applications of boundary layer research. The book is dispositioned as follows. The book consists of 15 chapters and two appendices. Some chapters are introductory, while others are at early postgraduate to advanced level. The scope is thus predicated on the collective strengths of the contributing chapter authors and specialized interests rather than common approaches and analysis. Chapter 1 considers the physics of fluid flow. Chapter 2 addresses the fundamental heat transfer modes, forced, natural, and mixed convection boundary layer flows, as well as some basics on boundary layer separation. Also, Chapter 3 briefly examines some fundamentals of compressible flows. Chapter 4 reviews and examines boundary layer equations, perturbation approaches, and turbulent boundary layers. Chapter 5 covers a broad overview of a very disciplined technique of laminar boundary layer analysis now commonly referred to as the Merk–Chao–Fagbenle method. In another regard, a recently developed technique based on homotopy analysis is applied in Chapter 6 to the solution of boundary layer problems. In Chapters 7 and 8, new numerical techniques to solve convective boundary layer problems are presented. Chapter 9 is concerned with mixed convection in rotating elliptic coolant channels. Chapter 10 broadly reviews common numerical methods in boundary layer analysis applied to the Blasius equation. The finite-element technique is also discussed in Chapter 10. A selection of convective fluid boundary layer problems is reviewed in Chapter 11. A new type of advanced fluid known as nanofluid, which may have profound implications in the near future, is surveyed in Chapter 12. Applications of fluid boundary layers and thermodynamics are discussed in Chapter 13, while Chapter 14 addresses non-Newtonian boundary layer transfer. The book is concluded with Chapter 15 on energy and climate change, emphasizing developing nations. Appendices A and B provide further some fundamentals of the subject of fluids.

Acknowledgments

The successful completion of this seminal monograph is, in no small measure, due to the efforts of those who gave generously of their time in reading, criticizing, and modifying the early drafts of the book, and who helped proofread the final copy. We are also stupendously and extraordinarily grateful to the authors who remained steadfast with this seminal book project and who provided the several interesting chapters contained therein. The numerous reviewers who also offered their time to critique some of the chapters in this book are greatly thanked and acknowledged. This undertaking has required extra patience from everyone, especially the chapter authors. We greatly acknowledge the support of Joanna Collet and Maria Convey of Elsevier for their dedicated patience and support throughout the process of developing this book. We are also grateful to our respective institutions for providing the enabling environment for this work to materialize.

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O.C. Okoye, B.O. Bolaji Department of Mechanical Engineering, Federal University, Oye Ekiti, Ikole Ekiti Campus, Ekiti State, Nigeria

1.1 Introduction Fluids are made up of molecules that are continuously in motion. Most engineering applications deal with the average or macroscopic effects of several fluid molecules. A fluid is regarded as a substance which is infinitely divisible, that is, a continuum, and the focus are not on the behavior of the respective molecules of the fluid. The continuum is a theoretical continuous medium used to replace the molecular structure, when dealing with the relationships of fluid flow on an analytical or mathematical basis [14]. The continuum hypothesis treats the fluid as being infinitely divisible without character change. Therefore, material properties like viscosity, density or thermal conductivity, and variables such as pressure, temperature and velocity, can be defined at a mathematical point as the limit of the mean of the given variable or quantity across the molecular fluctuations [22]. The method of continuum mechanics is very useful in offering physical explanation and mathematical description of different transport phenomena without the need to fully understand the internal micro- and nanostructures of fluids [20]. Classical fluid mechanics is based on the concept of a continuum. When considering the behavior of fluids under normal conditions, the continuum assumption holds true. In situations in which the mean free path of the molecules is of the same order of magnitude as the least significant characteristic dimension, the continuum assumption becomes invalid. In such situations, for example, rarefied gas flow, the concept of a continuum is discarded and the microscopic and statistical viewpoints are adopted [6,32,40]. Solid structures, as well as fluid flow fields [20], are assumed to be continua insofar as the local material properties are defined as averages across material elements or volumes significantly larger than the microscopic length scales of the solid or fluid while being small in comparison with the macroscopic structure. Mathematically, for the continuum hypothesis to be valid, the following condition must be satisfied [12,22]: d  V 1/3  L, where d is the length scale which is representative of the microstructure of the fluid, most often a molecular length scale, V 1/3 is the characteristic linear dimension of the averaging volume, L is the macroscopic scale which is typical for spatial gradients in the variables that were averaged. The size of the flow domain usually dictates the scale of L. Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00009-8 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Applications of Heat, Mass and Fluid Boundary Layers

Adopting the continuum hypothesis, the macroscopic laws of classical continuum physics are used to give a mathematical description of fluid motion, as well as heat transfer in systems that are not isothermal. These laws are: the law of conservation of mass, the law of conservation of linear and angular momentum, and the law of conservation of energy (i.e., the first law of thermodynamics). While the second law of thermodynamics is not directly used in the derivation of the governing equations, it provides constraints on the permissible forms of the constitutive models which relate the gradients of velocity in the fluid to short-range forces acting across the surfaces inside the fluid [22,23]. Most theoretical studies in fluid dynamics are based on the concept of a perfect fluid. A perfect fluid is one that is both frictionless and incompressible. When perfect fluids move, two fluid layers in contact with each other do not experience tangential forces (i.e., shearing stresses). However, they act on each other with normal forces (i.e., pressure) only. Thus, the perfect fluid has no internal resistance to change the shape of the fluid. The mathematical theory of the motion of a perfect fluid is very well developed, and gives a description of real fluid motion which is satisfactory in many situations, for instance, the formation of jets of liquid in air. However, the theory of perfect fluids cannot explain the drag of a body. In real fluids, the inner layers of the fluid transmit both tangential and normal stresses, and this is also true near a solid wall which is in contact with the fluid. Thus, the results obtained from the theory of a perfect fluid are not acceptable in such situations. The tangential or friction forces in a real fluid are related to the viscosity of the fluid [16,38]. The viscosity of real fluids can be considerably affected by shear rate, temperature, pressure, molecular structure, molecular weight, and time of shearing. The viscosity of a gas increases with temperature, but the viscosity of a liquid decreases with temperature. There are no inviscid fluids in reality [6,28,29]. Nevertheless, in some situations, the effects of viscosity are relatively small when compared with other effects. In such situations, therefore, viscous effects can often be neglected. For instance, viscous forces which are developed in flowing water may be several orders of magnitude less than the forces as a result of other influences like gravity or pressure differences [28]. There are two ways of analyzing the motion of a mass of fluid upon which forces act in certain conditions. The equations of the motion of a fluid have been obtained based on any of the given methods of analysis. One method of analysis is to track the history of all particles of the fluid, while the other method investigates the velocity, density, and pressure at every point of space which the fluid occupies at all points in time. These obtained equations are respectively known as the Lagrangian and Eulerian forms of the hydrokinetic equations [44].

1.2 The basic equations of viscous flow Viscous flow equations have been available for over a century. The complete form of these equations is impossible to solve even with the high computing power of modern digital computers. The equations for turbulent flow cannot be solved with current

Physics of fluid motion

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mathematical techniques since the boundary conditions are randomly dependent on time [5,43]. Derivations of the equations of fluid dynamics [20] give a more in-depth understanding of the physics which all of the terms in the final equations mathematically represent. It also provides insight into the assumptions (or the shortcomings) of a given mathematical model [1]. White [43] noted that the exact number of fundamental equations of compressible viscous flow is a matter of personal choice, as some relations are more fundamental than others. In this chapter, the system is considered to have only three fundamental relations along with four auxiliary relations. The fluid is assumed to be uniform and homogeneous in composition. The fundamental equations are the three laws of conservation for physical systems, namely: • The law of conservation of mass (continuity) • The law of conservation of momentum (Newton’s second law) • The law of conservation of energy (the first law of thermodynamics). These three fundamental equations are used to obtain three unknowns, which are [33]: • The thermodynamic pressure, p • The absolute temperature, T , and • The velocity vector, V . Assuming local thermodynamic equilibrium, p and T are considered to be the two independent thermodynamic variables needed. In the final forms of the conservation equations, four additional thermodynamic variables are included. They are the enthalpy (or internal energy), density, thermal conductivity, and dynamic viscosity. The thermodynamic pressure, p, and the absolute temperature, T , can, therefore, be used to uniquely obtain these additional four variables. The basic equations which will be derived are largely general and based on the following assumptions [43]: • • • • •

The fluid constitutes a (mathematical) continuum. The particles of the fluid are basically in thermodynamic equilibrium. The conduction of heat is according to Fourier’s law. Body forces are only a result of gravity. There are no internal sources of heat.

1.2.1 Law of conservation of mass: the continuity equation The law of conservation of mass relates the density field to the velocity field. An infinitesimal control volume will be used to derive the law of conservation of mass in both the rectangular and cylindrical coordinate systems [24].

1.2.2 Derivation of the law of conservation of mass in the rectangular coordinate system Consider an infinitesimal cube having dimensions dx, dy, and dz (see Fig. 1.1). At the center of the cube, O, the density is ρ, and the velocity has components u, v, and w,

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 1.1 Differential control volume in the rectangular coordinate system [35].

in the x, y, and z axes, respectively. By using the truncated Taylor series expansion of the density multiplied by the normal component of velocity at the midpoint of all the six faces of the cube gives [4]: Center of front face: (ρw) ∼ = ρw + ∼ Center of rear face: (ρw) = ρw − Center of Center of Center of Center of

∂(ρw) dz ∂z 2 , ∂(ρw) dz ∂z 2 , ∂(ρu) dx ∼ right face: (ρ u) = ρ u + ∂x 2 , ∂(ρu) left face: (ρu) ∼ = ρu − ∂x dx 2 , ∂(ρv) dy ∼ top face: (ρv) = ρv + ∂y 2 , ∂(ρv) dy bottom face: (ρv) ∼ = ρv − ∂y 2 .

The mass flow rates through each face of the cube are shown in Fig. 1.2. They were obtained by calculating the product of the density, the surface area of each face, and the normal component of velocity at the midpoint of each face. The rate of change of mass of the control volume as it shrinks to a point is [4]:  ∂ρ ∂ρ dv ∼ dxdydz = ∂t ∂t where dxdydz = volume of the cube. Substituting the appropriate terms into the integral form of the continuity equation and simplifying gives [6,28,34]: ∂ρ ∂(ρu) ∂(ρv) ∂(ρw) + + + = 0. ∂t ∂x ∂y ∂z

(1.1)

Eq. (1.1) is the differential form of the conservation of mass equation in the rectangular coordinate system. It is also referred to as the continuity equation. Eq. (1.1) is valid for both steady and unsteady flow, as well as for both incompressible and compressible fluids [28]. In vector notation, Eq. (1.1) is written as [4,7,28]: ∂ρ + ∇ · ρV = 0. ∂t

(1.2)

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Figure 1.2 Mass flow rate through each face of the control volume [4].

Figure 1.3 Differential control volume in cylindrical coordinate system [35]. (A) Isometric view; (B) Projection on rθ plane.

1.2.3 Derivation of the law of conservation of mass in the cylindrical coordinate system The differential control volume for the cylindrical coordinate system is shown in Fig. 1.3. At the center, O, of the control volume, the density is ρ and the components of the velocity in the r, θ , and z directions are Vr , V θ , and Vz , respectively. Using the truncated Taylor series expansion, the mass flux through the six faces of the control volume is given in Table 1.1. The components of velocity in the r, θ, and z directions are assumed to be in the positive direction. Thus, the net rate of mass flux going out through the control surface is [26,41]:   ∂ρVr ∂ρVθ ∂ρVz + +r drdθdz. ρVr + r ∂r ∂θ ∂z

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Applications of Heat, Mass and Fluid Boundary Layers

Since the volume of the fluid element is “rdθdrdz” and the density within the control volume is ρ, the rate at which mass changes within the control volume is: ∂ρ rdθdrdz. ∂t The differential equation for the conservation of mass in the cylindrical coordinate system becomes ρVr + r

∂ρVr ∂ρVθ ∂ρVz ∂ρ + +r +r = 0, ∂r ∂θ ∂z ∂t

or ∂ρVz ∂ρ ∂(rρVr ) ∂ρVθ + +r +r = 0. ∂r ∂θ ∂z ∂t By dividing through by r, the equation becomes ∂ρ 1 ∂(rρVr ) 1 ∂(ρVθ ) ∂(ρVz ) + + +r = 0. ∂t r ∂r r ∂θ ∂z

(1.3)

In vector notation, Eq. (1.3) is ∂ρ + ∇ · ρV = 0. ∂t

1.3 Momentum equation By applying Newton’s second law to an infinitesimal particle of fluid having mass dm, the differential form of the momentum equation can be derived. Newton’s second law for a finite system is F=

dP , dt

where F is the net force acting on the system and P is the linear momentum of the system. Rewriting Newton’s second law for an infinitesimal system having mass dm gives [6] dF = dm

dV . dt

Since the acceleration of an element of fluid having mass dm, which moves in a velocity field, is given by a=

∂V ∂V ∂V ∂V DV = +u +v +w , Dt ∂t ∂x ∂y ∂z

Table 1.1 Mass flux through the control surface of a cylindrical differential control volume [6].  Surface ρV · dA            ∂ρ ∂Vr dr dr dr dr dr r Vr − ∂V r − dr Inside (−r) = − ρ − ∂ρ ∂r ∂r 2 2 2 dθ dz = −ρVr rdθdz + ρVr 2 dθ dz + ρ ∂r r 2 dθ dz + Vr ∂r r 2 dθ dz             ∂ρ ∂Vr dr dr dr dr dr r Outside Vr + ∂V r + dr (+r) = ρ + ∂ρ ∂r ∂r 2 2 2 dθ dz = ρVr rdθdz + ρVr 2 dθ dz + ρ ∂r r 2 dθ dz + Vr ∂r r 2 dθ dz            ∂ρ dθ ∂ρ dθ dθ drdz = −ρV drdz + ρ ∂Vθ dθ drdz + V θ Front Vθ − ∂V (−θ ) = − ρ − ∂θ θ θ ∂θ ∂θ ∂θ 2 2 2 2 drdz            ∂ρ dθ ∂ρ dθ dθ drdz = ρV drdz + ρ ∂Vθ dθ drdz + V θ Back Vθ + ∂V (+θ ) = ρ + ∂θ θ θ ∂θ ∂θ ∂θ 2 2 2 2 drdz            dz V − ∂Vz dz rdθdr = −ρV rdθdr + ρ ∂Vz dz rdθdr + V ∂ρ dz rdθdr Bottom (−z) = − ρ − ∂ρ z z z ∂z 2 ∂z 2 ∂z ∂z 2 2            dz V + ∂Vz dz rdθdr = ρV rdθdr + ρ ∂Vz dz rdθdr + V ∂ρ dz rdθdr Top (+z) = ρ + ∂ρ z z z ∂z 2 ∂z 2 ∂z ∂z 2 2 Then, or

      

    

 ∂Vz ∂ρ ∂ρ ∂ρ ∂Vθ r +V + ρ + V + r ρ + V drdθ dz ρV · dA = ρVr + r ρ ∂V r z θ ∂r ∂r ∂θ ∂θ ∂z ∂z   ∂ρVz ∂ρVθ r ρV · dA = ρVr + r ∂ρV ∂r + ∂θ + ∂z drdθ dz

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 1.4 Stresses in the x direction on a fluid element [35].

Newton’s second law can be written as [6]   DV ∂V ∂V ∂V ∂V dF = dm = dm +u +v +w . Dt ∂t ∂x ∂y ∂z

(1.4)

In order to obtain expressions for the forces acting on the fluid element in the three coordinate directions, consider the element of fluid shown in Fig. 1.4. In the figure only the x component of the stresses which result in the x component of the surface forces are shown. The fluid element is of mass dm and volume dxdydz. The stresses at the center of the fluid element are σxx , τyx , τzx , and the stresses (obtained using the truncated Taylor’s series expansion about the fluid element’s center) which act on all the six faces of the fluid element in the x direction are shown in Fig. 1.4. The net surface force in the x direction is obtained by summing the forces in that direction:



∂σxx dx ∂σxx dx dFSx = σxx + dydz − σxx − dydz ∂x 2 ∂x 2



∂τyx dy ∂τyx dy dxdz − τyx − dxdz + τyx + ∂y 2 ∂y 2

∂τzx dz ∂τzx dz dxdy. dxdy − τzx − + τzx + ∂z 2 ∂z 2 Simplifying this equation gives

∂τyx ∂σxx ∂τzx + + dxdydz. dFSx = ∂x ∂y ∂z If the force due to gravity is assumed to be the only body force acting on the fluid element, then the net force in the x direction, dFx , is as presented below. The expressions for the force components in the y and z directions are derived in a similar way,

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and are presented as:

dFx = dFBx + dFSx

∂τyx ∂σxx ∂τzx = ρgx + + + ∂x ∂y ∂z

dxdydz,



∂τxy ∂σyy ∂τzy + + dxdydz, dFy = dFBy + dFSx = ρgy + ∂x ∂y ∂z

(1.5a)

(1.5b)



∂τxz ∂τyz ∂σzz dFz = dFBz + dFSz = ρgz + + + dxdydz. ∂x ∂y ∂z

(1.5c)

Substituting the expressions for the components of the force in the x, y, and z directions into Eq. (1.4) gives the general differential equations of motion for fluids. The equations are valid for any fluid which satisfies the continuum assumption. These equations are presented as [6,28]:

∂τyx ∂σxx ∂τzx ∂u ∂u ∂u ∂u + + =ρ +u +v +w , (1.6a) ρgx + ∂x ∂y ∂z ∂t ∂x ∂y ∂z

∂τxy ∂σyy ∂τzy ∂v ∂v ∂v ∂v ρgy + + + =ρ +u +v +w , ∂x ∂y ∂z ∂t ∂x ∂y ∂z ρgz +



∂τxz ∂τyz ∂σzz ∂w ∂w ∂w ∂w + + =ρ +u +v +w . ∂x ∂y ∂z ∂t ∂x ∂y ∂z

(1.6b)

(1.6c)

1.3.1 Constitutive relations for the equation of motion for Newtonian fluids The constitutive relations applicable to Newtonian fluids are [8]: σii = −P + 2μ

∂ui + l∇ · V , ∂xi



∂uj ∂ui + τij = τj i = μ ∂xj ∂xi

(1.7)

,

(1.8)

where i and j are the indices that represent the components of the Cartesian coordinates. For Newtonian fluids, the shear stresses are proposed to be directly proportional to the time rate of deformation of an element of a fluid with the viscosity coefficient m being the factor of proportionality [19]. From Stoke’s assumption, the coefficient of bulk viscosity, l, is given by [8,9,43] 2 l = − μ. 3

(1.9)

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Applications of Heat, Mass and Fluid Boundary Layers

Thus, in the rectangular coordinate system [19] and [8]:

∂v ∂u + , τxy = τyx = μ ∂x ∂y

∂w ∂v τyz = τzy = μ + , ∂y ∂z

∂u ∂w τzx = τxz = μ + , ∂z ∂x

(1.10a) (1.10b) (1.10c)

2 ∂u σxx = −p − μ∇ · V + 2μ , 3 ∂x

(1.11a)

2 ∂v σyy = −p − μ∇ · V + 2μ , 3 ∂y

(1.11b)

2 ∂w , σzz = −p − μ∇ · V + 2μ 3 ∂z

(1.11c)

where p is the local thermodynamic pressure. The thermodynamic pressure is linked to the temperature and density by the equation of state [6]. In most applications, p is the only variable of importance. For a fluid with constant density, the second term in the normal stress equations is always zero. Only in cases of very large gradients of velocity in the orientation of the stress, in the last term of the normal stress equations is p significantly different from σ . In the analysis of a normal shock wave, for instance, all three terms in the shear stress equations are significant [8,19]. In concise form, the Cartesian stress tensor elements can be written as

∂uj 2 ∂uk ∂ui σij = − p + μ + δij + μ . 3 ∂xk ∂xj ∂xi The Newtonian fluid stress tensor elements in cylindrical coordinates are presented below:

∂ur τrr = μ 2 + l∇ · V , (1.12) ∂r 

 1 ∂uθ ur + + l∇ · V , (1.13) τθθ = 2 r ∂θ r

∂uz + l∇ · V , (1.14) τzz = μ 2 ∂z   ∂  uθ  1 ∂ur + , (1.15) τrθ = τθr = μ r ∂r r r ∂θ   1 ∂uz ∂uθ + , (1.16) τθz = τzθ = μ r ∂θ ∂z

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11

 τzr = τrz = μ

 ∂ur ∂uz + . ∂z ∂r

(1.17)

1.3.2 Equations of motion for Newtonian fluids: Navier–Stokes equations Substituting the expressions for the stresses into the differential equations of motion (i.e., Eq. (1.4)) gives the Navier–Stokes equations presented below [8,30]: 

 

 Du ∂p ∂ ∂u 2 ∂ ∂u ∂v ρ = ρgx − + μ 2 − ∇ ·V + μ + Dt ∂x ∂x ∂x 3 ∂y ∂y ∂x 

 ∂ ∂w ∂u + μ + , ∂z ∂x ∂z 

 

 Dv ∂p ∂ ∂u ∂v ∂ ∂v 2 ρ = ρgy − + μ + + μ 2 − ∇ ·V Dt ∂y ∂x ∂y ∂x ∂y ∂y 3 

 ∂ ∂v ∂w + μ + , ∂z ∂z ∂y 

 

 Dw ∂p ∂ ∂ ∂w ∂u ∂v ∂w ρ = ρgz − + μ + + μ + Dt ∂z ∂x ∂x ∂z ∂y ∂z ∂y 

 ∂ ∂w 2 + μ 2 − ∇ ·V . ∂z ∂z 3 For incompressible flow having constant viscosity, the Navier–Stokes equations become [4,9,27,28]:

2

∂p ∂u ∂u ∂u ∂ u ∂ 2u ∂ 2u ∂u + + +u +v +w = ρgx − +μ , (1.18a) ρ ∂t ∂x ∂y ∂z ∂x ∂x 2 ∂y 2 ∂z2

∂v ∂v ∂v ∂v ρ +u +v +w ∂t ∂x ∂y ∂z ρ

2

∂p ∂ v ∂ 2v ∂ 2v + + = ρgy − +μ , (1.18b) ∂y ∂x 2 ∂y 2 ∂z2

∂w ∂w ∂w ∂w +u +v +w ∂t ∂x ∂y ∂z

= ρgz −

2

∂p ∂ w ∂ 2w ∂ 2w + + +μ . ∂x ∂x 2 ∂y 2 ∂z2 (1.18c)

The Navier–Stokes equations for constant viscosity and density are presented below in the cylindrical coordinate system [9,17,18]: (r component)   vθ2 ∂vr ∂vr vθ ∂vr ∂vr + vr + − + vz ρ ∂t ∂r r ∂θ r ∂z

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Applications of Heat, Mass and Fluid Boundary Layers

= ρgr −

  1 ∂ 2 vr ∂p 2 ∂vθ ∂ 1 ∂ ∂ 2 vr − +μ + , (1.19) [rvr ] + 2 ∂r ∂r r ∂r r ∂θ 2 r 2 ∂θ ∂z2

(θ component)

∂vθ ∂vθ vθ ∂vθ vr vθ ∂vθ + vr + + + vz ρ ∂t ∂r r ∂θ r ∂z

  1 ∂ 2 vθ 1 ∂p 2 ∂vr ∂ 1 ∂ ∂ 2 vθ + 2 +μ + , = ρgθ − [rvθ ] + 2 r ∂θ ∂r r ∂r r ∂θ 2 r ∂θ ∂z2 (1.20) (z component)

∂vz ∂vz vθ ∂vz ∂vz ρ + vr + + vz ∂t ∂r r ∂θ ∂z 

 ∂p ∂vz 1 ∂ 2 vz ∂ 2 vz 1 ∂ = ρgz − + +μ r + 2 . ∂z r ∂r ∂r r ∂θ 2 ∂z2

(1.21)

1.3.3 Constitutive equations for non-Newtonian fluids Non-Newtonian fluids like large molecular weight polymers which do not follow the constitutive equations for Newtonian fluids are common in the chemical and plastics industry. Substances such as paints, toothpastes, and lubricants show non-Newtonian behavior. Such fluids display effects like climbing of a rod which rotates in a stationary fluid container, turning into semisolid under the application of an electric or magnetic field and die swell when leaving a tube [24]. As [10] noted, with the continuous increase in the use of plastics in the modern society, being able to predict their behavior is of immense economic value in manufacturing processes. The manner in which non-Newtonian viscosity varies with shear rate is remarkable, and cannot be neglected in modeling stress which acts when molten polymers and many biomedical and industrial materials flow. In such flow systems, the viscosity changes in a very considerable manner. The stress–velocity relationship for Newtonian fluids is not valid for such systems, since the viscosity changes. Therefore, these systems need a different non-Newtonian constitutive equation [27]. Various theoretical models have been proposed within the past century; however, prediction of the flow behavior of non-Newtonian fluids is not adequate. In practice, the parameters of a particular constitutive model are obtained from conducting some simple experiments. More experiments are then used to validate the predictions of the given model on other flow geometries. In most cases, the predictions are correct for a handful of simple flows with character very similar to those from which the variables of the given constitutive model were obtained. More studies are still required about the behavior of non-Newtonian fluids [13]. Non-Newtonian fluids display certain characteristics like normal stress effects and shear thinning, so that one single constitutive relation is not adequate for the description of the various phenomena [39]. Morrison [27] noted that the study of non-

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13

Newtonian effects is a complete branch of knowledge on its own, and a considerable number of constitutive models have been proposed for non-Newtonian fluids. Morrison [27] discusses only two classes of non-Newtonian constitutive equations, namely, inelastic and viscoelastic. Most of the nomenclature employed in the field is based on simple experiments in which the substance is subjected to simple extension or in simple shear. Classifications on the basis of such experiments are presented in Table 1.2. As presented there, between elastic solids and Newtonian fluids, a succession of effects is shown having different forms which find application in drag reduction and drug delivery among other applications [10]. The constitutive equation for an incompressible and isotropic non-Newtonian fluid is given in Eq. (1.22). Fluids which obey this equation are called second-order Rivlin– Erickson fluids [42]: T = −pI + μ1 A1 + μ2 (A1 · A1 ) + μ3 A2 ,

(1.22)

where T is the complete stress tensor, An are Rivlin–Erickson tensors, μ1 is Newtonian viscosity, μ2 is cross-viscosity, μ3 is elastico-viscosity. For flows characterized by only stress and rate of deformation, the most widespread form of constitutive equation is given in Eq. (1.23). This relation describes Stokesian fluids, and in certain instances Reiner–Rivlin fluids [10]:   τij = −p + μ ∇ · V δij + μδij + μ dik dkj ,

(1.23)

where μ is additional viscosity coefficient. The constitutive equation for advanced Bingham fluids, or occasionally viscoplastic fluids, is [37]: τij = Tij

if Tij Tjk ≤ T2 , or

τij = Tij + (−p + ∇ · V ) δij + μdij

1 if Tmn Tmn > T2 . 2

For the latter equation, Tij = √

2T dij , 2Tmn Tmn

(1.24)

where Tij is yield stress tensor component, T is yield stress based on the von Mises yield criterion.

Table 1.2 Classification of non-Newtonian fluids [10]. Type of fluid Elastic solids

Classification Hookean

Stress/rate of deformation behavior Linear stress–strain relation

Examples Most solids below the yield stress

Plastic solids

Perfectly plastic Bingham plastic Visco-plastic

Ductile metals stressed above the yield point Iron oxide suspensions Drilling mud, nuclear fuel slurries, mayonnaise, toothpaste, blood

Yield dilatanta Visco-elastic

Strain continues with no additional stress Behaves Newtonian when threshold in exceeded Yield like the Bingham plastic, but the relation between stress and rate of deformation is not linear Dilatant when threshold shear is exceeded Exhibits both viscous and elastic effects

Shear thinning

Apparent viscosity reduces as shear rate increases

Dilatant or shear thickening

Apparent viscosity increases as shear rate increases

Some colloids, clay, milk, gelatin, blood, liquid cement, molten polystyrene, polyethylene oxide in water Concentrated solutions of sugar in water, suspensions of rice or cornstarch, solutions of certain surfactants

Rheopectic

Electrorheologic

Apparent viscosity increases the longer stress is applied Apparent viscosity decreases the longer stress is applied Becomes dilatant when an electric field is applied

Magnetorheologic

Becomes dilatant when an magnetic field is applied

Melted chocolate bars, single – or polycrystalline suspensions in insulating fluids Colloids with nanosize silica particles suspended in polyethylene glycol

Linear stress-rate of deformation relationship

Water, air

Power-law fluids

Time-dependent viscosity

Thixotropic Electromagnetic

Newtonian fluids

a Dilatant here refers to shear thickening as stress increases.

Egg white, polymer melts, and solutions

Some lubricants Nondrip paints, tomato ketchup

Physics of fluid motion

15

1.3.4 The law of conservation of energy (the first law of thermodynamics) From the first law of thermodynamics applied to the system, the sum of the work and heat supplied to the system will give rise to an increase in the energy of the system as presented mathematically by dEt = dQ + dW,

(1.25)

where Et is the total energy possessed by the system, Q is heat added to the system, and W is work done on the system. For a flowing fluid particle, the total energy is the sum of the internal, kinetic, and potential energy. Therefore, the energy per unit volume in this case is

1 Et = ρ e + V 2 − g · r , 2

(1.26)

where e is the internal energy per unit mass and r is the displacement of the particle. Writing the energy equation as a time rate of change, which follows the particle of a fluid, gives: DQ DW DEt = + , Dt Dt Dt

(1.27)



DEt DV De =ρ +V − g ·V . Dt Dt Dt

(1.28)

or

The heat and work transfer on a differential fluid element is shown in Fig. 1.5. For some materials, Fourier’s law is the constitutive relation for thermal energy diffusion within the molecules of the material [8,10]. Assuming that heat transfer to the element is given by Fourier’s law yields q = −k∇T .

(1.29)

Considering heat flow in the x-direction as shown in Fig. 1.5, the heat flow into the element is “qx dydz” and the heat flow out of the element is

∂qx dx dydz. qx + ∂x

16

Applications of Heat, Mass and Fluid Boundary Layers

Figure 1.5 Exchange of heat and work on the left- and right-hand sides of an element of fluid.

Similarly, by obtaining the heat flow in the y- and z-directions, respectively, the net heat transfer to the element is

∂qy ∂qx ∂qz − + + dxdydz. (1.30) ∂x ∂y ∂z If internal heat generation is neglected, dividing Eq. (1.30) by the element of volume dxdydz gives [43] DQ = − div q = + div (k∇T ) . Dt

(1.31)

From Fig. 1.5, the net rate of work done on the element of fluid per unit volume is [11] DW = − div w Dt   ∂  ∂  = uτxx + vτxy + wτxz + uτyx + vτyy + wτyz ∂x ∂y  ∂  + uτzx + vτzy + wτzz , ∂z or   DW = ∇ · V · τ ij . Dt

(1.32)

Decomposing the expression above yields     ∂ui . ∇ · V · τij = V · ∇ · τij + τij ∂xj

(1.33)

But from the momentum equation [8,43], ρ

DV = ρg + ∇ · τij , Dt

(1.34)

Physics of fluid motion

17

so that the first term on the right-hand side of Eq. (1.5) can be written as [43]

DV ρ V − g ·V . Dt DQ DW t Substituting the appropriate terms for DE Dt (Eq. (1.28)), Dt (Eq. (1.31)), and Dt (Eq. (1.33)) into Eq. (1.27) gives a form of the first law of thermodynamics commonly used for fluid motion [43]:

ρ

∂ui De . = div (k∇T ) + τij Dt ∂xj

(1.35)

A more well-known form of Eq. (1.35) can be obtained by noting that [8] τij

∂ui ∂ui = −p div V + τij , ∂xj ∂xj

(1.36)

where the elements of the viscous stress dyadic tensor for Newtonian fluids in the rectangular coordinate system are [8]

∂uj ∂ui  + (1.37) + δij div V , τij = μ ∂xj ∂xi from the continuity equation (Eq. (1.2)), the following can be obtained: D p Dp p Dρ =ρ − . p div V = − ρ Dt Dt ρ Dt Joining Eqs. (1.36) and (1.38) gives

D ∂ui Dp p Dp ρ = e+ = + div (k∇T ) + τij + div (k∇T ) + μΦ. Dt ρ Dt ∂xj Dt

(1.38)

(1.39)

The last term in Eq. (1.39) is the viscous dissipation term, and Φ is the dissipation function. For a Newtonian fluid, the dissipation function in the Cartesian coordinate system is [3,8,38])  2 

    ∂v ∂w 2 ∂w ∂v 2 ∂v ∂u 2 ∂u 2 + + + + + + Φ =2 ∂x ∂y ∂z ∂x ∂y ∂y ∂z  2  2 ∂u ∂w 2 ∂u ∂v ∂w + − . (1.40) + + + ∂z ∂x 3 ∂x ∂y ∂z The dissipation function in the cylindrical coordinate system is [3] 

   ∂vr 2 1 ∂vθ ∂vz 2 vr 2 ∂  vθ  1 ∂vr 2 Φ =2 + + + + + r ∂r r ∂θ r ∂z ∂r r r ∂θ

18

Applications of Heat, Mass and Fluid Boundary Layers



1 ∂vz ∂vθ + + r ∂θ ∂z

2



∂vr ∂vz + + ∂z ∂r

2

  2 1 ∂ (rvr ) 1 ∂vθ ∂vz 2 − . + + 3 r ∂r r ∂θ ∂z (1.41)

In boundary-layer flows, the enthalpy is, in general, more useful than the internal energy; and the term Dp Dt in Eq. (1.39) is usually negligible, while the term p div V in Eq. (1.36) cannot be neglected. Normally, the viscous dissipation term is neglected unless the system has large velocity gradients [3].

1.3.5 The second law of thermodynamics: entropy production The second law of thermodynamics is essentially [20] given by Stotal = Ssystem + Ssurrounding ≡ Sgen > 0.

(1.42)

The entropy, S, gives an indication of the extent of molecular material randomness, and for a process to take place or for a device to function, Sgen (entropy production) must be greater than zero. Heat transfer causes the change of entropy. The greater the value of Sgen , the lower the efficiency of the device, process, or system [20]. The change of entropy is given by [21,43] T ds = de + pdv = de −

p dρ. ρ2

(1.43)

For a particle of fluid, the rate of change of entropy is [21] T

DS De p Dρ = − 2 . Dt Dt ρ Dt

(1.44)

Putting the internal energy and continuity equations into Eq. (1.44) gives [21] ρ

DS ∅ ∅ 1 ∂qi ∂  qi  qi ∂T + =− + . =− − 2 Dt T ∂xi T ∂xi T T T ∂xi

(1.45)

By utilizing Fourier’s law of heat conduction, this equation becomes ρ



∂  qi  k ∂T 2 ∅ DS + . =− + 2 Dt ∂xi T ∂xi T T

(1.46)

The gain of entropy as a result of reversible heat transfer is the first term on the right-hand side of the equation. The second term represents the entropy production because of heat conduction while the last term represents the entropy production as a result of viscous heat generation. Since the second law of thermodynamics demands that the entropy production as a result of irreversible phenomena be positive, μ and k are greater than zero. For inviscid flow which does not conduct heat, entropy is conserved along the paths of the fluid particles [21].

Physics of fluid motion

1.4

19

Velocity slip and temperature jump

If the fluid that makes contact with a solid surface is a liquid, the molecules of the liquid are very closely packed and their mean free path is very small, so that the particles of a fluid which makes contact with the surface are basically in equilibrium with the surface. Thus, the fluid particles stick to the surface and get into thermal equilibrium with the surface. Both boundary conditions are referred to as no-slip and no-temperature jump conditions, respectively. For the case in which the fluid is a gas, and the molecules of the gas have large mean free path, the no-slip, no-temperature jump boundary conditions will not hold [36,43]. The slip velocity, uw , is uw ≈

3 μ τw , 2 ρa μ

(1.47)

where a is the speed of sound in the gas. Dividing by the free stream velocity, U , and rearranging gives uw 3 U 2τw = 0.75Ma Cf , ≈ U 4 a ρU 2

(1.48)

where Ma is Mach number of the free stream and Cf is the flow’s skin-friction coefficient. For turbulent flow, Cf is not greater than 0.005, and this value decreases as the Mach number increases, so that it may be concluded that for turbulent boundary layer uw is approximately equal to zero. Thus, the no-slip condition applies, and for laminar boundary layer, the skin friction coefficient is approximately [6] −1

Cf ≈ 0.6Rex 2 ,

(1.49)

where Rex is the local Reynolds number. Combining Eqs. (1.48) and (1.49) gives for laminar boundary layer [31] 0.4Ma uw . ≈ √ U Rex

(1.50)

Thus, for large Mach numbers and small Reynolds numbers, considerable slip can be obtained. As the flow moves further downstream over the solid boundary, the value of Rex increases, so that slip is no longer significant [2]. Similar to velocity slip, if the mean free path for the flow of a gas is large in comparison with the flow dimensions, the effect of temperature jump occurs. This effect occurs because some gas molecules on the solid surface do not come into thermal equilibrium with it [15,43]. The expression of the kinetic theory for the temperature jump, Tgas − Tw , is

2 2γ lk dT Tgas − Tw ≈ , (1.51) −1 α γ + 1 μcp dy w

20

Applications of Heat, Mass and Fluid Boundary Layers

where Tgas is the temperature of the gas, Tw is the temperature of the solid surface, l is the mean free path of gas molecules, γ is the specific heat ratio, and α is the thermal-accommodation coefficient, defined as follows [15]: α=

E i − Er , E i − Ew

(1.52)

where Ei is the energy of the molecules which strike the surface, Er is the energy of the molecules which were reflected from the surface, and Ew is the energy which the molecules would possess if the molecules attained the temperature of the surface and had the same amount of energy as the surface. The values of α must be obtained experimentally, and experimental results reveal that the value of α is approximately one [15,43]. Thus, taking the value of α as one and substituting appropriate values for l and dT dy (from Fourier’s law); applying Reynolds analogy and taking γ = 1.4 for air, gives the following relation for the temperature jump divided by the driving temperature difference controlling the wall heat transfer, Tr − Tw : Tgas − Tw = 0.87MaCf . T r − Tw

(1.53)

Therefore, it is seen that in turbulent flow, temperature jump is insignificant. In laminar flow, temperature jump is very small, except in regions that are very close to the leading edge of a flow which has a high Mach number. Usually, the no-slip and no-temperature jump boundary conditions are used in regular analysis of viscous flow of gases [25].

References [1] B. Alkahtani, M.S. Abel, E.H. Aly, Analysis of fluid motion and heat transport on magnetohydrodynamic boundary layer past a vertical power law stretching sheet with hydrodynamic and thermal slip effects, AIP Advances 5 (2015) 127228.1–127228.11. [2] F.G. Awad, S.M.S. Ahamed, P. Sibanda, M. Khumalo, The effect of thermophoresis on unsteady Oldroyd-b nanofluid flow over stretching surface, PLoS ONE 10 (8) (2015) 1–23, https://doi.org/10.1371/journal.pone.0135914. [3] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nd ed, Wiley, New York, 2002. [4] Y.A. Çengel, J.M. Cimbala, Fluid Mechanics: Fundamentals and Applications, McGrawHill, Boston, 2006.

Physics of fluid motion

21

[5] J.G.M. Eggels, F. Unger, M.H. Weiss, J. Westerweel, R.J. Adrian, R. Friedrich, F.T.M. Nieuwstadt, Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment, Journal of Fluid Mechanics 268 (1994) 175–209. [6] R.W. Fox, A.T. McDonald, Introduction to Fluid Mechanics, 5th ed, Wiley, New York, 2001. [7] A.Y. Gelfgat, Visualization of three-dimensional incompressible flows by quasi-twodimensional divergence-free projections in arbitrary flow regions, Theoretical and Computational Fluid Dynamics 30 (4) (2016) 339–348. [8] S.M. Ghiaasiaan, Convective Heat and Mass Transfer, Cambridge University Press, Cambridge, 2011. [9] L.A. Glasgow, Transport Phenomena: An Introduction to Advanced Topics, Wiley, New Jersey, 2010. [10] W.P. Graebel, Advanced Fluid Mechanics, Academic Press, Amsterdam, 2007. [11] V.G. Gupta, J. Ajay, A.K. Jha, Convective effects on MHD flow and heat transfer between vertical plates moving in opposite direction and partially filled with a porous medium, Journal of Applied Mathematics and Physics 4 (2016) 341–358. [12] N.G. Hadjiconstantinou, A.T. Patera, Heterogeneous atomistic-continuum representations for dense fluid systems, International Journal of Modern Physics 8 (1997) 967–976. [13] R. Haldenwang, R. Kotze, R. Chhabra, Determining the viscous behavior of nonNewtonian fluids in a flume using a laminar sheet flow model and ultrasonic velocity profiling (UVP) system, Journal of the Brazilian Society of Mechanical Sciences and Engineering 34 (3) (2012) 276–284. [14] S. Han, New conception in continuum theory of constitutive equation for anisotropic crystalline polymer liquids, Natural Science 2 (9) (2010) 948–958. [15] J.P. Holman, Heat Transfer, 10th ed, McGraw-Hill, Boston, 2010. [16] R.N. Ibragimov, Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications, Physics of Fluids 23 (2011) 123102.1–123102.8. [17] R.N. Ibragimov, D.E. Pelinovsky, Incompressible viscous fluid flows in a thin spherical shell, Journal of Mathematical Fluid Mechanics 11 (1) (2009) 60–90. [18] D.D. Joseph, Potential flow of viscous fluids: historical notes, International Journal of Multiphase Flow 32 (2006) 285–310. [19] W.M. Kays, M.E. Crawford, Convective Heat and Mass Transfer, 3rd ed, McGraw-Hill, New York, 1993. [20] C. Kleinstreuer, Modern Fluid Dynamics: Basic Theory and Selected Applications in Macro- and Micro-Fluidics, Springer, Dordrecht, 2010. [21] P.K. Kundu, I.M. Cohen, Fluid Mechanics, 2nd ed, Academic Press, San Diego, 2002. [22] G.L. Leal, Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes, Cambridge University Press, Cambridge, 2007. [23] G.W. Milton, J.R. Willis, On modifications of Newton’s second law and linear continuum elastodynamics, Proceedings of the Royal Society A (2007) 855–880. [24] M. Miyan, Different phases of Reynolds equation, International Journal of Applied Research 2 (1) (2016) 140–148. [25] B. Mohajer, V. Aliakbar, M. Shams, A. Moshfegh, Heat transfer analysis of a microspherical particle in the slip flow regime by considering variable properties, Heat Transfer Engineering 36 (6) (2015) 596–610. [26] Y. Morinishi, O.V. Vasilyev, T. Ogi, Fully conservative finite difference scheme in cylindrical coordinates for incompressible flow simulations, Journal of Computational Physics 197 (2) (2004) 686–710.

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[27] F.A. Morrison, An Introduction to Fluid Mechanics, Cambridge University Press, Cambridge, 2013. [28] B.R. Munson, et al., Fundamentals of Fluid Mechanics, 7th ed, Wiley, New Jersey, 2013. [29] Y. Nakayama, R.F. Boucher, Introduction to Fluid Mechanics, Butterworth–Heinemann, Oxford, 1999. [30] D.A. Nelson, G.B. Jacobs, D.A. Kopriva, Effect of boundary representation on viscous, separated flows in a discontinuous-Galerkin Navier–Stokes solver, Theoretical and Computational Fluid Dynamics 30 (4) (2016) 363–385. [31] A. Noor, R. Nazar, K. Jafar, I. Pop, Boundary-layer flow and heat transfer of nanofluids over a permeable moving surface in the presence of a coflowing fluid, Advances in Mechanical Engineering 6 (2014) 1–10. [32] S.T. O’Connell, P.A. Thompson, Molecular dynamics-continuum hybrid computations: a tool for studying complex fluid flows, Physical Review E 52 (1995) 5792–5795. [33] P.H. Oosthuizen, D. Naylor, An Introduction to Convective Heat Transfer Analysis, McGraw-Hill, New York, 1999. [34] T.J. Pedley, Lectures on plankton and turbulence: introduction to fluid dynamics, Scientia Marina 61 (1) (1997) 7–24. [35] P.J. Pritchard, Fox and McDonald’s Introduction to Fluid Mechanics, 8th ed, Wiley, New Jersey, 2011. [36] M.M. Rashidi, N.F. Mehr, Effects of velocity slip and temperature jump on the entropy generation in magnetohydrodynamic flow over a porous rotating disk, Journal of Mechanical Engineering 1 (3) (2012) 4–14. [37] P. Saramito, A new constitutive equation for elastoviscoplastic fluid flows, Journal of NonNewtonian Fluid Mechanics 145 (2007) 1–14. [38] H. Schlichting, Boundary-Layer Theory, 7th ed, McGraw-Hill, New York, 1979. [39] J.H. Spurk, N. Aksel, Fluid Mechanics, 2nd ed, Springer-Verlag, Berlin, 2008. [40] V.L. Streeter, Fluid Mechanics, 3rd ed, McGraw-Hill, New York, 1962. [41] R. Verzicco, P. Orlandi, A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates, Journal of Computational Physics 123 (1996) 402–414. [42] Z.U. Warsi, Fluid Dynamics: Theoretical and Computational Approaches, 3rd ed, CRC Press, Boca Raton, 2006. [43] F.M. White, Viscous Fluid Flow, 2nd ed, McGraw-Hill, New York, 1991. [44] P.K. Yeung, Lagrangian investigations of turbulence, Annual Review of Fluid Mechanics 34 (2002) 115–142.

Mechanisms of heat transfer and boundary layers

2

Sufianu Aliua , O.M. Amoob , Felix Ilesanmi Alaoc,d , S.O. Ajadie of Benin, Department of Mechanical Engineering, Benin City, Nigeria, b Department of Mechanical Engineering, University of Ibadan, Ibadan, Nigeria, c Department of Mathematical Sciences, University of Texas, Richardson, TX, United States, d Federal University of Technology, Akure, Nigeria, e Faculty of Science, Department of Mathematics, O.A.U, Ile Ife, Nigeria a University

2.1 Introduction Fluid mechanics is a discipline that employs the fundamental laws of mechanics and thermodynamics to characterize and understand the motion of fluids. Both mathematics and physics provide a sound basis to treat all that flows. In what follows, heat transfer and its governing equations, as well as boundary layer flows and applications are discussed.

2.2

Heat transfer

By a simple experiment, it is understood that something flows from hot objects to cold ones. It is imagined that all bodies contain an invisible fluid called caloric. The flow that occurs between hot and cold substance is called heat. Heat is energy in transit. Heat transfer may be defined as the microscopic transfer of random motions due to a temperature difference. Thermal energy is related to the temperature of matter. The temperature is the average microscopic kinetic energy of the perturbation (random component) motion of the molecules. For a given material and mass, the higher the temperature, the greater its thermal energy. This energy is transferred by the interaction of a system with its surrounding since it is a property of a substance. However, from a thermodynamic aspect, this deals with the end states of the processes and provides no valid information on the mechanisms causing the process. An example thereof is known as heat transfer. Therefore, heat transfer is the study of the exchange of thermal energy through a body or between bodies which occurs when there is a temperature difference. When two bodies are at different temperatures, thermal energy transfers from the one with a higher temperature to the one with a lower temperature. For example, heat constantly flows from the bloodstream to the surrounding air. The warmed air buoys off the body to warm the area being occupied, e.g., a room. When you leave the room, some small buoyancy-driven or convective motion of the air will continue because the walls can never be perfectly isothermal. Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00010-4 Copyright © 2020 Elsevier Ltd. All rights reserved.

24

2.3

Applications of Heat, Mass and Fluid Boundary Layers

Modes of heat transfer

The fundamental modes of heat transfer, which are conduction, convection, and radiation, are discussed in this section.

2.3.1 Conduction Conduction is the microscopic temperature gradient of the root-mean-square velocity of a substance. This substance may be solid, liquid, or gaseous. The conduction process takes place at the molecular level and involves the transfer of energy from the more energetic molecules to those with a lower energy level. This transfer of energy can be easily visualized within gases, where it is observed that the average kinetic energy of molecules in the higher-temperature regions is greater than that of those in the lower-temperature regions. The more energetic molecules, being in constant and random motion, periodically collide with molecules of a lower energy level and exchange energy and momentum. In this manner, there is a continuous transport of energy from the high-temperature regions to those of lower temperature. In liquids, the molecules are more closely spaced than in gases, but the molecular energy exchange process is qualitatively similar to that in gases. In solids that are nonconductors of electricity (dielectrics), heat is conducted by lattice waves caused by atomic motion. In solids that are good conductors of electricity, this lattice vibration mechanism is only a small contribution to the energy transfer process, the principal contribution being due to the motion of free electrons, which move in a similar way to molecules in a gas. At the macroscopic level, the heat flux (i.e., the heat transfer rate per unit area normal to the direction of heat flow) q  is proportional to the temperature gradient [1,2] q  = −k

dT dx

(2.1)

where the proportionality constant k is a transport property known as the thermal conductivity and is a characteristic of the material. The negative sign is a consequence of the fact that heat is transferred in the direction of decreasing temperature. Eq. (2.1) represents the one-dimensional form of Fourier’s law of heat conduction. Recognizing that the heat flux is a vector quantity, we can write a more general statement of Fourier’s law (i.e., the conduction rate equation) as q  = −k∇T

(2.2)

where ∇ is the three-dimensional del operator and T is the scalar temperature field from the conduction heat transfer equation above. It is seen that the heat flux vector q  actually represents a current of heat (thermal energy) that flows in the direction of the steepest temperature gradient.

Mechanisms of heat transfer and boundary layers

25

2.3.2 Convection Convection is another mode of heat transfer that relates to the transfer of heat from a bounding surface to a fluid in motion or the heat transfer across a flow plane within the interior of the flowing fluid. Convection heat transfer is the energy transfer occurring within a fluid due to the combined effects of conduction and bulk fluid motion. Convection can be classified in one of two ways, namely free convection and forced convection. However, in practice, these two modes are designed to occur together in what is referred to as mixed convection. We talk of forced convection when the flow is caused by external means. If the fluid motion is induced by a pump, a blower, a fan, or some similar device, the process is called forced convection. In contrast, for free or natural convection, the flow is induced by buoyancy forces, which are due to density differences caused by temperature variations in the fluid. An example is the free convection heat transfer that occurs from hot components on a vertical array of circuit boards in air. Air that makes contact with the components experiences an increase in temperature and hence a reduction in density. Since it is now lighter than the surrounding air, buoyancy forces induce a vertical motion for which an inflow of cooler ambient air replaces warm air ascending from the boards. While we have presumed pure forced convection and pure natural convection conditions, another corresponding condition known as mixed (combined) forced and natural convection may exist, as mentioned earlier. For example, if velocities associated with the flow are small and buoyancy forces are large, a secondary flow that is comparable to the imposed forced flow could be induced. In this case, the buoyancy-induced flow would be normal to the forced flow and could have a significant effect on convection heat transfer from the components. Mixed convection would result if a fan were used to force air upward between the circuit boards, thereby assisting the buoyancy flow, or downward, thereby opposing the buoyancy flow. Typically, the energy that is being transferred is the sensible or internal thermal energy of the fluid. However, for some convection processes, there is, also, latent heat exchange. This latent heat exchange is generally associated with a phase change between the liquid and vapor states of the fluid. Two special cases of interest in this regard are boiling and condensation. For example, convection heat transfer results from fluid motion induced by vapor bubbles generated at the bottom of a pan of boiling water or by the condensation of water vapor on the outer surface of a cold water pipe. In convective processes involving heat transfer from a boundary surface exposed to a relatively low-velocity fluid stream, it is convenient to introduce a heat transfer coefficient h defined below, which is known as Newton’s law of cooling [1,2] q  = h(Tw − Tf ).

(2.3)

Here Tw is the surface temperature and Tf is a characteristic fluid temperature. For surfaces in unbounded convection, such as plates, tubes, bodies of revolution, etc., immersed in a large body of fluid, it is customary to define h in Eq. (2.3) with Tf as the temperature of the fluid far away from the surface. For bounded convection, including such cases as fluids flowing in tubes or channels, across tubes in bundles, etc., Tf is usually taken as the enthalpy-mixed-mean temperature, customarily identified as Tin .

26

Applications of Heat, Mass and Fluid Boundary Layers

The heat transfer coefficient defined by Eq. (2.3) is sensitive to the physical properties of the fluid, and the fluid velocity. However, there are some special situations in which h can depend on the temperature difference, T = Tw − Tf . For example, if the surface is hot enough to boil a liquid surrounding it, h will typically vary as T 2 ; or in the case of natural convection, h varies as some weak power of T . 1 As q  = Aq , the thermal resistance in convection heat transfer is given by Rth = hA , which is the resistance at a surface-to-fluid interface. At the wall, the fluid velocity is zero, and the heat transfer takes place by conduction. Therefore, we may apply Fourier’s law to the fluid at y = 0, as q  = −k

∂T |y=0 , ∂y

(2.4)

where k is the thermal conductivity of the fluid. By combining this equation with Newton’s law of cooling, we then obtain   ∂T  k ∂y |y=0 q =− , (2.5) h= Tw − Tf T w − Tf so that we need to find the temperature gradient at the wall to evaluate the heat transfer coefficient. For convective processes involving high-velocity gas flows (high subsonic or supersonic flows), a more meaningful and useful definition of the heat transfer coefficient is given by q  = h(Tw − Taw ).

(2.6)

Here, Taw , commonly called the adiabatic wall temperature or the recovery temperature, is the equilibrium temperature the surface would attain in the absence of any heat transfer to or from the surface and in the absence of radiation exchange between the surroundings and the surface. In general, the adiabatic wall temperature is dependent on the fluid properties and the properties of the bounding wall. Generally, the adiabatic wall temperature is reported in terms of a dimensionless recovery factor r defined as Taw = Tf + r

V2 . 2cp

(2.7)

The value of r for gases normally lies between 0.8 and 1.0. It can be seen that for low-velocity flows, the recovery temperature is equal to the free-stream temperature, Tf .

2.3.3 Radiation Thermal radiation is the energy emitted by matter that is at nonzero temperature. Regardless of the form of matter, the emission may be attributed to changes in the

Mechanisms of heat transfer and boundary layers

27

electron configurations of the constituent atoms or molecules. The energy of the radiation field is transported by electromagnetic waves (or photons). While the transfer of energy by conduction or convection requires the presence of a material medium, radiation does not. Radiation transfer occurs most efficiently in a vacuum. Radiative heat transfer does not require a medium to pass through; thus, it is the only form of heat transfer present in a vacuum. It uses electromagnetic radiation (photons), which travels at the speed of light and is emitted by any matter with a temperature above 0 degrees Kelvin (−273°C). Radiative heat transfer occurs when the emitted radiation strikes another body and is absorbed. We all experience radiative heat transfer every day; solar radiation, absorbed by our skin, is why we feel warmer in the sun than in the shade. The electromagnetic spectrum classifies radiation according to wavelengths of the radiation. Main types of radiation are (from short to long wavelengths): gamma rays, X rays, ultraviolet (UV), visible light, infrared (IR), microwaves, and radio waves. Radiation with shorter wavelengths is more energetic and contains more heat. X-rays, having wavelengths ∼ 10−9 m, are very energetic and can be harmful to humans, while visible light with wavelengths ∼ 10−7 m contains less energy and therefore has little effect on life. A second characteristic which will become important later is that radiation with longer wavelengths generally can penetrate through thicker solids. Visible light, as we all know, is blocked by a wall. However, radio waves, having wavelengths on the order of meters, can readily pass through concrete walls. The amount of radiation emitted by an object is given by qemitted = εσ.AT 4 ,

(2.8)

where A is the surface area, T is the temperature of the body, s is a constant called Stefan–Boltzmann constant, equal to 5.67 × 10−8 W/m2 K4 , and ε is a material property called emissivity. The emissivity has a value between 0 and 1 and is a measure of how efficiently a surface emits radiation. It is the ratio of the radiation emitted by a surface and the radiation emitted by a perfect emitter at the same temperature.

2.3.4 Thermodynamics of heat transfer Thermodynamics is an ancient subject that is concerned with the macroscopic effects of systems that are in or near equilibrium, such as their energy and entropy. Heat transfer is the relaxation process that tends to do away with temperature gradients (where ∇T → 0 is in an isolated system), but systems are often kept out of equilibrium by imposed boundary conditions. Heat transfer tends to change the local state according to the energy balance, which for a closed system is defined as Q ≡ E − W → Q = E|V ,non−dis = H |p,non−dis ,

(2.9)

i.e., heat Q (i.e., the flow of thermal energy from the surroundings into the system, driven by thermal nonequilibrium, not related to work or the flow of matter) equals the increase in stored energy E minus the flow of work W . For nondissipative systems (i.e., without mechanical or electrical dissipation), heat equals the internal energy

28

Applications of Heat, Mass and Fluid Boundary Layers

change if the process is at constant volume, or the enthalpy change if the process is at constant volume, both cases converging for a perfect substance model to Q = mc T . In thermodynamics, often one refers to heat in an isothermal process, but this is a formal limit for small gradients and large periods. Thus, in heat transfer, the interest is not on heat flow Q, but on heat flow-rate •

dQ dt

Q=

(2.10)

that should be referred to as heat rate because the “flow” characteristic is inherent to the concept of heat, contrary, for instance, to the concept of mass, to which two possible “speeds” can be ascribed: mass rate of change and mass flow rate. Heat rate, thence, is the energy flow rate at constant volume, or enthalpy flow rate at constant pressure. We define heat flux as the heat flow rate expressed as •

dT dQ = mc ≡ KA T , dt dt

Q=

(2.11)

where K is the global heat transfer coefficient (associated to a bounding area A and the average temperature jump T between the system and the surroundings), defined by (2.11); the inverse of K is named the global heat resistance coefficient, M≡

1 . K

(2.12)

In most heat-transfer problems, it is undesirable to ascribe a single average temperature to the system, and thus a local formulation must be used, defining the heat flow-rate density (or heat flux) as •

•

q=

dQ . dA

(2.13)

2.3.5 Heat transfer and entropy production The statistical mechanical interpretation of the second law of thermodynamics indicates that, for a system that is not in statistical equilibrium, its evolution towards equilibrium leads not to a decrease in entropy, but instead to an increase. In the universe, entropy increase is as a result of a process defined as the sum of the entropy changes of all elements of a system that are involved in the process. Therefore, the •

rate of entropy production of the universe denoted as SU n is as a result of the heat transfer process through a wall given below as [6] •

•

•

•

SU n = Sres1 + Swall + Sres2 .

(2.14)

Mechanisms of heat transfer and boundary layers

29

•

Since Swall = 0, Swall must be constant. The dots in Eq. (2.14) mean derivatives. Further, since the reservoir temperature is constant, we have •

Sres =

Q , Tres

(2.15)

where Qres1 is negative and has equal size as Qres2 , therefore, we have

• 1 1 SU n = |Qres1 | − . T2 T1

(2.16)

•

Hence the term in parentheses is a positive term, and so SU n > 0.

2.4

The boundary layer equations

When the viscous terms in the Navier–Stokes (NS) equation are neglected, the Euler equation is obtained. Analytical solutions of the Euler equation applied to practical flow problems fail to account for shear forces on solid walls and aerodynamic drag on bodies immersed in fluids. Prandtl in 1904 showed that viscous flows could be analyzed by dividing the flow about a solid surface into two distinct regions a very thin layer of flow near a solid wall (boundary layer) where viscous forces and rotationality cannot be neglected and an outer region where friction may be neglected. Viscosity exerts vital significance on subsonic fluid flows. Viscous effects produce a significant correction of the flow’s details. Prandtl’s boundary layer approximation, which in addition to being able to specify the no-slip condition at solid walls can explain the existence of crakes, predicts flow separation, shear force, and heat transfer. The boundary layer exerts a controlling influence on the bulk flow (where viscosity is negligible), where, for example, it can prompt the initiation of turbulence in the bulk flow. The boundary layer concept led to the introduction of certain approximations to the Navier–Stokes equation, which resulted in the boundary layer equations. Boundary layers are the characteristic properties of flows of high Reynolds (Re) numbers. This Re is sufficiently high as not to be turbulent whereby vorticity is confined within the boundary layer wall [4].

2.4.1 Dimensional analysis and similarity Dimensional analysis, similitude, and theory of models are useful theoretical constructs that set the stage for the planning of experimentation or small-scale modeling, and data consolidation and reduction. They are topics at the foundation of fluid mechanics. The technique allows us to present data in dimensionless form, which is very efficient, reducing the number of experiments or simulations required to validate or examine the relationships between variables and scale results, reduce a large number of parameters in a problem to a small number of dimensionless groups and also

30

Applications of Heat, Mass and Fluid Boundary Layers

provide theoretical results. Remarkable insights into the dominant phenomenon are achieved through dimensional analysis and reduction of equations into dimensionless numbers or groups. In thermal-fluid-mass problems, for example, dimensionless numbers emerge, some of which are common (e.g., Reynolds number) and convenient in reducing the number of independent and dependent parameters. The dimensionless numbers are made possible through the physical insight of the problem, normalization of the governing equations, and associated boundary conditions. The dimensionless numbers provide a form of generality that is independent of the methods used, that is theoretical, numerical, or experimental. The importance of similarity transformation in boundary layer theory has long been appreciated. Dimensional analysis and similarity transformations are par for the course in boundary layer analysis. Dimensional analysis, used for scaling, is a technique for reducing a complex physical system of equations to a much simpler one. Its applications go beyond fluid mechanics, and applicable to most fields. Similarity is essentially a coordinate transformation and serves as a quantitative research technique. It is a key area of fluid mechanics that allows a researcher to generalize experimentally determined flow results. Similarity can be of different type: geometric, kinematic, dynamic/thermodynamic, or thermal [8]. Taken together, dimensional analysis and similarity are a unified area albeit with different modifications of the same procedure of investigation, and whereby notable differences are due to the different magnitudes of preliminary information available about the process or problem being investigated. It aids in developing fundamental ideas and mathematical techniques of the method of generalized analysis and for the ultimate purpose of studying actual problems. Similarly transformed solutions thus describe similar phenomena. Similarity also leads to the establishing of dimensionless expressions. The success of dimensionless numbers in fluids is enabled by the generation of physically interpretable parameters that dominate the physics of the flow. Further computations of these are then enabled, and physical intuition is gained. The formal theoretical principle employed in dimensional analysis is the Buckingham’s π theorem, which is the cornerstone of dimensional analysis. Similarity solutions to partial differential equations, in general, form the basis for the majority of useful laminar boundary layer solutions. The most widespread use of similarity analyses in present-day literature occurs in the fields of fluid flow and heat transfer. Similarity methods to boundary layer flow problems is an intriguing area of applied mathematics. Two different flows are considered to be similar or exhibit similarity if their nondimensional numbers are the same. Finding useful scaling parameters to transform differential equations and boundary conditions is something of an art. The scale represents an estimate of its maximum order of magnitude that identifies the various unknowns in a problem. Scaling leads to nondimensionalization whereby the independent and dependent variables are chosen so that they are of order unity [7]. When a dynamically determined length scale is smaller than the system dimensions, then the similarity method is applicable. Stated yet another way, similarity transformations can be dynamic, geometric, or even thermodynamic. The Blasius-flat plate and the Hiemenz-stagnation point boundary layer problems, for example, are particularly of a class of similar solutions for flows without a characteristic length. Perhaps the

Mechanisms of heat transfer and boundary layers

31

most robust technique available for transforming boundary layer partial differential equations is the Lie group of transformations. Similarity is an indispensable tool to the analysis of fluid mechanical phenomena in general, and especially boundary layer processes. In summary, the techniques for nondimensionalizing are well known and understood and need not be dealt with here any further. However, for more in-depth information, the interested reader may refer to [12,13].

2.4.2 The velocity boundary layer equations The boundary layer is a two-dimensional phenomenon. As such, consideration is given to steady two-dimensional laminar flow in the xy-plane Cartesian coordinates. The following definitions are introduced: x is the distance in the stream direction measured along the wall from the point where the boundary layer starts; y is the distance perpendicular to the boundary layer measured from the wall; u is the fluid velocity component in the x-direction; v is the fluid velocity component in the y-direction. The general equations of motion for an incompressible viscous fluid with uniform transport properties are: (The continuity equation) ∂ (ρu) ∂ (ρv) + = 0; ∂x ∂y in particular, for uniform density, ρ we obtain ∂u ∂v + = 0. ∂x ∂y

(2.17)

(The momentum equation) ρu

∂u ∂u dp ∂ 2u + ρυ =− +μ 2. ∂x ∂y dx ∂y

(2.18)

As we approach the outer edge, the velocity component u gradually becomes the mainstream velocity U which varies only in the x-direction so the terms which include υ, ∂u ∂2u ∂y , and ∂y 2 become very small compared with terms which depend on x. Eq. (2.18) becomes ρU

du dP =− . dx dx

(2.19)

Upon inserting (2.19) into (2.18), we obtain u

∂u du ∂ 2u ∂u +v =U +υ 2, ∂x ∂y dx ∂y

where ν =

μ ρ

is the fluid kinematic viscosity.

(2.20)

32

Applications of Heat, Mass and Fluid Boundary Layers

The boundary conditions to be satisfied include: i. No slip at the boundary or u (x, y = 0)=0; ii. No mass flux through the wall or υ (x, y = 0) = 0; iii. Fluid velocity becomes that of the inviscid flow in the mainstream or u (x, y → ∞) = U . These equations and boundary conditions further assume Re ≡ Uυx  1. Eqs. (2.17) and (2.20) can be reduced to simpler forms that will apply in the boundary layer. The reduction is achieved by introducing dimensionless variables obtained by considering the fact that for large Reynolds numbers, distances and velocity components in the direction perpendicular to the boundary layer are smaller than corresponding quantities parallel to the boundary layer. We choose L and V as the reference length and velocity, respectively; L and V are constant and non-zero. We choose ρv 2 as our reference pressure. The dimensionless variables are defined as: x , L P P∗ = 2, ρv u ∗ u = , L y ∗ = (y/L) Ren , v ∗ = (v/V ) Rem , x∗ =

(2.21)

where the Reynolds number Re is ρ VμL and the exponents n and m are arbitrary positive numbers. Eq. (2.17), when written in terms of these variables, becomes ∗ ∂u∗ n−m ∂v + Re = 0. ∂x ∗ ∂y ∗

(2.22)

To obtain conclusions acceptable for flow in a boundary layer, exponents n and m are chosen to be equal. The continuity equation therefore becomes ∂u∗ ∂v ∗ + = 0. ∂x ∗ ∂y ∗

(2.23)

The asterisks imply that Eq. (2.23) is in dimensionless form. Expressed in terms of our dimensionless variables, it may be rewritten as u∗

∗ 2 ∗ ∂u∗ ∂P ∗ 1 ∂ 2 u∗ ∗ ∂u 2n−1 ∂ u + v = − + + Re . ∂x ∗ ∂y ∗ ∂x ∗ Re ∂x (∗)2 ∂y (∗)2

(2.24)

As Re becomes very large, the second term on the right vanishes. Taking m = n = 1/2 is realistic for a boundary layer. In terms of these variables, the momentum

Mechanisms of heat transfer and boundary layers

33

conservation equation in the y-direction may be arranged as u∗

∗ ∂v ∗ ∂P ∗ 1 ∂ 2v∗ ∂ 2v∗ ∗ ∂v + v = −Re + + . ∂x ∗ ∂y ∗ ∂y ∗ Re ∂x (∗)2 ∂y (∗)2

(2.25)

After dividing by Re, we find that for large Re this equation reduces to ∂P ∗ → ∞. ∂y ∗

(2.26)

The pressure P varies only in the stream direction, which is in the x-direction. We, therefore, write the equation for momentum balance in the boundary layer as u∗

∂u∗ ∂u∗ dP ∗ ∂ 2 u∗ + v ∗ ∗ = − ∗ + (∗)2 . ∗ ∂x ∂y dx ∂y

(2.27)

Eq. (2.27) is a dimensionless form of Eq. (2.20). Consideration is not given to the momentum in the perpendicular direction to the boundary layer as the terms of that equation are of a lower magnitude compared to the terms of Eq. (2.27). Also, the pressure does not vary in the transverse direction, which explains the use of a total derivative in Eq. (2.27).

2.4.3 The thermal boundary layer equation The fluid thermal properties are assumed to be uniform, and the flow is steady. The rate at which the thermal energy of volume elements changes per unit volume equals the rate at which thermal energy is conducted into the volume element. This yields ∂T ∂T ∂ 2T (2.28) +v =K 2 , ∂x ∂y ∂y   K is the fluid thermal diffusivity. where K = ρc Eq. (2.28) is the thermal energy equation for boundary layer flow. If the temperature of the wall is Tω and that of the fluid in the mainstream is T∞ then the boundary conditions satisfied by the temperature T are u

y = 0, y → ∞,

T = Tω , T → T∞ .

(2.29)

2.4.4 The concentration boundary layer equation Convective mass transfer involves the material transport between a solid boundary and a moving fluid or between two immiscible moving fluids that are separated by a mobile interface. Mass transfer, as it is called, is in many respects similar to heat transfer. It is

34

Applications of Heat, Mass and Fluid Boundary Layers

an important physical process in many engineering systems that involve the molecular and convective mixing of fluids. The convective mass transfer rate equation is NA = Kc CA ,

(2.30)

where the mass flux NA is the molar-mass flux of species A, CA is the concentration difference between the boundary surface concentration and the average concentration of the diffusing species in the moving fluid stream, and Kc is the convective mass transfer coefficient. Eq. (2.30) is analogous to the Newton´s law of cooling. An analogous differential equation to that of momentum and thermal boundary layer equations applies to mass transfer in a concentration boundary layer when no production of the diffusing components occurs and when the second derivative of CA with respect to x, ∂ 2 CA /∂x 2 , is much smaller in magnitude than the second derivative of CA with respect to y. The equation for steady, incompressible, constant diffusivity two-dimensional flow for the transfer in a single phase where the mass exchange takes place between a boundary surface and a moving fluid is given by ∂ 2 CA u∂CA ν∂CA , + = DAB ∂x ∂y ∂y 2

(2.31)

where DAB is the mass diffusivity. The corresponding boundary conditions are similar to those expressed in Eq. (2.29). Furthermore, in mass transfer, the dimensionless Sherwood number (Sh) replaces the Nusselt number (N u) in heat transfer, and the Schmidt number (Sc) replaces the Prandtl number (P r) in heat transfer as well.

2.4.5 More on convection boundary layer flows The conservation of mass and energy equations, Eqs. (2.17) and (2.28), for forced convection are also applicable to natural convection. The momentum equation requires some modifications to accommodate the effect of buoyancy. Let’s discuss natural convection flows briefly. Natural convection heat transfer describes a situation where the bulk movement of the fluid is driven by a temperature-related buoyancy effect rather than by an external forcing mechanism such as a pump, a fan, or a larger fluid movement, such as wind, which is independent of the heat transfer effect. In many situations, the forced convection effect so completely overwhelms any fluid movement from buoyancy that the natural convection effects can reasonably be ignored for analysis purposes. However, there are many other situations where the natural convection effects have some significant modifying influence on the forced flow (often termed “mixed convection”) and many situations where the natural convection effects are completely responsible for any fluid movement (pure natural convection). Natural convection can be broadly divided into “free” and “bounded” categories. Free natural convection refers to jets and plumes where there is bulk fluid movement driven by a buoyancy effect that is unconfined by any solid surface. Heat transfer occurs through conduction and mixing with the surrounding fluid. Examples of free

Mechanisms of heat transfer and boundary layers

35

Figure 2.1 Hydrodynamic and temperature profile for a natural convection flow on a vertical heated plate.

natural convection include the movement of air above a forest fire or candle flame, the thermal plume of a power plant discharge into a cooler lake, or the sinking plume of cold water descending below a melting iceberg. Bounded natural convection has some solid surface—usually the source or sink for heat transfer with the fluid—which confines or directs the flow in some way. Examples of bounded natural convection include air rising along a sunlight heated wall, air descending along the inside of a cold window pane, or fluid rising or descending around a horizontal heated or cooled pipe immersed in the fluid. The remainder of this document will focus on bounded natural convection. Figs. 2.1 and 2.2 illustrate the hydrodynamic and thermal boundary layers for natural and forced convection for a flow on a vertical heated plate. Focusing on the hydrodynamic boundary layers, both cases have zero velocity at the surface of the plate (no-slip condition). However, in the case of forced convection, the velocity increases out to the edge of the boundary layer where it matches the imposed free-stream flow velocity (V∞ ). For the case of natural convection, the fluid outside of the boundary layer is not moving, so the velocity profile within the boundary layer reaches a peak and then decreases to match the fluid velocity outside the boundary layer, which is zero. The thermal boundary layers appear qualitatively similar between the two cases because they are both constrained to match the free-stream temperature at the edge of the boundary layer and to match a plate surface temperature which is related to the convective heat transfer coefficient in each case. Consider, for example, a vertical heated plate immersed in a quiescent fluid with the assumptions that the natural convection flow is steady, laminar, two-dimensional, and the fluid is Newtonian with constant properties. The only exception, in this case, is the consideration of density difference ρ − ρ∞ which is what gives rise to the buoyancy force that causes the flow. This occurrence gives rise to what is called the Boussinesq approximation. Succinctly, the Boussinesq approximation describes the effects of buoyancy as simply as possible by accounting for flows where fractional density changes are small, and velocity changes are significant, albeit constrained by the assumption of incompressibility. The upward direction along the plate is regarded as x while the normal direction to the surface is regarded as y. Gravity acts in the x direction; x and y components of the velocity within the boundary layer are u = u (x, y) and v = v (x, y), respectively. The momentum conservation equation in the x direction can be obtained since pressure

36

Applications of Heat, Mass and Fluid Boundary Layers

Figure 2.2 Hydrodynamic and temperature profile for a forced convection flow.

forces, stresses, and gravity forces are involved as ρ

u∂u v∂u + ∂x ∂y

=μ

∂ 2 u ∂ρ − − ρg. ∂y 2 ∂x

(2.32)

The x momentum equation in the fluid only can be obtained by setting u = 0. This gives ∂ρ∞ = −ρ∞ g. ∂x

(2.33)

∂v Note that ν  u in the boundary layer. We take ∂x ≈ ∂v ∂y ≈ 0 assuming that there are no body forces, including gravity in the y direction. The force balance in the y direction gives ∂P ∂y = 0, which implies that the pressure in the boundary layer and the pressure in the quiescent fluid are equal. Hence,

∂ρ∞ ∂P = = −ρ∞ g. ∂x ∂x Substituting into (2.32) yields

∂u ∂u +v ρ u ∂x ∂y

=μ

∂ 2u + (ρ∞ − ρ) g. ∂y 2

(2.34)

We introduce the property of volume expansion coefficient β, which enables us to relate the variation of density with the temperature at constant pressure. The volume expansion coefficient β is defined as a measure of the change of a substance with the temperature at constant pressure: β=

1 υ

β∼ =−



∂v ∂T

=− p

1 ρ



∂ρ ∂T

1 p 1 P∞ − P =− ρ T ρ T∞ − T

P∞ − P = ρβ (T − T∞ ) ,

,

(2.35)

at constant P ,

(2.36)

p

(2.37)

Mechanisms of heat transfer and boundary layers

37

where P∞ and T∞ are the pressure and temperature of the quiescent fluid, respectively. Substituting Eq. (2.37) into (2.34) and dividing through by ρ gives the momentum equation for natural convection, namely u∂u v∂u ∂ 2u + = υ 2 + gβ (T − T∞ ) . ∂x ∂y ∂y

(2.38)

The buoyancy term, gβ(T − T∞ ), drives the fluid movement and also makes a solution of the natural convection problem significantly more involved than a corresponding forced convection problem. In two-dimensional forced convection, the buoyancy term is absent from the momentum equation, and conservation of mass and momentum may be used to solve for the velocity field, u and v. Then, these can be used as input to the energy equation to solve for the temperature field, T . In contrast, in natural convection the momentum equation explicitly includes temperature through the buoyancy term, so the equations are coupled and must be solved simultaneously. To obtain nondimensionalized equations, the independent and dependent variables of the governing equations for natural convection with the boundary condition are divided by suitable constant quantities. We choose a characteristic length, L, a reference velocity, V , and temperature difference Tw − T∞ to divide the variables as follows: x∗ =

x , L

y∗ =

y , L

u∗ =

u , V

v∗ =

v , V

and T ∗ =

T − T∞ . T w − T∞

Substituting them into the momentum equation and simplifying gives ∂u∗ u∗ ∗ ∂x

∂u∗ + v∗ ∗ ∂y



 gβ (Tw − T∞ ) L3 T ∗ 1 ∂ 2 u∗ = + . υ2 Rel2 ReL ∂y ∗2

(2.39)

We can then define the dimensionless Grashof number GrL which represents the convection effects as GrL =

gβ (Tw − T∞ ) L3 , υ2

(2.40)

where g is the acceleration due to gravity in m/s2 , β is the volume expansion coefficient measured in K1 , Tw is the wall temperature of plate, T∞ the fluid temperature at some point sufficiently far away from the surface, L the characteristic length of the geometry in m, and υ is the fluid kinematic viscosity in m2 /s. The Prandtl number, P r, is Pr ≡

v ; α

(2.41)

and the Nusselt number, N u, is N uL ≡

hL ; k

(2.42)

38

Applications of Heat, Mass and Fluid Boundary Layers

where h is the convective heat transfer coefficient, and k is the thermal conductivity of the fluid. Most pure natural convection correlations can be expressed in the form of N uL = f (GrL , P r).

(2.43)

Recalling the definition of Reynolds number, Re, which is a dimensionless number from fluid mechanics and forced convection given by ReL ≡

V∞ L , v

(2.44)

2  allows general discrimination between forced and natural convection. If GrL /ReL 1, then free convection effects can be ignored, and the problem can be treated as pure 2  1, then forced convection effects are negligible and forced convection. If GrL /ReL 2 ≈ 1, then mixed the problem can be treated as pure natural convection. If GrL /ReL convection conditions would prevail and neither forced nor natural convection effects may be neglected. For the case of an isothermal vertical plate of length L immersed in an infinite quiescent fluid with the laminar flow, there is an exact similarity solution to the mass, momentum, and energy equations. The result of that analysis is

hL 4 GrL 1/4 = N uL = f (P r) , (2.45) k 3 4

where h is the average convective heat transfer coefficient over the length of the plate, and 0.75P r 0.5 f (P r) =  0.25 . 0.609 + 1.221P r 0.5 + 12.38P r

(2.46)

This relationship applies for both hot and cold plates relative to the surrounding fluid temperature. For the case of a plate hotter than the fluid, the boundary layer flow will be upward, and for the case of a cooler plate, the flow will be downward. Just as in forced convection, natural convection flows can transition to turbulence. The parameter commonly used for evaluation for the transition to turbulence in natural convection is the Rayleigh number, sometimes termed the Grashof–Prandtl product: Rax ≡

gβ (Ts − T∞ ) x 3 . να

(2.47)

Turbulence is normally assumed for flow past a vertical plate for Rax > 109 and laminar flow exists for Rax < 109 . For a turbulent flow, empirical correlations are most often used to predict the heat transfer. An empirical correlation for a vertical isothermal plate in a quiescent fluid that is applicable over the range of both laminar and turbulent flows is  2 1/6 0.387RaL hL = 0.825 +  N uL = (2.48) 8/27 . k 1 + (0.492/P r)9/16

Mechanisms of heat transfer and boundary layers

39

Empirical correlations are available in heat transfer handbooks for natural convection heat transfer in many different physical and geometric configurations. As another example, an empirical correlation has been developed for natural convection from a long horizontal isothermal cylinder, using the cylinder diameter, D, as the characteristic length:  2 1/6 0.387RaD hD N uD = = 0.60 +  8/27 , k 1 + (0.559/P r)9/16

(2.49)

which is applicable for RaD < 1012 . Natural convection heat transfer contributes, or is the dominant mode, in many important cooling and heating applications. Empirical correlations and analytical solutions using the Nusselt, Grashof, Rayleigh, and Prandtl numbers have been developed to estimate heat transfer behavior for a wide variety of physical boundary conditions and geometric configurations. Last but not least, we discuss the case of mixed convection. The conservation of mass and energy equations, Eqs. (2.17) and (2.28), for forced convection are also applicable for mixed convection. The boundary layer equations for the mixed forced and free convection laminar flow are [5] ∂u ∂v + = 0, ∂x ∂y νd 2 u u∂u v∂u U du , + = + gγβ (T − T∞ ) + (Momentum) ∂x ∂y dx dy 2 u∂T v∂T κ∂ 2 T (Energy) , + = ∂x ∂x ∂y 2

(Continuity)

(2.50) (2.51) (2.52)

while the associated boundary conditions are u(x, 0) = v(x, 0) = 0; T (x, 0) = Tw (constant), u(x, ∞) = U (x); T (x, ∞) = T∞ (constant),

(2.53)

where U is the external velocity along the outer edge of the boundary layer, which is assumed to be known from potential flow theory or experiment. The above equations incorporate all the usual assumptions for an incompressible laminar boundary layer and the additional assumptions of small temperature differences and small pressure increases throughout the boundary layer. Variations in density are only allowable for producing buoyancy forces. In Eq. (2.51), γ = + cos α where, for any location on the surface of a body, α is the angle between the tangent to the surface at that point and the gravitational direction. Eqs. (2.17), (2.51), and (2.28) neglect the effect of the component of the buoyancy force normal to the surface of the body.

40

Applications of Heat, Mass and Fluid Boundary Layers

2.5

Internal boundary layer flows

The walls of the boundary constrain internal flows. The viscous boundary layers grow downstream as the upstream flow enters the tube at the entrance. The axial flow u(r, x) at the wall is retarded while the flow in the center of the tube is accelerated to fulfill the continuity requirements. At a small distance from the entrance, the boundary layers merge, and the flow in the tube becomes fully viscous. A little further at x = Le , the axial velocity becomes constant with x as the flow turns fully developed, u ≈ u (r) only. The thermal boundary layers for flow through a tube grow downstream within the region of flow regarded as the thermal entrance region. The flow in this region is known as the thermally developing flow. The region beyond the thermal entrance region where the dimensionless temperature profile expressed as (Ts − T )/(Ts − Tm ) remains constant is known as thermally fully developed region. The flow whose velocity and dimensionless temperature profiles remain unchanged are known as fully developed.

2.6

External boundary layer flows

The laminar flow past a flat plate is an important case of an external flow. The boundary layer of Eq. (2.20) can be solved exactly for u and v with a constant free stream velocity U . Blasius showed that the dimensionless velocity profile u/U is a function only of the single composite dimensionless variable η: η=y

U νx

1/2 (2.54)

,

thus u = f  (η) . U

(2.55)

He then introduced a stream function ψ(x, y) as u=

∂ψ ∂y

and

v=−

∂ψ . ∂x

(2.56)

The continuity equation is identically satisfied. The function f (η) is defined as f (η)=

ψ . U ( νx U )1/2

(2.57)

The velocity components become u=

 νx 1/2 df ∂ψ ∂ψ ∂η = =U ∂y ∂η ∂y U dη



U νx

1/2 =U

df , dη

(2.58)

Mechanisms of heat transfer and boundary layers

v=−

v=

1 2

41

 νx 1/2 df ∂η ∂ψ U  ν 1/2 f, = −U −− ∂x U dη ∂x 2 Ux

Uν x

1/2 η

df −f . dη

(2.59)

By differentiating u and the relations in the momentum equation, we obtain ∂u U d 2f =− η 2, ∂x 2x dη ∂u =U ∂y



U νx

1/2

d 2f , dη2

(2.60)

∂ 2u U 2 d 3f = . νx dη3 ∂y 2 Substituting these relations into the momentum equation and simplifying yields 2

d 2f d 3f + f 2 = 0. 3 dη dη

(2.61)

Eq. (2.61) is a third-order nonlinear differential equation. The system of partial differential equations is thus converted to an ordinary differential equation. The boundary conditions are expressed as f =0

at η = 0,

f  = 0 at η = 0,

and f  = 1 as η → ∞.

(2.62)

Numerical solution of Eq. (2.61) shows that at Uu = 0.99, η = 4.91. Substituting η = 4.91 and δ = y into Eq. (2.54) gives the value of the velocity boundary layer as 4.91 4.91 δ =  1/2 = √ . U Rex

(2.63)

νx

The shear stress on the wall is determined from (2.60) as  1/2 2  d f ∂u  U =μU . τw = μ   ∂y y=0 νx dη2 

(2.64)

η=0

It is similarly found from the numerical solutions of Eq. (2.61) that η = 0. Eq. (2.63) can, therefore, be expressed as τw = 0.332U

ρμU x

1/2 =

0.332ρU 2 . √ Rex

d2f dη2

= 0.332 at

(2.65)

42

Applications of Heat, Mass and Fluid Boundary Layers

The coefficient of average local skin friction is given by Cf,x =

τw ρU 2 2

0.664 =√ . Rex

(2.66)

The energy boundary layer equation can also be solved exactly for laminar flow past a flat plate. A dimensionless temperature θ θ (x, y) =

T (x, y) − Ts , T ∞ − Ts

(2.67)

where Ts and T ∞ are constant. The temperature profile for flow over an isothermal flat plate is similar just as for  1/2 the velocity profiles. The thermal boundary layer thickness is proportional to νx . U Using the chain rule and substituting the expressions for u and v into the energy equation gives df dθ ∂η 1 + U dη dη ∂x 2



Uy x

1/2

df η dη

d 2θ dθ ∂η =α 2 dη ∂x dη



∂η ∂y

2 .

(2.68)

Simplifying leads to 2

dθ d 2θ + Pr f = 0, dη dη2

(2.69)

where Pr = ν/α, and the boundary conditions are: θ = 0 at η = 0;

2.7

θ =1

as η → ∞.

(2.70)

Wake and jet boundary layers

The boundary layer equations are not only applicable to regions near a solid wall. They can also be applied to flows where two layers of fluid with different velocities meet. This process occurs in the wake behind a body or when fluid discharges through an orifice. In the wakes of flat plates, thin boundary layers form above and below a flat plate placed in a sufficiently high Reynolds number flow. The two velocity profiles around the plate merge to form a thin wake behind its trailing edge. The thickness of the wake increases with distance while the mean velocity decreases with distance. The governing equations for the Blasius boundary layer apply to wakes but with the boundary condition given as [3] u → U∞

as y → ±∞.

(2.71)

On the other hand, in jets, the emergence of a jet from an orifice is another example of a boundary layer flow in the absence of a solid wall. The jet mixes with surrounding

Mechanisms of heat transfer and boundary layers

43

Figure 2.3 Two-dimensional laminar and turbulent jet [26].

fluid as it emerges from the narrow slit. The boundary layer thickness of jets increases with distance while the mean velocity decreases with distance. The governing equations for the Blasius boundary layer apply to jets as well. The boundary condition is u→0

as y → ±∞.

(2.72)

Two-dimensional laminar and turbulent jets are shown in Fig. 2.3. Notably, boundary layer equations may be extended to other types of flows such as free shear flows, converging and diverging channel flows, some of which are seldom treated according to present day boundary layer literature. For further reading on these boundary layer flow types, the reader is referred to [3].

2.8

Hydrodynamic boundary layer stability

We now turn our attention to another important aspect of boundary layer research; boundary layer stability [18,19]. The most important feature of flow stability analysis is in the value of the growth increment of the unstable disturbances. A brief discussion of some aspects is discussed. It is well established both experimentally and theoretically that a laminar boundary layer may become turbulent at high Reynolds number because the steady flow is dynamically unstable to small disturbances above a certain critical Reynolds number. Accidental disturbances which appear in the flow (Tollmien–Schlichting waves) are self-excited, and these initially infinitesimal disturbances amplify to the point where the steady laminar motion is destroyed. A typical theoretical curve of neutral stability to infinitesimal disturbances is shown in Fig. 2.4. The illustration in Fig. 2.4 has been fully verified experimentally for a low-speed flow over an insulated flat plate, and we now query as to the effects of compressibility and heat transfer on this type of instability.

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 2.4 Neutral stability curve.

The effects of Mach number. Theoretical calculations indicate that the laminar boundary layer becomes less stable as the free stream Mach number is increased. In this respect, it is important to note that the minimal critical Reynolds number merely marks the onset of possible amplification of infinitesimal disturbances in the boundary layer and does not directly indicate that Reynolds number of transition at which the disturbance has multiplied nonlinearly to the point where the flow is turbulent. The effects of heat transfer. Theoretically, cooling of the stream leads to a substantial increase in the stability of the boundary layer; while heating the stream has the opposite effect. If the wall is maintained below a certain temperature, the boundary layer is stable at any Reynolds number, and the transition will presumably never occur if large amplitude disturbances were initially absent. Some experimental results on laminar boundary layer stability. Here we give a graphical plot of some experimental result on laminar boundary layer stability based on the work of Lee as quoted in [14]. See Figs. 2.5 and 2.6 for 1. Effect of Mach number (M∞ ), 2. Effect of heat transfer, 3. Region of stability at any Reynolds number. Remarks. The destabilizing effects of a hot surface and the stabilizing effects of a cold surface have been verified qualitatively by experiment at supersonic speeds with cones. These experiments showed that transition to turbulence occurred earlier as the cone temperature was raised and occurred later as the cone temperature was reduced. For a supersonic flow on a flat plate, the measured destabilizing effect of surface heating is shown in Fig. 2.7. This illustrates the powerful influence of this factor. The processes from the initial stability of the laminar boundary layer to the realization of a turbulent boundary layer are qualitatively vague and quantitatively nebulous.

Mechanisms of heat transfer and boundary layers

45

Figure 2.5 Relationship of Mach number, heat transfer, and Reynolds number [20,24,25].

Figure 2.6 Region of stability relative to Mach number, heat transfer, and Reynolds number [20,24,25].

This is regardless of the speed or even compressibility. It’s, however, expected that certain properties which are discussed below are of importance. Randomness. The initial disturbance spectrum is generally expected to be evenly random with the absence of any discrete peaks in both frequency and orientation. Nonlinearity. The processes leading to transition are fundamentally nonlinear. After initial stability, some of the important features of the nonlinear growth are the effect of the frequency and disturbance spectra through the distortion of the mean flow, generation of harmonics and heat or resonance phenomena. Another possible feature of the nonlinear processes is the attainment of a metastable equilibrium state at a finite amplitude, as suggested by [15]. This question has been examined in details by Stuart [16,17] for an incompressible plane Poiseuille flow. Because he could not fully evalu-

46

Applications of Heat, Mass and Fluid Boundary Layers

Figure 2.7 Behaviors noted after growth of infinitesimal disturbances [20,24,25].

ate the relevant terms in his equation for equilibrium amplitude, the author could not come to a definite conclusion regarding the existence of a finite amplitude equilibrium state. Three-dimensionality. The initial disturbance spectrum is very likely threedimensional in orientation. Even though not all orientations are amplified at once, a band of them becomes unstable within the region of an infinitesimal disturbance at a certain point.

2.8.1 Prediction of transition A significant attempt in accomplishing this was done by Smith and Gamberoni [18] who, for low speed flows, tried to correlate transition Reynolds number over plates, wings, and bodies with amplitude ratio of the most unstable frequency from its neutral point to the transition point. Many more authors worked in this line, and despite the apparent success of this procedure, it is defective both in principle and perhaps also in practice. The authors also acknowledged that the boundary layer is agitated by disturbances impressed upon it by external disturbances, surface roughness, noise, and vibration. The true flow is observed to be similar to forced vibration. As such, the process of transition from laminar to turbulent boundary layer is almost as baffling as the turbulence in the flow that follows it.

2.8.2 Boundary layer separation Prandtl’s work on boundary layers not only established the boundary layer concept, it also shed light on the mechanics of separation, and techniques to delay flow separation

Mechanisms of heat transfer and boundary layers

47

Figure 2.8 Effects of favorable and adverse pressure gradients on the fluid flow [20,24,25].

Figure 2.9 Flow patterns near the point of separation [20,24,25].

such as suction. When the pressure gradient along a body surface is negative (favorable pressure gradient), the pressure acts locally to accelerate the flow along the body and overcome the viscous force that acts to decelerate it. However, when the pressure gradient is positive (adverse pressure gradient), both pressure and viscous stress act to decelerate the flow, and the boundary layer thickness rapidly increases. The flow soon reverses such that ∂u 1.

3.2

Fundamental assumption

To carry out a compressible flow analysis, some necessary assumptions must be established, and these are: (a) The gas is continuous, i.e., the motion of individual molecules does not have to be considered, the gas being treated as a continuous medium. This assumption applies to flows in which the mean free path of the gas molecules is very small compared with all the important dimensions of the solid body through or over which the gas is flowing. This assumption will, of course, become invalid if the gas pressure, and hence, density, becomes very low as is the case with spacecraft operating at very high altitudes and in low-density flows that can occur in high-vacuum systems. (b) No chemical changes occur in the flow field. Chemical changes influence the flow because they result in a change in composition with resultant energy changes. This is also referred to as a pure substance. One common chemical change results from combustion in the flow field. Other chemical changes that can occur when there are large pressure and temperature changes in the flow field are dissociation and ionization of the gas molecules. These can occur, for example, in the flow near a space vehicle during reentry to the Earth’s atmosphere. The gas is perfect. This implies that, the gas obeys the perfect gas law, i.e., p R = RT = T . ρ m This is often regarded as the thermal equation of state and expressed as p = ρRT . (c) Gravitational effects on the flow field are negligible. This assumption is almost always quite justified for gas flows. (d) Magnetic and electrical effects are negligible. These effects would normally only be important if the gas was electrically conducting, which is usually only true if the gas or a seeding material in the gas is ionized. In the so-called magnetohydrodynamic (MHD) generator, a hot gas that has been seeded with a substance

58

Applications of Heat, Mass and Fluid Boundary Layers

Figure 3.2 Control volume used in derivation of continuity equation.

that easily ionizes and which therefore makes the gas a conductor is expanded through a nozzle to a high velocity. The high-speed gas stream is then passed through a magnetic field that generates an electromotive force. This induces an electrical current flow in a conductor connected across the gas flow normal to the magnetic field. (e) The effects of viscosity are negligible.

3.3

Basic equations of compressible fluid flow

The basic equations with applied principles of compressible fluid flow are as follows: 1. 2. 3. 4. 5. 6.

Conservation of mass (continuity equation) Conservation of momentum or momentum equation (Newton’s law) Conservation of energy or energy equation (first law of thermodynamics) Equation of state (ideal gas law) The second law of thermodynamics and Other various concepts taken from thermodynamics.

Together with these basics, knowledge of applied mathematics is also an essential ingredient for a compressible flow analysis. More of these basics are discussed in what follows.

The conservation of mass or continuity equation Rate of increase of mass of fluid in control volume = Rate of mass entering control volume − Rate of mass leaving control volume. Therefore, we define a control volume as a volume in space (geometric entity, independent of mass) through which fluid may flow. (See Fig. 3.2.) The continuity equation is obtained by applying the principle of conservation of mass to a flow through a control volume. There is no mass transfer across the control volume. The only mass transfer occurs through the ends of the control volume. In the case of a one-dimensional flow, mass per second = ρAV , and, since mass per second

On some basics of compressible fluid flows

59

Figure 3.3 Control volume for derivation of momentum equation & pressure force on a curved control volume surface.

is constant according to conservation of mass, ρAV = constant.

(3.1)

Differentiating (3.1), we obtain d(ρAV ) = 0, i.e., ρd(AV ) + AV dρ = 0.

(3.2)

Equivalently, we can rewrite (3.2) as ρ(AdV + V dA) + AV dρ = 0 or ρAdV + ρV dA + AV dρ = 0.

(3.3)

Dividing Eq. (3.3) by ρAV , we have dV dA dρ + + = 0. V A ρ

(3.4)

Eq. (3.4) relates the fractional changes in density, velocity, and area over a short length of the control volume, therefore Eq. (3.4) is known as the continuity equation in differential form.

The conservation of momentum or momentum equation (Euler equation) In Fig. 3.3, the flow is steady and the gravitational forces are being neglected. The only forces acting on the control volume are the pressure forces and the frictional force exerted on the surface of the control volume. Since momentum is a vector quantity, conservation of momentum must apply in any chosen directions. Therefore, it is defined as Net force on gas in control volume in direction considered = Rate of increase of momentum in the direction considered for fluid in control volume + Rate of momentum entering control volume in the direction considered − Rate of momentum leaving control volume in the direction considered.

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Applications of Heat, Mass and Fluid Boundary Layers

Therefore, the momentum equation for compressible fluids is similar to that for incompressible fluids. This is because the momentum equation is the change in momentum flux equated to the force required to cause this change: Momentum flux = mass flux × velocity = ρAV × V , but mass flux ρAV = constant (this is the continuity equation). Thus, momentum equation is completely independent of the compressibility effect and, for compressible fluids, the momentum equation in the x-direction is expressed as  (3.5) Fx = (ρAV Vx )2 − (ρAV Vx )1 Using the above diagram, we equivalently derive the momentum equation as follows: The net force on the control volume in the x-direction is 1 P A − (P + dP )(A + dA) + [P + (P + dP )][(A + dA) − A] − dFμ 2

(3.6)

where dx is too small, while dP and dA have been neglected. The mean pressure on the curved surface can be taken as the average of the pressure acting on the two end surfaces. Also dFμ is the frictional force. Properly arranging Eq. (3.6), we have the net force on the control volume in the x-direction as −AdP − dFμ .

(3.7) .

Since the rate at which momentum crosses any section of the duct is equal to m V , we get ρV A[(V + dV ) − V ] = ρV AdV .

(3.8)

We rewrite the above equation as −AdP − dFμ = ρV AdV .

(3.9)

Frictional force is assumed to be negligible. The Euler’s equation for steady flow through a duct becomes −

dP = V dV . ρ

Integrating (3.10), i.e., Euler’s equation, we have  V2 dp + = c ≡ Bernoulli’s equation. 2 ρ

(3.10)

(3.11)

Assuming that the density is constant, we have V2 P + = c. 2 ρ

(3.12)

On some basics of compressible fluid flows

61

Figure 3.4 Control volume for deriving the energy equation.

Eq. (3.12) is Bernoulli’s equation for an incompressible fluid.

The conservation of energy or energy equation (first law of thermodynamics) The energy equation is also referred to as Bernoulli’s equation. (See Fig. 3.4.) As the compressible fluid flow is steady, the Euler equation is given as dP + V dV + gdz = 0. ρ

(3.13)

Integrating both sides of Eq. (3.13), we have    dP + V dV + gdz = constant, ρ

(3.14)

or 

dP V2 + + gz = constant. ρ 2

(3.15)

In a compressible flow, since ρ is not constant, it cannot be taken outside the integration sign. In compressible fluids the pressure changes with density, depending on the type of process.

3.3.1 Energy equation for adiabatic and isothermal processes For an isothermal process, the energy equation is ρv = constant or

P = c1 , ρ

where c1 is a constant and v is specific volume defined as

(3.16) 1 ρ

so

P ρ=     c dP c1 dP P dP dP = = c1 = c1 loge p = loge p, p = ρ P P ρ c 1

(3.17)

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Applications of Heat, Mass and Fluid Boundary Layers

substituting c1 =

P ρ

.

Now substituting



dP P

into Eq. (3.17), we have

P V2 loge P + + gz = constant, ρ 2

(3.18)

which further leads to V2 P loge P + + z = constant. ρg 2g

(3.19)

Eq. (3.19) is called the energy equation for a compressible flow undergoing an isothermal process.

3.3.2 Energy equation for an adiabatic process By the principle of conservation of energy, the flows through the control volume are considered as in Fig. 3.3 above. Fluid enters at section 3.1 with velocity V1 , has enthalpy h1 per unit mass, must leave through section 3.1 with velocity V2 and enthalpy h2 , so we have h2 + V22 = h1 +

V12 + q − w. 2

(3.20)

The assumptions for the above Eq. (3.20) are: • Work done is zero; • Perfect gases are considered, i.e., h = cp T . Therefore, we have the steady flow energy equation as cp T +

V22 V2 = cp T1 + 1 + q. 2 2

(3.21)

Applying Eq. (3.21) to the flow through the control volume shown in Fig. 3.4 yields cp T +

V2 (V + dV )2 + dq = cp (T + dT ) + . 2 2

(3.22)

Expanding Eq. (3.22) and neglecting the higher-order terms, we have cp dT + V dV = dq.

(3.23)

Eq. (3.23) indicates that for a compressible flow changes in velocity will induce temperature changes, also additional heat causes a change in velocity as well as temperature changes. Thus Eq. (3.20) gives c p T2 +

V22 V2 = c p T1 + 1 , 2 2

(3.24)

On some basics of compressible fluid flows

63

while Eq. (3.28) gives the adiabatic flow as cp dT + V dV = 0.

(3.25)

Eq. (3.25) indicates that in adiabatic flow, an increase in velocity causes a decrease in temperature.

3.3.3 Equation of state The two equations of state are: the ideal gas equation of state and the isentropic flow equation of state. The first describes gases at low pressure (relative to the gas critical pressure) and high temperature (relative to gas critical temperature). The second applies to the ideal gases experiencing isentropic, i.e., adiabatic and frictionless flow. The ideal gas equation of state is given as ρ=

p . RT

(3.26)

In Eq. (3.26), air is the most commonly incurred compressible flow gas and its gas constant is Rair = 1716 ft2 /s2 The two additional useful ideal gas properties are the constant volume and pressure specific heats which are defined as Cv =

du dT

and Cp =

dh . dT

The additional specific heat relationship is R = C p − Cv ,

therefore the specific heat ratio K for air is 1.4.

An isentropic process, i.e., constant entropy process, and ideal gas obey the isentropic process equation of state p = constant. ρk A combination of this equation of state with the ideal gas equation of state, applying the result to different locations in a compressible flow field, gives p2 = p1



T2 T1

k (k−1)

=

ρ2 ρ1

k .

(3.27)

Eq. (3.27) may be applied to any ideal gas as it undergoes an isentropic process.

3.4 Entropy factors In a compressible fluid flow, a variable s, known as entropy, has to be introduced. Entropy puts a limitation on which flow processes are physically possible. The change

64

Applications of Heat, Mass and Fluid Boundary Layers

in entropy between any two points is given below as     T2 p2 s2 − s1 = cp ln − R ln . T1 p1

(3.28)

Recalling that R = cp − cv , this equation is written as s2 − s1 = ln cp



T2 T1



p2 p1

k−1  k

.

(3.29)

If the flow is isentropic, i.e., no change in entropy, Eq. (3.29) requires T2 = T1



p2 p1

k−1 k

.

(3.30)

Hence, by perfect gas law, we have T2 p 2 ρ 1 = . T 1 p1 ρ 2

(3.31)

This implies that for an isentropic flow we obtain p2 = p1



ρ2 ρ1

k .

(3.32)

In an isentropic flow, pρ k2 is constant. Suppose that Eq. (3.28) is applied between the inlet and the exit of a differential control volume, then it gives     T + dT p + dp − R ln . (3.33) (s + ds) − s = cp ln T p Neglecting minimal value, Eq. (3.33) gives ds = cp

dT dp −R . T p

Equivalently, we have (3.34) as

ds dT k − 1 dp = − . cp T k p

(3.34)

(3.35)

It is noted that in isentropic flow equation (3.34) gives cp dT =

RT dp. p

(3.36)

On some basics of compressible fluid flows

65

Applying perfect gas law, we have cp dT =

dp . ρ

(3.37)

The energy equation for isentropic flow gives cp dT + V dV = 0.

(3.38)

Applying Eq. (3.37), we have dp + V dV = 0. ρ

(3.39)

Comparing Eq. (3.39) with Eq. (3.10) shows that this result was obtained using similar conservation of momentum considerations. For more penetrating aspects and understanding on compressible fluid flow the reader may refer to refs. [1–8].

3.5 A note on applications of compressible fluid flow As stated earlier, the applications of compressible fluid flows are vast and significant. In both gas and steam turbines, the flow is treated as compressible especially in effectively characterizing their effects on blading, nozzles, and in cooling. Also, in reciprocating engines, the flow of gases through the valves and in the intake and exhaust systems must be treated as compressible. Combustion systems alike often warrant treating the flow field as compressible. Flow regulators where an obstruction of the flow in a duct controls the pressure drop is another case of compressible flow application. The operation of various compressors is yet another example. Hypersonic and supersonic flight of aircraft and missiles are classic cases of the consideration of compressibility effects.

References [1] P.H. Oosthuizen, W.E. Carscallen, Introduction to Compressible Fluid Flow, 2nd edition, CRC Press (Taylor & Francis Group), New York, 2013. [2] F.I. Alao, S.B. Folarin, Adomian approximation approach to thermal radiation with heat transfer effect on compressible boundary layer flow on a wedge, Ghana Mining Journal (2013) 70–77. [3] F.I. Alao, S.B. Folarin, Similarity solution of the influence of the thermal radiation and heat transfer on steady compressible boundary layer flow, Open Journal of Fluid Dynamics (2013) 82–85. [4] T.L. Bergman, A.S. Larine, F.P. Incropera, D.P. Dewitt, Introduction to Heat Transfer, 6th edition, John Wiley & Sons, USA, 2011.

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Applications of Heat, Mass and Fluid Boundary Layers

[5] D.F. Young, B.R. Munson, T.H. Okiishi, W.W. Huebsch, A Brief Introduction to Fluid Mechanics, 5th edition, John Wiley & Sons, INC, USA, 2010. [6] J.H. Lienhard IV, J.H. Lienhard V, A Heat Transfer Textbook, 3rd edition, Phlogiston Press, Cambridge, Massachusetts, USA, 2008. [7] W.M. Rohsenow, J.P. Hartnett, Y.I. Cho (Eds.), Handbook of Heat Transfer, 3rd edition, McGraw-Hill, USA, 1998. [8] L. Rosenhead (Ed.), Laminar Boundary Layers, Dover Publication, Inc., USA, 1963.

Boundary layer equations in fluid dynamics

4

Hafeez Y. Hafeez, Chifu E. Ndikilar Federal University Dutse, Dutse, Jigawa State, Nigeria

4.1 The continuity equation Let us now apply the conservation of mass principle to the control volume approach. Consider a volume V bounded by a fixed surface S in a flow portion of S which may coincide with the fixed impermeable boundaries but other portions of S will not. For an incompressible flow, it must have equal volumetric flow rate entering and leaving V through S, so that it is mathematically expressed by the flow equation  (4.1) V · en ds = 0, S

where en is outward normal to S and V = (u, v, w) is the velocity components. This equation states that the net volumetric outflow through S is zero with outflow taken as positive and inflow taken as negative. This represent the control volume form of the continuity equation. If we take V to be a very small volume in a flow, then by applying the second form of the divergence theorem, we get   · V dv = 0. ∇ (4.2) V

A conservation of mass statement for the same control is that the net mass flow rate out through S must be balanced by the rate of mass decrease within volume, i.e.,  dm ρ V · en ds = − , dt S 

ρ V · en ds = − S

d dt

 ρ dv.

(4.3)

V

Applying the divergence theorem to Eq. (4.3), we get    · (ρ V ) dv = − d ρ dv. ∇ dt V V Rearranging Eq. (4.4), we have

  · (ρ V ) + ∂ρ dv = 0, ∇ ∂t V Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00012-8 Copyright © 2020 Elsevier Ltd. All rights reserved.

(4.4)

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Applications of Heat, Mass and Fluid Boundary Layers

which will hold for an arbitrary choice of V , and so  · (ρ V ) + ∂ρ = 0. ∇ ∂t

(4.5)

Eq. (4.5) is now called the continuity equation. If the flow is incompressible, then the density is constant and Eq. (4.5) reduces to  · V = 0. ∇ For a steady two-dimensional incompressible flow, we have ∂u ∂v + = 0. ∂x ∂y

(4.6)

4.2 The momentum equations Let us now apply Newton’s second law to an infinitesimal element. Substantial derivative in 3D Cartesian coordinates is given by D ∂ ∂ ∂ ∂ = +u +v +w . Dt ∂t ∂x ∂y ∂z

(4.7)

 as Eq. (4.7) can be written in terms of the operator ∇ ∂ D  = + (V · ∇), Dt ∂t

(4.8)

 is called the convective derivative. where ∂t∂ is called the local derivative while V · ∇ By definition, acceleration is a =

D V ∂ V  = + V (V · ∇). Dt ∂t

(4.9)

If we include the pressure force, gravity, and viscous force in the derivatives, Newton’s second law gives 



−



pen ds + S

fρ dv =

ρ g dv + V

V

 V

D V ρ dv, Dt

(4.10)

where the pressure force p ds acts in the negative en direction for p > 0, g is directed towards the center of the Earth, and f is the viscous force per unit mass. By applying the divergence theorem, we get  − V

 dv + ∇p





fρ dv =

ρ g dv + V

V

 V

D V ρ dv. Dt

(4.11)

Boundary layer equations in fluid dynamics

69

If dv is chosen to be very small, then Eq. (4.11) reduces to   + ρ g + fρ = D V ρ. −∇p Dt

(4.12)

For an incompressible Newtonian fluid, f is defined as follows:

2 ∂ 2 V ∂ 2 V ∂ V + + f = ν∇ 2 V = ν , ∂x 2 ∂y 2 ∂z2

(4.13)

where ν is kinematic viscosity, and then Eq. (4.12) becomes 1 D V − ∇p + g + ν∇ 2 V = . ρ Dt

(4.14)

Eq. (4.14) is called Navier–Stokes equation for homogeneous and heterogeneous incompressible flows. Substituting Eq. (4.9) into Eq. (4.14), we get ∂ V 1  + g + ν∇ 2 V = + V (V · ∇), − ∇p ρ ∂t  = ( ∂ , ∂ , ∂ ). The components are as where g = (0, 0, −g), V = (u, v, w), and ∇ ∂x ∂y ∂z follows: (x-component) 2

∂u ∂u ∂u ∂ u ∂ 2u ∂ 2u ∂u 1 ∂p + 2+ 2 = +ν +u +v +w , − 2 ρ ∂x ∂t ∂x ∂y ∂z ∂x ∂y ∂z

(4.15)

(y-component) −

2

1 ∂p ∂v ∂v ∂v ∂ v ∂ 2v ∂ 2v ∂v + + +ν +u +v +w , = ρ ∂x ∂t ∂x ∂y ∂z ∂x 2 ∂y 2 ∂z2

(4.16)

(z-component) 2

1 ∂p ∂ w ∂ 2w ∂ 2w ∂w ∂w ∂w ∂w + 2 + 2 = − +ν +u +v +w . 2 ρ ∂x ∂t ∂x ∂y ∂z ∂x ∂y ∂z

(4.17)

For a steady two-dimensional incompressible flow, the momentum equations become u

2

∂u 1 ∂p ∂ u ∂ 2u ∂u + +v =− +ν , ∂x ∂y ρ ∂x ∂x 2 ∂y 2

(4.18)

u

2

∂v ∂v 1 ∂p ∂ v ∂ 2v + +v =− +ν . ∂x ∂y ρ ∂y ∂x 2 ∂y 2

(4.19)

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Applications of Heat, Mass and Fluid Boundary Layers

4.3 Coutte flow Coutte flow is described as the flow between two infinite plates that are a distance h apart. The upper plate moves at a speed U while the lower plate is stationary. If the flow is assumed to be parallel, the only nonzero component of the velocity u is in the direction of the velocity of the upper plate. The continuity equation requires that u = u(y) only. In the absence of any pressure gradient, the flow is known as a simple Coutte flow. Here, a general Coutte flow obtained by imposing a constant pressure gradient on the flow will be given in detail. Assuming steady two-dimensional incompressible flows with partial fully-developed flow, Eq. (4.18) reduces to 0=−

μ

1 ∂p ∂ 2u +ν 2, ρ ∂x ∂y

d 2 u dp = = constant, dx dy 2

(4.20)

where μ = ρν and 

1 dp d 2u dy = 2 μ ∂x dy

 dy,

du 1 dp = y + A, dy μ dx 

du 1 dp dy = dy μ dx

u(y) =



 y dy + A

dy,

1 dp y 2 + Ay + B, μ dx 2

(4.21)

with A and B being constants. The no-slip condition requires that u(0) = 0 and u(h) = U . (See Fig. 4.1.) And then u(0) = 0 =⇒ B = 0, u(h) =

U 1 dp h 1 dp h2 + Ah = U =⇒ A = − = . μ dx 2 h μ dx 2

Figure 4.1 Coutte flow.

Boundary layer equations in fluid dynamics

Thus the final solution for the velocity profile is

h2 dp y y y 1− . u(y) = U − h 2μ dx h h

71

(4.22)

If the pressure gradient is set to zero, the solution for a simple Coutte flow is obtained as y (4.23) u(y) = U , h where the velocity profile is a straight line. For the general Coutte flow, the shape of the velocity profile depends on the dimensionless pressure gradient

dp h2 − . (4.24) P= 2μU dx For P > 0, i.e., for a negative pressure gradient, the velocity is positive everywhere across the channel. For P < 0, the velocity over a part of the cross-section may become negative. In this case, the sign of the volume flow rate depends on the magnitude of the dimensionless pressure gradient.

4.4 Plane Poiseuille flow This is the case of a fully developed incompressible flow between two infinitely large parallel plates that are a distance h apart. Both upper and lower plates are stationary. As before, if the flow is assumed parallel in the x-direction, u = 0 and v = w = 0. Therefore, the continuity equation requires that u = u(y), which satisfies the fully developed condition and thus pressure is a function of x only. For a steady two-dimensional incompressible flow Eq. (4.18) reduces to (see Fig. 4.2) 0=−

∂ 2u dp +μ 2 , dx ∂ y

with the final solution being u(y) =

1 dp y 2 + Ay + B. μ dx 2

(4.25)

Since both plates are stationary, applying the no-slip condition u(0) = 0, u(h) = 0 yields u(0) = 0 =⇒ B = 0, u(h) = 0 =

1 dp 1 dp 2 h + Ah =⇒ A = − h. 2μ dx 2μ dx

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 4.2 Plane Poiseuille flow.

Then the final solution of velocity profile is  1 dp  2 y − hy . u(y) = 2μ dx

4.5

(4.26)

Hagen Poiseuille flow (pipe flow)

Now we derive the most popular application of the internal flows, commonly known as Hagen Poiseuille flow (or pipe flow). Since pipes have cylindrical geometry, we use the cylindrical form of the momentum equation. Let us assume an incompressible steady flow through a circular pipe without any appreciable body forces. Assuming a parallel flow in the z-direction, Vz = 0 but Vr = Vθ = 0. The continuity equation becomes ∂Vz ∂(rVr ) 1 ∂Vθ + + = 0, ∂r r ∂θ ∂z

(4.27)

which implies ∂Vz = 0. ∂z Therefore, let us focus on the momentum equation in the z-direction, namely

∂Vz ∂Vz Vθ ∂Vz ∂Vz ρ + Vr + + Vz ∂t ∂r r ∂θ ∂z

2 1 ∂ 2 Vz ∂ 2 Vz ∂ Vz 1 ∂Vz dp . + ρBz + μ + + + =− dz r ∂r ∂r 2 r 2 ∂θ 2 ∂z2 We can further assume

∂Vz ∂θ

= 0 because of cylindrical geometry and so

2

∂ Vz 1 ∂Vz dp +μ , 0=− + dz r ∂r ∂r 2 ∂V

0=−

dp μ ∂(r ∂rz ) + , dz r ∂r

(4.28)

Boundary layer equations in fluid dynamics

73

z d(r dV r dp dr ) = , dr μ dz



r 



z d(r dV 1 dp dr ) dr = dr μ dz

r dr,

dVz 1 dp r 2 = + A, dr μ dz 2 dVz 1 dp dr = dr 2μ dz

Vz (r) =



 r dr + A

1 dr, r

r 2 dp + Alnr + B, 4μ dz

(4.29)

where A and B are constants and Eq. (4.29) is the velocity profile. Since the pipe radius z is R, the boundary condition may be written as Vz (r = R) = 0, and so ∂V ∂r (r = 0) = 0. The second boundary condition is due to the flow symmetry at r = 0, whereas the first is due to the no-slip condition:

A r dp + (r = 0) =⇒ A = 0, 2μ dz r R 2 dp R 2 dp + B = 0 =⇒ B = − . Vz (r = R) = 4μ dz 4μ dz ∂Vz (r = 0) = ∂r



Substituting the constants (i.e., A and B) in Eq. (4.29), we get

R 2 dp r2 1− 2 . Vz (r) = − 4μ dz R

(4.30)

Eq. (4.30) is the final velocity profile of the pipe flow. The laminar velocity profile we just derived is parabolic in shape and given by

r2 Vz (r) = Vmax 1 − 2 , R

(4.31)

where Vmax = constant. However, as the flow Reynolds numbers increases beyond Recr = 2300, the fluid flow in the pipe can be considered turbulent.

74

4.6

Applications of Heat, Mass and Fluid Boundary Layers

Flow over porous wall

Fundamental solutions for the hydrodynamics of the flow through porous channels start from the Navier–Stokes equations for a fluid of constant density ρ and kinematic viscosity ν (in the absence of external forces). In this section, we consider the steady two-dimensional channel of incompressible laminar flow whose porous walls are separated by a distance 2h. Also we consider the case of simple but nonvanishing convection acceleration, i.e., nonlinear terms in the momentum equation. Several flows with uniform suction (or injection) at the wall will be considered.

4.6.1 Uniform suction on a plane A uniform flow (with a velocity U as y → ∞) is considered over a porous plate. At the wall, the no-slip condition is still satisfied, but the normal velocity is not zero. Furthermore, it is assumed that the normal velocity component is constant everywhere in the flow field, i.e., u = 0, but v = vw = constant. The continuity equation shows ∂p = 0. The that u = u(y) only, and we assume that the pressure is constant, i.e., ∂x momentum equation (4.18) becomes ρvw

du d 2u =μ 2, dy dy

(4.32)

which implies that d 2 u vw du − = 0, ν dy dy 2

(4.33)

where ν = μ/ρ. Eq. (4.33) is a differential equation, with auxiliary equation given by p2 −

vw p = 0, ν

whose roots are p1 = 0,

p2 =

vw . ν

The solution of Eq. (4.33) is of the form u(y) = Aep1 y + Bep2 y , where A and B are constants. Thus, u(y) = A + Be

vw ν

y

,

(4.34)

with the boundary conditions of u(0) = 0 and u(∞) = U . Since vw is constant, the convective acceleration is linear, that is, (see Fig. 4.3)

Boundary layer equations in fluid dynamics

75

Figure 4.3 Suction on a plane flow.

u(0) = A + B = 0 =⇒ B = −A,   vw u(∞) = A 1 − e ν ∞ = U =⇒ A = U. The final solution of Eq. (4.34) is therefore  vw  u(y) = U 1 − e ν y .

(4.35)

In order to have a physically realistic solution (to satisfy the condition u(∞) = U ), the wall velocity vw must be negative (wall suction), otherwise u would be unbounded for large y (otherwise A = U ). The boundary thickness can be found by noting that u = 0.99U at y = δ. This gives δ = −4.6

ν . vw

(4.36)

This thickness is constant, independent of y and U , because the convection toward the wall exactly balances the tendency of shear layer to grow due to viscous diffusion.

4.7 Flow between plates with bottom injection and top suction The flow between two porous plates at y = +h and y = −h, respectively, is consid∂p ered. The flow is driven by a pressure gradient ∂x . It is assumed that a uniform vertical flow is generated, i.e., the vertical velocity component is constant everywhere in the flow field, v = vw = constant. Again the continuity equation shows that u = u(y) only, while the momentum equation (4.18) becomes vw

du 1 dp d 2u =− +ν 2, dy ρ dx dy

(4.37)

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Applications of Heat, Mass and Fluid Boundary Layers

since ν = μ/ρ. Thus rearranging Eq. (4.37), we have d 2 u vw du 1 dp − = . 2 ν dy μ dx dy

(4.38)

The homogeneous part of Eq. (4.38) becomes d 2 u vw du − = 0. ν dy dy 2

(4.39)

As seen in the previous case, Eq. (4.39) has the solution u(y) = A + Be

vw ν

y

.

For a particular integral of Eq. (4.38), we set u(y) = ay 2 + by + c,

(4.40)

where a, b, and c are constants. This leads to d 2u du = 2ay + b =⇒ 2 = 2a. dy dy

(4.41)

Substituting Eq. (4.41) into Eq. (4.38), we get

vw 1 dp vw b − 2a y = . 2a − ν ν μ dx Comparing the coefficients, we get a = 0 =⇒ b = −

ν 1 dp . vw μ dx

(4.42)

Now Eq. (4.40) becomes u(y) = −

ν 1 dp y + c. vw μ dx

(4.43)

The final solution is the sum of Eq. (4.34) and Eq. (4.43), i.e., u(y) = D + Be

vw ν

y

−

ν 1 dp y. vw μ dx

(4.44)

Since vw is constant, the equation is linear. We retain the no-slip condition for the main flow (see Fig. 4.4): u(+h) − u(−h) = O(ϕ), u(h) = D + Be

vw ν

h

−

ν 1 dp h, vw μ dx

(4.45)

Boundary layer equations in fluid dynamics

77

Figure 4.4 Flow between plates with bottom injection and top suction.

u(−h) = D + Be−

vw ν

h

+

ν 1 dp h, vw μ dx

(4.46)

Subtracting Eq. (4.46) from Eq. (4.45), we get B=

2 vνw e

vw ν

h dp μ dx − vνw

h−e

h

=

h dp μ dx 2 sinh( vνw h)

2 vνw

=

ν h dp vw μ dx . sinh( vνw h)

(4.47)

Substituting Eq. (4.47) into Eq. (4.45), we get D=−

ν h dp vνw h vw μ dx e sinh( vνw h)

+

ν h dp . vw μ dx

(4.48)

Eq. (4.44) reduces to u(y) = −

ν h dp vνw h vw μ dx e sinh( vνw h)

vw

ν h dp ν y ν h dp ν y dp v μ dx e + + w − . vw vw μ dx sinh( ν h) vw μ dx

But the wall Reynolds number is Re = Rearranging Eq. (4.49), we get

vw ν h,

so

Re h

=

vw ν

=⇒

h Re

=

(4.49)

ν vw .

y   eRe − eRe h h2 1 dp y −1+ . u(y) = − Re μ dx h sinh(Re)

(4.50)

The final solution of Eq. (4.50) is y   2 y u(y) eRe − eRe h = −1+ , umax Re h sinh Re

(4.51)

2

h where umax = 2μ (− dp dy ) is the centerline velocity for imporous or Poiseuille flow. For very small Re (or small vertical velocity), the last terms in the parentheses of Eq. (4.51) can be expanded in a power series and sinh Re ≈ Re, i.e.,

 1 + Re + 2 y u(y) = −1+ umax Re h

(Re)2 2

+ · · · − (1 + Re yh + Re

(Re)2 y 2 2 h2

+ ···)

 ,

78

Applications of Heat, Mass and Fluid Boundary Layers

 Re(1 + u(y) 2 y = −1+ umax Re h

Re 2

−

y h

−

Re

Re y 2  2 h2 )

,

u(y) y2 =1− 2. umax h

(4.52)

Eq. (4.52) shows that Poiseuille solution is recovered. For very large Re (or large vertical velocity), Eq. (4.51) can be written as y   u(y) 2 y eRe − eRe h = − 1 + 2 Re , umax Re h e − e−Re y   2 y 1 − e−Re(1− h ) u(y) = −1+2 , umax Re h 1 − e−2Re

for Re → ∞ and

y h

> 1, except for y = +h, where we get

  u(y) 2 y = −1+2 , umax Re h   u(y) 2 y = 1+ . umax Re h

(4.53)

Thus a straight line variation which suddenly drops to zero at the upper wall is obtained. Also note that the average velocity decreases with an increasing Reynolds number, i.e., the function factor increases as we allow more cross flow through the walls. Some velocity profiles, which illustrate these conditions, are shown in Fig. 4.5 (we present plots of Eqs. (4.51) and (4.52)).

4.8 Flow in a porous duct The previous examples in the last section were linear due to the assumption of constant cross flow velocity. If we impose more realistic boundary conditions, e.g., suction at both walls, then the net mass flow will change with x, and at the very least we must have u = u(x, y) and v = v(y) to satisfy the continuity relation. This means ∂u that both the axial convective acceleration terms u ∂x and v ∂u ∂y will be products of variables and thus become nonlinear. The solution may be vexed, with both existence and uniqueness problems, and it is instructive to consider an example. (See Fig. 4.6.) Let the duct be uniformly porous, i.e., assume the wall velocity vw is constant, independent of x, then the average velocity u in the duct will vary linearly with x because of the mass flow through the walls. The two most studied geometries are the circular tube and channel flow between parallel plates. In practice, there must always

Boundary layer equations in fluid dynamics

79

Figure 4.5 Velocity profiles for flow between parallel plates with bottom injection and top suction.

be an entrance region, and the main velocity continually varies. It is a controversial question as to whether a “fully developed” condition can be achieved. For the present example let us consider a channel flow between uniformly parallel plates. We assume without proof that we are far downstream of the entrance and that the boundary conditions are: at y = h, u = 0, v = vw ,

(4.54)

at y = −h, u = 0, v = −vw ,

(4.55)

i.e., both walls have either equal suction (vw > 0) or equal injection (vw < 0). Let u(0) denote the average axial velocity at an initial section (x = 0). Then it is clear from a gross mass balance that u(x) will differ from u(0) by the amount vhw x. This

Figure 4.6 Flow in a porous duct.

80

Applications of Heat, Mass and Fluid Boundary Layers

observation led Berman [4] to formulate the following relation for the stream in the channel:     ψ(x, y) = hu(0) − vw x f y ∗ , (4.56) where y ∗ = yh , ψ(x, y) is a stream function, u(0) is the initial average axial velocity, and f is a dimensionless function to be determined. The velocity components follow immediately from the definition of ψ:

  ∂ψ     vw x u x, y ∗ = (4.57) = u(0) − f  y ∗ = u(x)f  y ∗ , ∂y h     ∂ψ v x, y ∗ = − = vw f y ∗ = v(y). ∂x

(4.58)

Thus, the function f and its derivative f  represent the shape of the velocity profile; these are independent of x, and the flow is thus termed similar. The stream function must now be made to satisfy the momentum equations (4.18) and (4.19) for steady flow: 2

∂u ∂u 1 ∂p ∂ u ∂ 2u u + +v =− +ν , (4.59) ∂x ∂y ρ ∂x ∂x 2 ∂y 2

u

2

∂v 1 ∂p ∂ v ∂ 2v ∂v + +v =− +ν . ∂x ∂y ρ ∂y ∂x 2 ∂y 2

(4.60)

Since y ∗ = yh , Eqs. (4.59) and (4.60) become 2

1 ∂p 1 ∂ 2u ∂ u ∂u v ∂u = − + + + ν , ∂x h ∂y ∗ ρ ∂x ∂x 2 h2 ∂y ∗2

(4.61)

2

1 ∂p ∂ v 1 ∂ 2v ∂v v ∂v =− +ν + + u . ∂x h ∂y ∗ ρh ∂y ∗ ∂x 2 h2 ∂y ∗2

(4.62)

u

Using (4.57) and (4.58) in (4.61) and (4.62), the momentum equations reduce to



 ν  vw x vw   1 ∂p 2 − 2f = u(0) − ff − f , (4.63) − ρ ∂x h h h −

v2 νvw 1 ∂p = w ff  − 2 f  . ∗ ρh ∂y h h

(4.64)

Now differentiating (4.64) with respect to x, we get ∂ 2p ∂ 2p = = 0. ∗ ∂x∂y ∂x∂y

(4.65)

Boundary layer equations in fluid dynamics

81

Differentiating (4.63) with respect to y ∗ , we get



 ν  vw x d vw   ∂ 2p 2 − 2f = u(0) − ff − f . ∂x∂y ∗ h dy ∗ h h From Eqs. (4.65) and (4.66), we obtain

 ν  d vw   2 − 2f ff − f = 0, dy ∗ h h

(4.66)

(4.67)

 vw   ν ff − f  f  − 2 f  = 0. h h Letting the suction Reynolds number be Re = expression, we get   f  + Re f  f  − ff  = 0.

hvw ν

and substituting into the above (4.68)

The boundary conditions on f (y ∗ ) are now f (1) = 1, f (−1) = −1, f  (1) = 0, f  (−1) = 0.

(4.69)

These show that f (y ∗ ) is antisymmetric about y ∗ = 0, so that at the centerline v =  0 and ∂u ∂y = 0, or f (0) = f (0) = 0. Eq. (4.68) has no known analytic closed-form solution, but it can be integrated once. Integrate (4.68) once with respect to y ∗ , we get   f  + Re f  2 − ff  = K = const. (4.70) Hence, the solution of the equations of motion and continuity are given by nonlinear fourth-order differential equation (4.68) subject to the boundary condition (4.69).

4.9 Approximate analytic solution (perturbation) The nonlinear ordinary differential equation (4.68) subject to condition (4.69) must in general be integrated numerically. However, for the special case where Re is small, approximate analytic results can be obtained by using a regular perturbation approach. Note that the perturbation method is used because Eqs. (4.68) and (4.70) are nonlinear and, by using this technique, we get a linear approximated version of the true equations. In this solution, f (y ∗ ) may be expanded in powers of Re as: ∞      f y∗ = Ren fn y ∗ ,

(4.71)

n=0

where fn (y ∗ ) satisfies the symmetric boundary conditions f0 (0) = f0 (1) = f0 (0) = 0, f0 (1) = 1

(4.72)

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Applications of Heat, Mass and Fluid Boundary Layers

and fn (0) = fn (1) = fn (0) = 0, fn (1) = 1.

(4.73)

Here fn are independent of Re. Substituting (4.71) in (4.68), we get       f0 + Ref1 + Re2 f2 + Re f0 + Ref1 + Re2 f2 f0 + Ref1 + Re2 f2    − f0 + Ref1 + Re2 f2 f0 + Ref1 + Re2 f2 = 0. Equating the coefficients of Re, we get f0 = 0,

(4.74)

f1 + f0 f0 − f0 f0 = 0,

(4.75)

f2 + f0 f1 + f1 f0 − f0 f1 − f1 f0 = 0.

(4.76)

The solution of (4.74) is of the form   Ay ∗3 By ∗2 f0 y ∗ = + + Cy ∗ + D, 6 2 where A, B, and C are constants. Applying the boundary condition (4.72) to the above equation, we get   1  f0 y ∗ = 3y ∗ − y ∗3 . 2

(4.77)

The solutions of Eqs. (4.75) and (4.76) subject to the boundary condition (4.73) are given respectively as    1  ∗7 y − 3y ∗3 − 2y ∗ , f1 y ∗ = − 280

(4.78)

  f2 y ∗ =

(4.79)

  1 14y ∗11 − 385y ∗9 + 198y ∗7 + 876y ∗3 − 703y ∗ . 1293600 Hence, the first order perturbation solution for f (y ∗ ) is       f  y ∗ = f0 y ∗ + Ref1 y ∗ ,   1  Re  ∗7  y − 3y ∗3 − 2y ∗ . f 1 y ∗ = 3y ∗ − y ∗3 − 2 280 The second-order perturbation of solution for f (y ∗ ) is         f 2 y ∗ = f0 y ∗ + Ref1 y ∗ + Re2 f2 y ∗ ,

(4.80)

Boundary layer equations in fluid dynamics

83

  1  Re  ∗7  f 2 y ∗ = 3y ∗ − y ∗3 − y − 3y ∗3 − 2y ∗ 2 280  Re2  14y ∗11 − 385y ∗9 + 198y ∗7 + 876y ∗3 − 703y ∗ . (4.81) + 1293600 The results above reduce to those of Berman [4], and we now get a linear approximated version of the true equations above. The first-order expression for the velocity components are:     vw x   ∗  u x, y ∗ = U (0) − f y h

    Re  vw x 3  1 − y ∗2 1 − 2 − 7y ∗2 − 7y ∗4 , (4.82) = U (0) − h 2 420     Re  ∗7     1 ∗ 3y − y ∗3 − y − 3y ∗3 − 2y ∗ . (4.83) v x, y ∗ = vw f y ∗ = vw 2 280 For pressure distribution, from Eq. (4.63), we get          h2 ∂p vw x    ∗  = U (0) − f y + Re f  2 y ∗ − f y ∗ f  y ∗ , ρν ∂x h and, since f  (y ∗ ) + Re(f  2 (y ∗ ) − f (y ∗ )f  (y ∗ )) = K, from (4.70) we have     ∂p Kρν Kμ vw x vw x = 2 U (0) − = 2 U (0) − . (4.84) ∂x h h h h Now, from Eq. (4.64), we have     μvw   ∗  ∂p = f y − ρν 2 f y ∗ f  y ∗ . ∗ ∂y h Since dp = dp =

∂p ∂x dx

+

(4.85)

∂p ∗ ∂y ∗ dy ,

   ∗  ∗ ∗ Kμ μvw   ∗  vw x 2 f y f y dy . y − ρν dx + f U (0) − h h h2

(4.86)

Integrating (4.86), we get 

p x, y

∗



   ρν 2 2  ∗  Kμ μvw    ∗  vw x 2 = p(0, 0) − f y + 2 U (0)x − + f y − f  (0) . 2 2h h h (4.87)

The pressure drop in the major flow direction is given by    Kμ vw x 2  − U (0)x . p(0, x) − p x, y ∗ = 2 2h h

(4.88)

84

Applications of Heat, Mass and Fluid Boundary Layers

Figure 4.7 Velocity profiles for flow between parallel plates with equal injection or suction.

4.10 Numerical solution The approximate results of the previous section are not reliable when the Reynolds number is not small. To obtain the detailed information on the nature of the flow for different values of Reynolds number (e.g., for Re = 0, 10, 20, 30), a numerical solution to the governing equations is necessary. The Runge–Kutta program in Appendix C is used to solve Eq. (4.70) numerically. One initial condition and constant K are unknown, i.e., started at y ∗ = 1, f  (1) and K were guessed and the solution was double-iterated until f (−1) = −1 and f  (−1) = 0. The most complete sets of profiles are shown in Figs. 4.7 and 4.8. The velocity profiles have been drawn for

Figure 4.8 Velocity profiles for flow between parallel plates with equal injection or suction.

Boundary layer equations in fluid dynamics

85

different values of Reynolds number (i.e., for Re = 0, 10, 20, 30). The shapes change smoothly with Reynolds number and show no odd or unstable behavior. Suction tends to draw the profiles toward the wall. From Fig. 4.7, it observed that for Re > 0 in the region 0 ≤ y ∗ ≤ 1, f (y ∗ ) decreases as the Reynolds number Re increases. Also from Fig. 4.8, it observed that, for Re > 0, f  (y ∗ ) decreases with an increase of the Reynolds number in the range of 0 ≤ y ∗ ≤ 1.

4.11 The boundary-layer equations As Prandtl showed for the first time in 1904, usually the viscosity of a fluid only plays a role in a thin layer (along a solid boundary, for instance). Prandtl called such a thin layer “Uebergangsschicht” or “Grenzschicht”; the English terminology is “boundary layer” or “shear layer” (Dutch is “grenslaag”). When a viscous fluid flows along a fixed impermeable wall, or passes the rigid surface of an immersed body, an essential condition is that the velocity at any point on the wall or other fixed surface is zero. The extent to which this condition modifies the general character of the flow depends on the value of the viscosity. If the body is of streamline shape and if the viscosity is small without being negligible, the modifying effect appears to be confined within narrow regions adjacent to the solid surfaces; these are called boundary layers. Within such layers the fluid velocity changes rapidly from zero to its mainstream value, and this may imply a steep gradient of shearing stress. As a consequence, not all the viscous terms in the equation of motion will be negligible, even though the viscosity, which they contain as a factor, is itself very small. A more precise criterion for the existence of a well-defined laminar boundary layer is that the Reynolds number should be large, though not so large as to imply a breakdown of the laminar flow.

4.11.1 Boundary layer governing equations The Navier–Stokes equations are considered sufficiently general to describe the Newtonian fluids appearing in hydro- and aerodynamics. Solving these equations is a complex job, also with computational means (despite the fast computers available nowadays). Fortunately, the equations contain terms that can be neglected in large parts of the flow domain. This allows the equations to be simplified, and herewith to reduce the effort for solving them. The terms that describe the viscous shear stresses offer such a possibility for simplification. These terms are only of interest in local areas of high shear (boundary layer, wake). Outside these areas “nonviscous” equations can be used. We begin with the derivation of the equations that describe the flow in shear layers, like boundary layers and wakes. The starting point is the Navier–Stokes equations for a steady two-dimensional incompressible flow, where the density ρ is assumed constant. The equations are for-

86

Applications of Heat, Mass and Fluid Boundary Layers

mulated in a Cartesian coordinate system (x, y) with velocity components (u, v). It is further assumed that the x-axis coincides (locally) with the solid boundary. The equations of motion for a steady two-dimensional incompressible flow are ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

∂u ∂x ∂u u ∂x ∂v u ∂x

+ v ∂u ∂y + v ∂v ∂y

+

= − ρ1 = − ρ1

∂v ∂y ∂p ∂x ∂p ∂y

= 0, 2

∂ u + ν( ∂x 2 + ∂2v + ν( ∂x 2

+

∂2u ), ∂y 2 ∂2v ), ∂y 2

(4.89)

where the x and y variables are respectively the horizontal and vertical coordinates, u and v are respectively the horizontal and vertical fluid velocities, and p is the fluid pressure. A wall is located in the plane y = 0. We consider dimensionless variables x =

x y u v L  p U , t = t , , y  = , u = , v  = ,p = L δ U U δ L ρU 2

(4.90)

where L is the horizontal length scale and δ is the boundary layer thickness at x = L, which is unknown. We will obtain an estimate for it in terms of the Reynolds number Re; U is the flow velocity, which is aligned in the x-direction parallel to the solid boundary. The dimensionless form of the governing equations is ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

∂u ∂x 

+

∂v  ∂y 

= 0,

2 ∂ 2 u ∂p  ∂u ν ∂ 2 u  ∂u + u ∂x + UνL Lδ 2 ∂(y  + v ∂y  = − ∂x  + U L  )2 , ∂(x  )2 ∂v  ∂v  ∂v  L 2 ∂p  ν ∂ 2 v ν L 2 ∂ 2 v   ∂t  + u ∂x  + v ∂y  = −( δ ) ∂y  + U L ∂(x  )2 + U L ( δ ) ∂(y  )2 ,

∂u ∂t 

(4.91)

where the Reynolds number for this problem is Re =

UL . ν

(4.92)

Inside the boundary layer, viscous forces balance inertia and pressure gradient forces. In other words, inertia and viscous forces are of the same order, so −1 Re 2 ν L 2 = O(1) =⇒ δ = O . UL δ L

(4.93)

Now we drop the primes from the dimensionless governing equations and with Eq. (4.93) have ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

∂u ∂t 1 ∂v ( Re ∂t

∂u + u ∂x ∂v + u ∂x

∂u ∂v ∂x + ∂y = 0, ∂p 1 ∂2u ∂2u + v ∂u ∂y = − ∂x + Re ∂x 2 + ∂y 2 , ∂p 1 ∂2v ∂2v + v ∂v ∂y ) = − ∂y + Re2 ( ∂x 2 + ∂y 2 ).

(4.94)

Boundary layer equations in fluid dynamics

87

In the limit Re → ∞, the equations above reduce to ⎧ ∂u ∂v ⎪ ∂x + ∂y = 0, ⎨ ∂p ∂u ∂u ∂u ∂2u ∂t + u ∂x + v ∂y = − ∂x + ∂y 2 , ⎪ ⎩ 0 = − ∂p ∂y .

(4.95)

Notice that according to Eq. (4.95), the pressure along the y-component is constant across the boundary layer. In terms of dimensional variables, the system of equations above assumes the form ⎧ ∂u ∂v ⎪ ∂x + ∂y = 0, ⎨ ∂p ∂u ∂u ∂u ∂2u (4.96) ∂t + u ∂x + v ∂y = − ∂x + ∂y 2 , ⎪ ⎩ 1 ∂p 0 = − ρ ∂y . These are the boundary-layer equations, with which the flow in a shear layer can be approximately described. The corresponding boundary conditions are: at y = 0 (wall)

u=v=0

at y = δ(x) (outer stream)

(no-slip), u = U (x)

(4.97) (patching).

(4.98)

This is the no-slip condition which prevents the shear stress at the wall to become infinite. When the flow is studied at a molecular level, this condition has to be adapted; in practice this is only relevant for highly rarefied gases.

4.12

Influence of boundary layer on external flow

The boundary layer is very thin and at first instance this suggests that the external flow around the “clean” body be computed, i.e., without boundary layer. Based on the external streamwise velocity, thereafter the boundary layer can be computed. Finally, the external flow has to be corrected for the presence of the boundary layer.

4.12.1 Momentum thickness θ and momentum integral Momentum thickness is the distance that, when multiplied by the square of the free stream velocity, equals the integral of the momentum defect. Alternatively, the total loss of momentum flux is equivalent to the removal of momentum through a distance θ . It is a theoretical length scale to quantify the effects of fluid viscosity near a physical boundary. A shear layer of unknown thickness grows along the sharp flat plate in Fig. 4.9. The no-slip wall condition retards the flow, making it into a rounded profile u(y), which merges into the external velocity U which is constant at a “thickness” y = δ(x). By utilizing the control volume, we find (without making any assumptions about

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Figure 4.9 Growth of a boundary layer on a flat plate.

laminar versus turbulent flow) that the drag force on the plate is given by the following momentum integral across the exit plane:  δ(x) D(x) = ρb u(U − u) dy, (4.99) 0

where b is the plate width into the paper and the integration is carried out along a vertical plane x = constant. Eq. (4.99) was derived by Kármán, who wrote it in the convenient form of the momentum thickness θ as  δ(x) D(x) = ρbU 2 θ = ρb u(U − u) dy, (4.100) 0

 θ= 0

δ



u u 1− dy, U U

(4.101)

where Eq. (4.101) is called the momentum thickness θ . Momentum thickness is thus a measure of total plate drag. Drag also equals the integrated wall shear stress along the plate  x D(x) = b τw (x) dx, 0

D(x) = bτw (x). dx Meanwhile, the derivative of Eq. (4.100), with U = constant, is D(x) dθ = ρbU 2 . dx dx

(4.102)

(4.103)

By comparing this with Eq. (4.102), we arrive at the momentum-integral relation for a flat-plate boundary-layer flow: τw (x) = ρU 2

dθ . dx

It is valid for either laminar or turbulent flat-plate flow.

(4.104)

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89

To get a numerical result for laminar flow, von Kármán assumed that the velocity profiles had an approximately parabolic shape

2y y 2 − 2 , u(x, y) ≈ U δ δ

0 ≤ y ≤ δ(x),

(4.105)

which makes it possible to estimate both momentum thickness and wall shear θ=

 δ 0

2y y 2 − 2 δ δ

τw (x) = μ

1−

2y y 2 2 + 2 dy ≈ δ, δ 15 δ

∂u 2μU |y=0 ≈ . ∂y δ

(4.106)

By substituting (4.106) into (4.104) and rearranging, we obtain δdδ ≈ 15

ν dx, U

(4.107)

where ν = μρ . We can integrate from 0 to x, assuming that δ = 0 at x = 0, the leading edge, to get 1 2 15νx δ = , 2 U or

1 5.5 ν 2 δ = . ≈ 5.5 1 x Ux Rex2

(4.108)

This is the desired thickness estimate. It is all approximate, of course, part of von Kármán’s momentum-integral theory, but it is accurate, being only 10% higher than the known exact solution for a laminar flat-plate flow, for which we gave xδ ≈ 5.01 . Rex2

By combining Eqs. (4.108) and (4.106), we also obtain a shear-stress estimate along the plate as cf =

2τw ≈ ρU 2



8 15

Rex

1

2

=

0.73 1

.

(4.109)

Rex2

Again this estimate, in spite of the crudeness of the profile assumption (4.105), is 1/2 only 10% higher than the known exact laminar-plate-flow solution cf = 0.664/Rex . The dimensionless quantity cf , called the skin-friction coefficient, is analogous to the friction factor f in ducts.

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Figure 4.10 Displacement effect of a boundary layer.

4.12.2 Displacement thickness δ ∗ The displacement thickness for the boundary layer is defined as the distance the surface would have to move in the y-direction to reduce the flow passing by a volume equivalent to the real effect of the boundary layer. Another interesting effect of a boundary layer is its small but finite displacement of the outer streamlines. As shown in Fig. 4.10, outer streamlines must deflect outward a distance δ(x)∗ to satisfy conservation of mass between the inlet and outlet  h  δ ρU b dy = ρub dy, δ = h + δ∗. (4.110) 0

0

δ∗

The quantity is called the displacement thickness of the boundary layer. To relate it to u(y), cancel ρ and b from Eq. (4.110), evaluate the left integral, and subsequently add and subtract U from the right integrand:  δ  δ   Uh = (U + u − U )dy = U h + δ ∗ + (u − U )dy, (4.111) 0

0

or ∗

δ =

 δ 0

u dy. 1− U

(4.112)

Thus the ratio of δ ∗ /δ varies only with the dimensionless velocity-profile shape u/U . Introducing our profile approximation (4.105) into (4.112), we obtain by integration the approximate result 1 δ ∗ ≈ δ, 3

δ∗ 1.83 . ≈ 1/2 x Rex

(4.113)

These estimates are only 6% away from the exact solutions for a laminar flat-plate 1/2 flow given as δ ∗ = 0.344δ = 1.721x/Rex . Since δ ∗ is much smaller than x for large Rex and the outer streamline slope V /U is proportional to δ ∗ , we conclude that the velocity normal to the wall is much smaller than the velocity parallel to the wall. This is a key assumption in boundary-layer theory. We also conclude from the success of these simple parabolic estimates that von Kármán’s momentum-integral theory is effective and useful.

Boundary layer equations in fluid dynamics

4.13

91

The flat-plate boundary layer

The classic and most often used solution of boundary-layer theory is for flat-plate flow, as in Fig. 4.9, which can represent either laminar or turbulent flow.

4.13.1 Laminar flow For a laminar flow passing the plate, the boundary-layer equations (4.95) can be solved exactly for u and v, assuming that the free-stream velocity U is constant (dU/dx = 0). The solution was given by Prandtl’s student Blasius. With an ingenious coordinate transformation, Blasius showed that the dimensionless velocity profile u/U is a function only of the single composite dimensionless variable y[U/(νx)]1/2 :

u = f  (η), U

U η=y νx

1

2

,

(4.114)

where the prime denotes differentiation with respect to η. Substitution of (4.114) into the boundary-layer equations (4.95) reduces the problem, after much algebra, to a single third-order nonlinear ordinary differential equation for f , namely 1 f  + ff  = 0. 2

(4.115)

The boundary conditions (4.97) become: at y = 0, at y → ∞,

f (0) = f  (0) = 0,

(4.116)

f  (∞) → 1.0.

(4.117)

This is the Blasius equation, for which accurate solutions have been obtained only by numerical integration. Since u/U approaches 1.0 only as y → ∞, it is customary to select the boundarylayer thickness δ ∗ as that point where u/U = 0.99. And this occurs at η ≈ 5.0: δ0.99

U νx

1

2

≈ 5.0,

or δ 5.0 ≈ x Rex1/2

(Blasius).

(4.118)

With the profile known, Blasius, of course, could also compute the wall shear and displacement thickness cf =

0.664

, 1/2

Rex

δ∗ 1.721 . = 1/2 x Rex

(4.119)

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4.13.2 Turbulent flow There is no exact theory for turbulent flat-plate flow, although there are many elegant computer solutions of the boundary-layer equations using various empirical models for the turbulent eddy viscosity. The most widely accepted result is simply an integral analysis similar to our study of the laminar-profile approximation (4.105). We begin with Eq. (4.103), which is valid for a laminar or turbulent flow. We recall it here for convenient reference: τw (x) = ρU 2

dθ . dx

(4.120)

From the definition of cf , Eq. (4.109), this can be rewritten as cf = 2

dθ . dx

(4.121)

The turbulent profiles are nowhere near parabolic and the flat-plate flow is very nearly logarithmic, with a slight outer wake and a thin viscous sublayer. Therefore, just as in turbulent pipe flow, we assume that the logarithmic law (4.122) holds: 1 yu∗ u ≈ ln + B, u∗ κ ν

∗

u =



τw ρ

1

2

.

(4.122)

Over the full range of turbulent smooth wall flows, the dimensionless constants κ and B are found to have the approximate values κ ≈ 0.41 and B ≈ 5.0. Eq. (4.122) holds all the way across the boundary layer. At the outer edge of the boundary layer, y = δ and u = U , and Eq. (4.122) becomes 1 δu∗ U ≈ ln + B. u∗ κ ν

(4.123)

But the definition of the skin-friction coefficient, Eq. (4.109), is such that the following identities hold: 1 1 cf 2 2 2 δu∗ U ≡ , . (4.124) ≡ Reδ ∗ u cf ν 2 Therefore Eq. (4.123) is a skin-friction law for a turbulent flat-plate flow:

2 cf

1

2

 1  cf 2 ≈ 2.44 ln Reδ + 5.0. 2

(4.125)

It is a complicated law, but we can at least solve it for a few values and list them in Table 4.1. We can forget the complex log-friction law (4.125) and simply fit the numbers in the table to a power-law approximation −1/6

cf ≈ 0.02Reδ

.

(4.126)

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Table 4.1 A few values of cf for different values of Reδ . Reδ cf

104 0.00493

105 0.00315

106 0.00217

107 0.00158

This we shall use as the left-hand side of Eq. (4.121). For the right-hand side, we need an estimate for θ (x) in terms of δ(x). If we use the logarithmic-law profile (4.122), we shall be up to our hips in logarithmic integrations for the momentum thickness. The turbulent profiles can be approximated by a one-seventh-power law

u U

≈ turb

1/7 y , δ

(4.127)

the momentum relation (4.101) in dimensionless form becomes

 2 δu u 1− dy. CD = L 0 U U

(4.128)

With this simple approximation (4.127), the momentum thickness (4.128) can easily be evaluated as 1/7   δ 1/7  7 y y θ≈ (4.129) 1− dy = δ. δ δ 72 0 Now substitute Eqs. (4.126) and (4.129) into von Kármán’s momentum law (4.121) to obtain

d 7 −1/6 cf ≈ 0.02Reδ =2 δ , dx 72 or −1/6

Reδ

= 9.72

dδ d(Reδ ) = 9.72 . dx d(Rex )

(4.130)

Separate the variables and integrate, assuming δ=0 at x = 0: 6/7

Reδ ≈ 0.16Rex ,

or

δ 0.16 . ≈ x Rex1/7

(4.131)

Thus the thickness of a turbulent boundary layer increases as x 6/7 , far more rapidly than the laminar increase x 1/2 . Eq. (4.131) is the solution to the problem because all other parameters are now available. For example, combining Eqs. (4.131) and (4.126), we obtain the friction variation cf ≈

0.027 1/7

.

Rex

In conclusion, further insights may be obtained from refs. [1–12].

(4.132)

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References [1] J.M. McDonaugh, Lectures in Elementary Fluid Dynamics, Department of Mechanical Engineering and Mathematics, University of Kentucky, Lexington, KY, 2009, pp. 40506–0503. [2] J.B. Chabi Orou, Lecture Note on Fluid Mechanics, African University of Science and Technology, Abuja, 2012. [3] Frank White, Viscous Fluid Flow, 2nd edition, 1996. [4] A.S. Berman, Laminar flow in channels with porous walls, J. Appl. Phys. 24 (1953). [5] S. Ganesh, S. Krishnambal, Magnetohydrodynamics flow of viscous fluid between two parallel plates, J. Appl. Sci. 6 (11) (2006) 2420–2425. [6] H. Schlichting, Boundary Layer Theory, 7th ed., McGraw-Hill, New York, 1979. [7] F.M. White, Viscous Fluid Flow, 2nd ed., McGraw-Hill, New York, 1991. [8] L. Rosenhead (Ed.), Laminar Boundary Layers, Oxford University Press, London, 1963. [9] T. von Kármán, On laminar and turbulent friction, Z. Angew. Math. Mech. 1 (1921) 235–236. [10] G.B. Schubauer, H.K. Skramstad, Laminar Boundary Layer Oscillations and Stability of Laminar Flow, Natl. Bur. Stand. Res. Pap. 1772 (April 1943), see also J. Aeronaut. Sci. 14 (1947) 69–78, NACA Rep. 909, 1947. [11] W. Rodi, Turbulence Models and Their Application in Hydraulics, Brookfield Pub., Brookfield, VT, 1984. [12] Frank M. White, Fluid Mechanics, fourth edition, McGraw-Hill, New York, 1998.

The Merk–Chao–Fagbenle method for laminar boundary layer analysis

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R. Layi Fagbenlea , Leye M. Amoob , S. Aliuc , A. Falanad a Mechanical Engineering Department and Center for Petroleum, Energy and Law, CPEEL (A John D. and Catherine T. McArthur Center of Excellence), University of Ibadan, Ibadan, Oyo State, Nigeria, b Stevens Institute of Technology, Hoboken, NJ, United States, c University of Benin, Benin City, Department of Mechanical Engineering, Benin City, Nigeria, d Department of Mechanical Engineering, University of Ibadan, Ibadan, Nigeria

5.1 Introduction Fluid mechanics in general and boundary layers in particular are mathematically complex. Such complexity at times not only advances the study and understanding of fluids, but also advances the applied mathematics discipline. Mathematics continues to allow for much needed conclusions to be drawn from several disciplines. To this end, numerous mathematicians continue to make significant contributions to the discipline of fluid dynamics. Boundary layer problems involve a rapid change in the value of a physical variable over a limited region of space, and they constitute a particular class of singular perturbation problems. In this regard, nearly all boundary layer problems involve differential equations in which the highest derivative term is multiplied by a small parameter. Also, the boundary layer is always considered as semiinfinite, the major reason being freedom from having to consider end boundary effects where all imponderables and imaginable can be expected. Considering an infinite surface might be so difficult as to distract from the main interest of the enquiry in the first instance. That said, there is nothing that prohibits the younger generation of researchers from confronting this problem, considering their advantage of exposure to relatively larger body of knowledge than preceding generations. Hydro- or fluid dynamics is governed by nonlinear partial differential equations (PDEs), which are very difficult to solve analytically. To the best of our knowledge, no general closed-form solution to these equations exists. The governing equations of the boundary layer are primarily based on a simplification of the system of the second-order nonlinear partial differential equations (PDEs), which are known as the Navier–Stokes (NS) equations of motion for viscous flows. The simplification offered by Prandtl in 1908 is generally referred to as Prandtl Boundary Layer (PBL) equations. Unlike the NS equations, which are elliptical, boundary layer equations are parabolic in nature, and the techniques used to solve them are based on the laws of similarity in boundary layer flows. Three primary methods may be used to solve boundary layer problems: the similarity or differential method (most common approach), the integral method, and the Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00013-X Copyright © 2020 Elsevier Ltd. All rights reserved.

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full numerical solution method [1]. Many special cases of nonlinear PDEs have led to appropriate changes in variables or stretching transformations, depending on the task they are intended to accomplish. Some transformations linearize the system of equations under consideration, while others transform the system to one for which a solution exists. The transformations that reduce a system of PDEs to a system of ordinary differential equations (ODEs) by exploiting an inherent symmetry of the problem are often regarded as “similarity transformations.” The similarity method is the original Blasius method which was developed to solve boundary layer problems analytically. Blasius introduced and employed an independent variable called the similarity variable to Prandtl’s boundary layer equations [2,3]. This was based on the premise that the velocity is geometrically similar along the flow direction, where conservation PDEs are converted to ODEs. The similarity transformation captures the boundary layer growth and significantly simplifies the analysis and solution of the governing equations. The finding of a similarity variable that is suitable for the transformation to take place is an art rather than a science, and it requires having good insights into the problem. The numbers of independent variables in the PDEs are carefully converted into a single independent variable (known as the similarity variable). The original initial boundary conditions are also equally transformed into appropriate boundary conditions in the new combined variable. The similarity transformation technique is an indispensable tool to the analysis of fluid mechanical behavior in general and especially boundary layer processes. Asymptotic techniques allow us to make simple a complex system, which then provides for an enlightened form of empiricism that we refer to as similarity. Several methods and approaches have been developed to find similarity variables, for example, the Vaschy– Buckingham Pi theorem [4–6]. The most rigorous and systematic approach of finding similarity variables is based on the Lie group of transformations [7–14]. The premise of the Lie-group approach is that each variable in the initial equation is subjected to an infinitesimal transformation. The demand that the equation is invariant under these transformations leads to the determination of the potential or possible symmetries. This approach has been routinely applied to boundary layer equations. Apropos boundary layer theory, the authors of [15] provided a comprehensive account of classical methods, including several possible outcomes depending on the perspective of the problem to be solved. The Clarkson–Krustal direct method, which is used to find similarity reductions, was employed in [16] to unsteady boundary layer equations. It is important to note that the similarity variable found is not unique or peculiar to one problem only; it can be applied to other similar problems wherever appropriate. Furthermore, Hansen [17] discussed the “stretching variable” method used to find similarity transformations. Overall, similarity problems reduce the original PBL equations to a form that is invariant with respect to affine transformations. The local flow field is then resolved through analytical/numerical solutions of the PDEs governing the boundary layer. Characteristically, the velocity profiles of boundary layer flows yield a suite of homothetic curves and plots. Why are they typically homothetic? Regarding the velocity profile, for example, we normalize by Uu∞ and this tends to or approaches unity. Similarly, regarding the temperature profile, we normalize by freestream temperature, or T − T∞ , and this tends to or approaches zero. Integral methods, in another

The Merk–Chao–Fagbenle method for laminar boundary layer analysis

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regard, yield closed-form solutions by assuming a profile of velocity, temperature, and concentration mass transfer. It involves the integration of the equations from the wall to free stream, thus yielding an overall performance that includes the growth of the boundary layer. Lastly, the full numerical method uses well proven numerical schemes and practical simulation codes with high speed computers to solve several boundary layer problems. It should be remarked that some studies in the literature discuss their results as exact solutions. Caution in this regard is important. Generally, when we speak of “exact solutions” of fundamental equations, such as the NS equations, and this could be the full NS equations or any of their approximated forms, so long as the obtained solutions obtained by any technique are indeed as exact as they come, that is, there is no better solution found. The exactness refers to the solution of the equation itself. If the equation in question has been an approximation of a more robust equation, then the claim of exactness of solution should be only to the approximate solution.

5.2 Historical perspectives of series-based methods and the Merk–Chao–Fagbenle (MCF) procedure Several approaches have been used to develop and solve many transport problems to which boundary-layer theory is applied. Series-based expansion techniques which date to Blasius [18] were subsequently refined by Hiemenz in 1911 [19], Howarth in 1935 [20], and Görtler in 1957 [21]. These were characteristically established to solve nonsimilar problems. They are advantageous because many coefficients can be solved for once and for all. Moreover, they have a universal character, and subsequent terms beyond the first term solution are valid and corrective of the first term solution, which shows good agreement with the findings of several studies. Universal functions, generally, are independent of the wall temperature of particular problems. A series approach, such as the method used and reviewed here, is one of a few methods used to analyze nonsimilar boundary layer problems. Nonsimilar problems refer to when and where the governing equations and boundary conditions cannot both be expressed in terms of a single variable or parameter. Other notable methods include approximate integral methods and numerical modeling using finite difference and finite element methods. A cogent, rapidly converging, yet simple procedure that makes use of universal functions for calculating boundary layer transfer was first proposed by Merk, as part of his PhD studies [22] at Shell laboratory in Amsterdam (1958) and later corrected and validated by Chao and Fagbenle [23,24]. The term universal functions also implies the functions of dimensionless variables that are independent of a particular problem. That is, the basis of the method is to make the solution depend on certain tabulated universal functions independent of the geometry/body section. Universal functions continue to be fundamental in many methods based on boundary layer theory. Several other types of universal functions are widely used in both laminar and turbulent boundary layer flows to solve nonisothermal and conjugate heat transfer problems.

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Applied to metrology and planetary boundary layers, universal functions of boundary layer theory were also considered in a review by Dorfman [25]. Before the development of Merk’s methodology, the convergence of previous power law and parametric series methods was slow. However, the method has been well established as being rapid in convergence (perturbation solutions converge rapidly), accurate, and powerful, which are key factors in its effectiveness. The method is theoretically sound and convenient, particularly because its validity does not depend on ad hoc assumptions about the nature of the boundary layer. Even the use of one term in the expansion of the series provides a very good approximation to the surface coefficients of heat and mass transfer and shows good agreement with previous experimental and theoretical works [26,27]. In fact, in [27], 15 methods for predicting laminar heat transfer were evaluated, and the MCF method was found to be accurate to within 1–3% of experimental data. For example, for the case of a circular cylinder, the method effectively predicted heat transfer over the first 70% of the cylinder, which agreed exceedingly well with experiment, though the method broke down, as can be expected, as separation was approached at about 80% of the cylinder. Notably, boundary layer theory is not amenable to separation problems. However, the well-known Goldstein singularity serves as an indicator of separation at or near the point where it occurs [28]. The methodology is inherently a perturbation series. Perturbation works well when some parameter is either large or small. We note that an asymptotic series need not be convergent and a convergent series need not be asymptotic. Convergence simply defines a unique limiting sum; this, however, does not give any information on the rate of convergence or how well the sum of a fixed number of terms approximates the limit. Indeed, an asymptotic expansion can be of more value than a convergent expansion. Perturbation further enhances the accuracy of the method throughout the range of all the parameters involved. Aesthetically, the Merk–Chao–Fagbenle procedure is more acceptable than other methods reported in the literature [29]. The use of this method consists of expanding the solution to a problem in a series form in one of the independent variables. The dependence on other independent variables remains unspecified, thus allowing the coefficients of the series to be functions of those variables. The transformations occur such that the equations for velocity, temperature, and concentration functions take a form in which asymptotic expansion (matching of the boundary layer equations (inner) to the inviscid/potential flow (outer) solutions) can be applied. The variable within which the expansion takes place, and the manner in which the series is chosen must be right, which is fundamental to obtaining an accurate solution by using the very first term of the series. Thus, the chief virtues of the method are as follows: – The rapid rate of convergence and; – After the first term, which itself is qualitatively accurate, successive approximations/terms of the series are corrective (aiding in understanding and validating the first term) and provide a more accurate solution to the problem. The second virtue is particularly important in light of several low-quality works that have saturated the literature in recent past. The second virtue can be used to verify the correctness of such works through the utilization of a series-based method. We note that as much insight (compared to other non-series methods) or more can be gained physically through the utilization of a series approximation.

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Now one may say that this is a dated methodology. Well, we argue that there is nothing dated about the principles of conservation of mass, momentum, and energy equations for continua or the Navier–Stokes equations (or any of their simplified forms) for viscous flows as they enjoy continued use in every sphere of life, and to treat thermal–fluid boundary layer problems. More so, Navier–Stokes-based equations are fundamental to all fluid–thermal–mass problems. The present-day boundary layer literature is full of tricks, incorrect nondimensionalization, arbitrary scaling, erroneous results etc. The MCF method, on the other hand, is free from such tricks and, in many regards, does not allow for it, hence it is a disciplined methodology applied for varying boundary layer problems. Several works have applied and assessed the performance of the Merk series after its modification by Chao and Fagbenle [30–37]. Most methods in the relevant literature are accurate in most engineering applications by using the Merk–Chao–Fagbenle method [38]. Though initially applied in the study of Newtonian fluids, the method has also been applied to non-Newtonian power-law fluids [38–44]. In [45–47], the authors conclude that “the convergence of the Merk–Chao series is excellent.” The authors in [48] recently extended the work of [43] to include the effects of viscous dissipation. The method has also been applied to rotating bodies, as examined in [49–51]. Regarding the effects of viscous dissipation in mixed convection flow for uniform surface heat flux and uniform wall temperature, the methodology was also employed to obtain heat transfer rate and skin friction coefficient in [52]. Other notable applications of the method are in [53–58]. The method is as suitable for nonsimilar flows as it is for similar flows [54,55,29], which constitute an important feature that is also partly responsible for the success of the method. That is, similarity solutions are also obtainable. It is a “wedge” method for obtaining locally similar solutions of the boundary layer equations. For a similarity solution, the method would yield the first-order or first-term approximation. In this regard, the structure of the similarity solution depends on the exact functional dependence of wall profiles of velocity, temperature, and concentration on the abscissa coordinate. Nonsimilar flows are particularly intriguing. Merk was indeed the first to propose an asymptotic expansion to account for boundary layer nonsimilarity [59]. The essence of applying perturbation indicates that the problem being solved is of a nonsimilar character. There are known problems for which similarity solution does not exist such as boundary layer flow over a flat plate with suction or blowing, flow over a sphere or circular cylinder. Similar solutions are generally possible for the entire region of accelerated and retarded flows whenever the external velocity is proportional to a power of the distance from the leading edge of the flow, in what is known as powerlaw or “wedge” flows, when, Uo ∼ x m , and where m is related to the wedge angle and illustrates what happens when the flow accelerates (m > 0) or decelerates (m < 0). Another way to put this is that whenever the potential flow outside the boundary layer is of the form Uo ∼ x m or Uo ∼ eαx where α is positive, the velocity profiles through the boundary layer at different values of x are geometrically similar.1 With regards to dynamic similarity, as long as the Re number is preserved, and regardless of the fluid 1 Goldstein, H (1939). Proc. Camb. Phil. Soc. 35, 338.

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type, the solutions of the flows past similar boundaries are identical in form. The characteristic Blasius solution corresponds to m = 0, while for Hiemenz solution, m = 1. In similar flows, the external velocity varies with the surface distance x or Uo = Cx m , where C is a constant, and where the boundary conditions are independent of x. In nonsimilar flows, the external velocity rarely varies according to this relation; instead, they are functions of x and y. The surface boundary conditions might not satisfy the requirements of similarity even if the external velocity does. Thus, nonsimilar flows are physically and technologically important, occur more frequently in daily life, and are more useful than similar flows. Nonsimilarity and high-order terms arise due to the form of the stream velocity, external vorticity, curvature of the body, surface mass transfer effects, induced pressure gradients, or a combination of these and other effects. Nonsimilar flows are situations in which the problem does not fully admit a similarity solution, leading to the need for perturbation or other approaches to obtain solutions. However, regions of such practical flows are regarded as “locally similar,” such as near the forward stagnation point of blunt bodies. Thus, the considerable effort expended on similar flows, which is evident in the literature, is worthwhile. Nonetheless, it is noteworthy that many boundary layer problems cannot be solved by the assumption of flow similarity. However, similar flow problems are predominantly studied, which is evident in the literature, because they generally involve the least mathematical complexity. That is, mathematical simplifications, and indeed tricks, aid in solving “similar” flow problems. Thus, this work observed Rogers’ comments in his book [60] regarding issues of accuracy and the computational difficulties associated with the MCF methodology; the authors here disagree with this assertion and believe that it indicates a poorly researched and understood conclusion of the methodology that is not supported elsewhere. In fact, the opposite of his conclusions has been elucidated and supported in several studies of the methodology, as cited in this work. The local nonsimilarity method, which was introduced by Sparrow et al. [61], has foundations in the Merk–Chao–Fagbenle method, with the exception that they developed their approximation by introducing a “g” function (defined as g {ξ, η} = ∂f ∂ξ ). However, considering other local similarity methods, the Sparrow method is locally autonomous. Furthermore, the recently developed semianalytical/numerical homotopy analysis/perturbation method (HAM) [62] is advanced as an alternative to perturbation methods. HAM is a generalized Taylor-series method. It is interestingly unique, whereby the inherent advantage is in the direct applicability to both linear and weakly and strongly nonlinear equations without requiring linearization, discretization, or perturbation (i.e., dependent on a large or small parameter). The other advantage is that the series of the method is convergent. However, the HAM approach is complex in expression, and does not yield any new boundary layer insights. Nonetheless, the HAM approach has been improved by many researchers, which is evidenced in the literature. However, HAM based methods can be said to reestablish the usefulness and strengths of series-based expansion methods. Suffice it to say the perturbation methods, regardless of whether they rely on a small parameter or not, remain the de facto approach to the great world of fluid motion

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problems [63]. The key elements in the choice of laminar boundary layer methods can be stated as follows: – The magnitude or how much information is desired from the problem; – The precision or accuracy of results obtained, i.e., is 10% or 15% adequate or is there a desire to obtain solutions that are accurate to within 1–3% of other “exact” solutions or experimental results and; – Simplicity of expression and speed of convergence of results. It is noteworthy that some authors refer to the methodology as the “Merk–Chao series,” while others refer to it as the “Chao–Fagbenle series.” However, the Merk method of 1959 did not garner much attention or use until 1973 and 1974 when Chao and Fagbenle published the modified and validated method in which the error in the governing differential equations was corrected for the universal functions associated with the method. Therefore, it can correctly be described as “the modified Merk method/series of Chao and Fagbenle”, or simply the Merk–Chao–Fagbenle (MCF) method. Regarding the latter designation, due credit is given to the original author and to the subsequent authors’ whose significant modification led to an increased use of the methodology. Thus, the latter designation is short and distinctive. Although we recognize that Merk’s work emanated from a refinement of the procedure by Meksyn [64], the latter’s method is now obsolete. Interested readers would nonetheless gain some fundamental insights and understanding from Meksyn’s book [64]. Several other published works and PhD dissertations [65,66,86] have employed the methodology in one capacity or another, and the results have been found to compare favorably with previous theoretical and experimental studies. To present a logical exposition of this methodology, the basic boundary layer equations are first expressed and various classical applications are discussed thereafter, such as crossflow over circular cylinders and Sear’s airfoil, where unpublished data using the MCF procedure is presented. All of the applications discussed continue to be of importance in practical engineering. Table 5.1 shows a series of historical events leading to the development of the MCF procedure. The table is especially based on timeline of events rather than year of publication. The table also demonstrates the challenges associated with keeping abreast of research developments and timely accessibility to publications, some of which still persists today in the digital age.

5.3

Mathematical formulations of the Merk–Chao–Fagbenle method

The fundamental governing equations of the MCF procedure are similar to others in the literature for forced convection, which can be represented as ∂ ∂ (ru) + (rv) = 0, ∂x ∂y ∂u ∂u ∂ 2u ∂u +v =u +v 2, u ∂x ∂y ∂x ∂y

(5.1) (5.2)

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Table 5.1 Historical events of the MCF procedure. Author and Year Blasius (1908) von Mises (1927) Falkner–Skan (1931) Mangler (1943)

Meksyn (1947–1948) Gortler (1952–1957) Merk (1959)

Smith (1963)

Evans(1968)

Bush (1964)

Remarks

References

Initiated the concept of asymptotic series solutions in boundary layers A change of variables procedure converting thin differential force momentum equations into transient thermal conduction with variable thermal diffusivity Extended the Blasius-type series technique to wedge flows

[18]

Transformation of thin axisymmetrical boundary layer flows by von Mises change of variables procedure. Applied to a special case for the transformation of rotationally symmetric boundary layers A transformation of variables that leads to exact solutions of wedge flows and good approximation for arbitrary shapes Proposed a series solution, which would converge more rapidly than the Blasius-type series in (β, η) coordinates Further developed Gortlers technique, leading to the first to asymptotically expand the full boundary layer equations in terms of a small parameter. However, the author inadvertently missed an important term in the series solution. Some principal virtues of the technique are (i) the rapid rate of convergence and (ii) successive series solutions are corrective Applies features of the Merk technique for finite difference solution in (β, η) coordinates and used for test case examples only A book predominantly devoted to nonsimilar boundary layer flows. Performs a different perturbation expansion compared to that of Merk; also missing an important term similar to Merk (repeating Merk’s error) and solved numerically Bush’s work comes after Evans’ because Evans was unaware of Bush’s work before publishing his book. Bush published a two-page paper correcting the missing term of Merk & Evans, and changed the series expansion parameter (2ξd/dξ). However, the author obtained solutions by asymptotically expanding an approximate solution rather than obtaining an approximate solution to the exact first-order equation.

[68]

[67]

[3]

[64]

[21] [22]

[69]

[29]

[70]

continued on next page

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103

Table 5.1 (continued) Author and Year Chao & Fagbenle (1973–1974)

Sparrow, Quack & Boerner (1970)

Remarks

References

With both authors unaware of Bush’s work, until completing their own work, they corrected the missing term in Merk’s work, changed series expansion parameter, fully appraised Merks technique to four terms, numerical analysis and application to various geometries. Several works have employed the technique since its full appraisal [refs]. Thereafter referred to in literature as the Merk–Chao–Fagbenle (MCF) method. The work of Bush and Chao & Fagbenle are similar A local non-similarity method, solved locally for any value of Though published prior to the full appraisal of the MCF method in 1974, the technique remarkably produces a first order approximation of the MCF method

[23,24]

[61]

∂T ∂T ∂ 2T +v =α 2 , ∂x ∂y ∂y ∂P = 0, where P = P (x) , ∂y

u

(5.3) (5.4)

with the associated boundary conditions u (x, 0) = v (x, 0) = 0 (noslip) , T (x, 0) = Tw (constant, Dirichlet) . u (x, ∞) = U (x) , T (x, ∞) = T∞ (constant) .

(5.5)

The reader will note that at infinity, we do not enforce v per singular perturbation theory. From the continuity equation (5.1), r is the radial distance from a surface element to the axis of symmetry and is a function of x only. For nonrotationally symmetric (or two-dimensional flows) bodies, r is to be dropped (i.e., r = L). Forced convection steady Dirichlet boundary layer transfer was first considered towards correcting the Merk series as demonstrated in [23,24]. Further considerations and results will be discussed here in what follows.

5.3.1 Forced convection over circular cylinder in crossflow A considerable amount of theoretical analysis for the two-dimensional boundary layer flow over the front portion of a circular cylinder can be found in the literature. Also, a limited amount of experimental data is available. Hence, the circular cylinder in crossflow is a nonsimilar flow that provides good opportunity for ascertaining the usefulness as well as limitations of the MCF method. A basic input to the MCF procedure is the information on the mainstream velocity distribution. The potential velocity distribution considered here is 2x U = 2 sin , (5.6) U∞ D

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Table 5.2 Coefficients values in Eq. (5.7). Author Hiemenz [19]

Re 18,500

A1 3.6314

A3 2.1709

A5 1.5144

Sogin & Subramanian [73]

122,000

3.6400

3.2000

0

Schmidt & Wenner [74] (adjusted by Eckert [75])

170, 000

3.6310

3.2750

0.1679

since the finite difference solution of [71] and the difference-differential solution of [72] are based on such distribution. To ensure a meaningful comparison with experiment, it is imperative that such spurious effects as flow instability, free stream turbulence, etc., do not come into play in the measured data. Several semiempirical representations of the measured mainstream velocity distributions have been proposed; they are of the following form: x  x 3  x 5 U = A1 − A5 . − A3 U∞ D D D

(5.7)

The coefficients A1 , A3 , and A5 as recommended by various investigators are listed in Table 5.2. Evans [29] posited following an examination of the behavior of the local heat and mass transfer rates in the region of adverse pressure gradient near or just before separation that when Re was greater than about 130,000, instability seemed to set in. Here, we considered all three cases. A comparison of these empirical representations, including the ideal potential flow distribution, is made in Table 5.3. It is observed that there is very minimal difference among the experimental curves for  ranging up to about 30◦ , beyond which the ratio UU∞ for the higher Re becomes less. Fig. 5.1 shows the variations of ξ and  with  while Fig. 5.2 exhibits the variations of 2ξ d dξ and 2

4ξ 2 ddξ . It is imperative to note that the pressure minimum occurs when  becomes zero. A common feature exhibited by the four curves in each plot is that, at small , they are slowly varying functions of . As the region of minimum pressure is approached, they become very sensitive to small changes in . When the latter occurs, 2 d2 and when absolute magnitudes of 2ξ d dξ and 4ξ dξ are large, the reliability of the MCF method is no longer certain.

5.3.2 Skin friction, displacement thickness, momentum thickness, and velocity distribution In Fig. 5.3, the calculated results for skin friction, displacement and momentum thicknesses when the external   velocity is described by the potential flow theory are shown. In the figure, cf =

μ

∂u ∂y y=0 1 2 2 ρU∞

and Re =

U∞ D ν . The skin friction data shows good agree-

ment with that calculated from the Blasius series as well as with the results of Terrill

The Merk–Chao–Fagbenle method for laminar boundary layer analysis

Table 5.3 Mainstream velocity distribution, UU , for circular cylinder in crossflow. ∞

Figure 5.1 Variation of ξ and  over the front portion of a circular cylinder in cross flow.

105

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2

2d  Figure 5.2 Variations of 2ξ d dξ and 4ξ dξ over the front portion of a circular cylinder in crossflow.

[72] and of Schonauer [71] for  up to about 85◦ beyond which the present results become unreliable. The Blasius series data were calculated from     x 3 u  x 5 u 1 x 3 2 5 2 3/2 f3 (0) + 6 f5 (0) , (5.8) cf Re1/2 = u1 f12 (0) + 4 2 L L u1 L u1 where u1 , u3 , u5 , etc., are the coefficients in the symmetrical external velocity distribution: x   x 3  x 5 U = u1 + u5 + ..., (5.9) + u3 U∞ L L L and f12 (0), f32 (0), f52 (0) are wall derivatives of the associated universal functions. These wall derivatives have been evaluated by Tifford [76] and a tabulation was reproduced in Schlichting’s text [77]. For a circular cylinder, L = D. The dimensionless displacement thickness results of the MCF method agree well with those by Terrill and by [71] in the range 0 <  < 65◦ , while the dimensionless momentum thickness results agree well up to about  = 85◦ . The primary reason for the deterioration in the accuracy of the MCF procedure for regions far from the front

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107

Figure 5.3 Skin friction, displacement thickness and momentum thickness over a cylinder in crossflow based on potential velocity distribution.

stagnation point is the rapid increase in the magnitudes of the coefficients 2ξ d dξ ,  2 2 4ξ 2 ddξ , and 2ξ d in such regions. This gives rise to the semidivergent sedξ ries of insufficient number of terms and, consequently, the results become uncertain. Skin friction results are shown in Fig. 5.4, along with results obtained from the 4-term MCF and Blasius series. In this case, data obtained from the MCF series are in good agreement with the Blasius series method and with the results of [69] for locations up to  = 35◦ downstream of the forward stagnation point. Beyond this point, the Blasius series exhibits significant discrepancy. Nonetheless, the MCF method continues to agree well with the results of [69] until  = 70◦ . For locations between  = 50◦ and  = 70◦ , there is a maximum deviation of about 3.3%. For  > 700 , the MCF series also becomes unreliable for reasons already stated. The Blasius series yields results of skin friction which are very good for all locations on the front half of the cylinder when the external stream velocity distribution is described by the potential theory, whereas the method yields good results only up to about  = 300 forward of the stagnation point when measured profiles are employed. The calculated skin friction is sensitive to the coefficients used in the polynomial representation of the external velocity distribution and any inaccuracies in its description appear to become magnified especially for regions far from the stagnation point. Included in Fig. 5.4 are the results for displacement and momentum thickness calculated

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 5.4 Skin friction, displacement thickness and momentum thickness over a cylinder in crossflow based on Hiemenz velocity distribution.

from the MCF procedure. Such data were not reported by [69] and hence no comparison could be made. The inferiority of Evans’ 2-term results as compared to the 2-term results of the MCF procedure is demonstrated in Tables 5.4 and 5.5. In these tables, and in Table 5.6, the results from [76] and [71] were evaluated from interpolation curves based on their 5-significant digit tabulation. In Table 5.5, the data from [69] were read directly from their plots. Since that plot was drawn at a compressed scale, the fourth digit of the data listed is doubtful. For  > 650 , even the third digit is uncertain because the curve drops very rapidly in that region. The MCF methodology thus has a wider range of applicability when the external flow is described by the potential theory than when described by measured profiles. The velocity profiles at the front stagnation point and for  = 29.47◦ and 59.25◦ are displayed in Fig. 5.5 when the external stream velocity distribution is that of [74]. It is interesting to note that the profiles at  = 0◦ and 29.47◦ are very similar. The velocity profiles in the boundary layer for potential external stream have also been calculated. At  = 0◦ and 60◦ , they agree with the curves shown in Schlichting’s book [77] within reading accuracy. However, at  = 90◦ , the MCF series becomes divergent and the results uncertain.

The Merk–Chao–Fagbenle method for laminar boundary layer analysis

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Table 5.4 Skin friction, 12 cf Re1/2 , for flow over a circular cylinder with the mainstream velocity according to potential theory.

Table 5.5 Skin friction, 12 cf Re1/2 , for flow over a circular cylinder with the mainstream velocity according to the Hiemenz velocity profile.

5.3.3 Heat transfer results In Fig. 5.6, for P r = 0.7, the heat transfer results of the MCF method, those of [78], and those of the integral method solution of [75] are presented, when the external flow is represented by the Schmidt and Wenner velocity distribution [79]. The curve labeled Eckert is based on the following equation [75]: NuRe−1/2 = 0.9449 − 0.7695

 x 2 D

− 0.3474

 x 4 D

.

(5.10)

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Applications of Heat, Mass and Fluid Boundary Layers

Table 5.6 Displacement and momentum thickness results for flow over a circular cylinder with the mainstream velocity according to potential theory.

Figure 5.5 Velocity distribution in boundary layer over a cylinder in crossflow.

Excellent agreement is indicated between the MCF method and those of [78] and [75] for  < 550 , beyond which the discrepancy becomes progressively larger. The heat transfer results of the MCF procedure were also compared with those of [80] for the case of potential external velocity distribution and for P r = 0.7, 0.1, and 0.01.

The Merk–Chao–Fagbenle method for laminar boundary layer analysis

111

Figure 5.6 Heat transfer over the forward portion of an isothermal cylinder in crossflow.

These are displayed in Fig. 5.7. This figure again demonstrates the influence of the  2 d 2 d2 coefficients 2ξ d on the accuracy of the MCF procedure. For dξ , 4ξ dξ , and 2ξ dξ low P r numbers, the wall temperature derivative for i = 1, 2, and 3 are relatively small compared to θ01 (, 0) and hence dampen the effect of these coefficients as they attain large values in regions far from the forward stagnation point. The results here start to deviate from the Frossling–Newman curve at   65◦ for P r = 0.7 and at   80◦ for P r = 0.01. Table 5.7 shows the heat transfer data for further comparison of the MCF method, the method of [78,80] and of [75]. The external velocity distribution is that of [74]. Included in the table are the 2-term results of Evans [29] which once again point to errors in his analysis.

5.3.4 Comparison with experimental heat transfer data While there is reasonably good agreement among the various theoretical methods for predicting heat transfer from a circular cylinder in crossflow, particularly in regions not far from the forward stagnation point, the same is not true when comparison is made for theoretical results and experimental data. Close to the forward stagnation point, measured values of the heat transfer group NuRe−1/2 are usually higher than theoretical prediction, while for  > 60o , but prior to separation, measured values are in many cases lower than the theoretical prediction. Since there is excellent agreement

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Figure 5.7 Heat transfer over the front portion of an isothermal cylinder in crossflow for three P r numbers.

among the various theoretical methods for the forward stagnation point, it is natural to seek the reason for the discrepancy between experiment and theory. Three major reasons were advanced in [29] which include (i) the effects of instability and of turbulence, (ii) the choice of suitable mean values for the transport properties of the medium in the reduction of the experimental heat transfer data, and (iii) the inaccuracies in the measured distribution of the external mainstream velocity around the cylinder. The effects of instability and of turbulence can hardly explain the 5% deviation between theory and experiment at the forward stagnation point since such deviation remains even in the low turbulence and low Re number experiments (15,550, 39,800, and 64,450). The authors in [79] conducted their experiments with the cylinder wall maintained at 100◦ C and the free stream at 24◦ C. It appears that they evaluated the transport and other properties at the arithmetic mean of the two temperatures. Table 5.8 shows changes in these properties in the said temperature range; some are more pronounced

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113

Table 5.7 Nusselt number, NuRe−1/2 , for flow over a circular cylinder with mainstream velocity according to the Schmidt and Wenner velocity profiles, P r = 0.7.

Table 5.8 Property changes over temperature range 24◦ C to 100◦ C for air at 1 atmosphere. Property % change

ρ −21

1/2

μ 26

ν 60

k 19

cp 0.4

α 51

(ν 1/2 )/(k) 6

than others, although ν k and hence NuRe−1/2 undergoes only a 6% increase. The percentage changes listed in the table are with reference to the values at 24◦ C. It should be remarked that most theoretical predictions on heat transfer, including the present results, are based on the assumption of constant fluid properties. As Table 5.8 shows, this would not be the case for an experimental examination such as that of [79]. In the absence of an exact solution, Evans [29] argued that the choice of a multiplying factor to make experimental data agree with theoretical predictions at the front stagnation point is in order. The factor 0.9551 applied to the mean of eight sets of heat transfer data of [79] happens to accomplish this if the velocity distribution according to Eq. (5.7) above is used in the theoretical analysis. The experimental data adjusted by Evans in this manner is therefore taken as the basis of comparison between theory and experiment in this regard. The results are summarized in Table 5.9. The Nusselt number is defined as N u = hD/k, and the factor 0.9551 was applied to both the maximum and the minimum as well as mean of eight sets of experimental data. An important point to observe is the significant difference between the maximum and minimum. The data selected in this study by Evans were in the Re number range of 8290 ≤ Re ≤ 101, 300, thus the flow was surely laminar. In view of this, any theoreti-

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Table 5.9 Adjusted values of experimentally determined heat transfer parameter, NuRe−1/2 , for circular cylinder in crossflow. The factor 0.9551 was applied to the original experimental data of [E. Schmidt and K. Wenner. Heat transfer on the circumference of a heated cylinder in transverse flow, NACA TM 1050, 1943]; Re = U∞D ν .

cal prediction that falls within the maximum–minimum boundaries can be considered acceptable. The heat transfer results here, based on four external mainstream velocity distributions, are thus compared with the adjusted maximum and minimum of the experimental data as depicted in Fig. 5.8. The predicted curves are computed from the 4-term MCF series, with the use of Euler transformation in summing the series where required. As is revealed in Table 5.7, the application of Euler transformation yields heat transfer rates which are lower than the results of Frossling and Eckert. These “low” values, however, are in good agreement with the adjusted experimental data. The predictions based on the [79] velocity profiles and on the velocity profiles from [73]exhibit the best agreement. Predictions based on potential velocity distribution grossly overestimate the heat transfer rates, drawing attention to the known fact that a reliable description of the external velocity is a prerequisite, if the accurate prediction of the heat transfer coefficient is desired.

5.3.5 Calculated temperature profiles The temperature profiles for P r = 0.7, based on the [79] velocity distribution, are ploty√ ted in Fig. 5.9(A) for three angular positions with D Re as the transverse coordinate. As expected, the temperature gradient becomes less steep as one moves downstream from the forward stagnation point. In Fig. 5.9(B), the data was replotted using η as the coordinate. In so doing, the data are, to a very large extent, compressed to fall on a single curve, suggesting that, for 0 <  < 60◦ , η is very close to the similarity variable. To illustrate the influence of P r on the temperature distribution in the boundary layer, three groups of temperature profiles for P r = 10, 0.1, and 0.01 have been calculated using universal functions given in [23]. The results, based also on the velocity distribution from [79], are shown in Fig. 5.10. As is well known, the thermal boundary layer thickness increases with a decrease in Prandtl number as is clearly shown in Fig. 5.10.

5.3.6 Comparison with experimental mass transfer data The velocity profiles in [73] measured the local mass transfer rates from a 4.2 in diameter naphthalene cylinder to air at ordinary temperature and pressure, flowing

The Merk–Chao–Fagbenle method for laminar boundary layer analysis

115

Figure 5.8 Comparison of predicted heat transfer parameter, NuRe−1/2 , with experiments. Calculation based on the MCF series using several external velocity distributions.

normal to the cylinder axis. The measurements were made at Re =122,000, 218,000, and 342,000. Their pressure drag data clearly revealed that the flow was subcritical at the lowest Re number and was in the midst of the critical zone at the highest Re. At Re =218,000, the pressure drag coefficient began to exhibit a reduction. To effect a meaningful comparison with the MCF theory, we consider the data for the two lower Re numbers. These authors also reported data when the turbulence intensity in the free stream was artificially raised by the use of a grid. For the purpose here, only the portion of data obtained under a turbulence level of about 1% was used for comparison. Under the conditions of the experiment in [73], the usual boundary layer assumptions are valid. Since the naphthalene vapor concentration was of the order of 0.1%, its effect on the thermodynamic and transport properties of air could be ignored. Also, the frictional dissipation and the normal velocity at the surface of the cylinder were both negligibly small. Consequently, the governing differential equations for the concentra-

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 5.9 Temperature distribution in boundary layers over an isothermal cylinder in crossflow (P r = 0.7, Schmidt and Wenner velocity distribution).

Figure 5.10 Effect of P r number on temperature distribution in boundary layers over an isothermal cylinder in crossflow.

tion boundary layer and its associated boundary conditions are completely analogous to those for the thermal boundary layer. Hence, the product of the Sherwood number

The Merk–Chao–Fagbenle method for laminar boundary layer analysis

117

Figure 5.11 Mass transfer from naphthalene cylinder in airflow. Comparison between experimental data and predictions of the MCF series.

and the reciprocal of the square root of the Reynolds number is given as ShRe−1/2 =

 r U d 1 θ (, 0; Sc) (2ξ )−1/2 θ01 (, 0; Sc) + 2ξ L U∞ dξ 1 

2 d 2 1 2d  1 +4ξ θ3 (, 0; Sc) + . . . , θ (, 0; Sc) + 2ξ dξ 2 dξ (5.11)

where Sh = bD D , with b being the mass transfer coefficient and D the diffusion coefficient of naphthalene vapor in air. 2 d2 The MCF variables, ξ , , 2ξ d dξ , and 4ξ dξ , equally correspond to the experimental outer stream velocity distribution at Re = 1.22x105 from [73]. Hence a straightforward application of Eq. (5.11) leads immediately to the information on the mass transfer parameter ShRe−1/2 . The results are depicted in Fig. 5.11. A tabulation (Table 5.10) is also included to illustrate the behavior of convergence of the MCF series. As in the case of corresponding heat transfer calculations, the Euler transformation has to be used for locations where  exceeds 55◦ .

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Applications of Heat, Mass and Fluid Boundary Layers

Table 5.10 Convective mass transfer from a circular cylinder. Sherwood number, ShRe−1/2 . Schmidt number = 1/0.395. Velocity profile according to [73] for Re = −122, 000.

At the lower Reynolds number of 1.22 × 105 , the experimental data are about 5% lower than theoretical prediction, while at the higher Re = 2.18 × 105 , they are about 2% higher at the front stagnation point. In the region 0 ≤  ≤ 70◦ , the effect of instability clearly shows up in the data. The location of minimum pressure was approximately at  = 71◦ . Notably, the authors in [73] also compared their data with the theoretical prediction of the MCF series, employing only one term in the series. They concluded that the calculations after the MCF series are reliable from the point of stagnation all the way to the predicted point of separation. The present results indicate that the close agreement described by [73] is fortuitous. Close to the point of 2 d2 separation, 2ξ d dξ , 4ξ dξ , etc., are not small and, accordingly, the use of just one term in MCF series cannot be justified in general.

The Merk–Chao–Fagbenle method for laminar boundary layer analysis

119

2

2d  Figure 5.12 Variation of ξ , , 2ξ d dξ , and 4ξ dξ over a sphere. Outer stream velocity distribution based

on measurements of [81].

5.4

Forced convection flow over a sphere

This example offers an opportunity to also examine the application of the MCF procedure to a rotationally symmetrical boundary layer. For comparison both experimental and theoretical Nusselt number results of [78] for P r = 1/0.395 = 2.56 are available. Furthermore, Merk himself [22] had also selected this example for study and obtained good agreement between his one-term prediction and data in [78]. With the availability of three additional terms in the MCF series, it provides an insight of the reason for the exceptionally good on-term result for this case. For a sphere of diameter, D, we have 1 1 2x r = sin ϑ = sin , D 2 2 D

(5.12)

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Applications of Heat, Mass and Fluid Boundary Layers

Table 5.11 Nusselt number, N uRe−1/2 , for flow over a sphere with mainstream velocity according to [81]; P r = 1/0.395.

where ϑ is the polar angle measured from the forward stagnation point. In the analysis of [78], the mainstream velocity distribution was deduced from the measured static pressure data reported by [81] for Re = 157, 200. It is represented by x  x 5  x 7  x 3 U =3 + 4.7391 − 5.4181 . − 3.4966 U∞ D D D D

(5.13)

Using Eq. (5.13) and for P r = 2.56, the author of [78] obtained from his general series solution the following expression for the local heat transfer coefficient: N uRe−1/2 = 1.8615 − 2.1477

 x 2 D

+ 2.4609

 x 4 D

+ ...,

(5.14)

which will later be compared with the results of the MCF procedure here. Using the relationships given by Eqs. (5.12) and (5.13), the variation of ξ , , 2ξ d dξ , and 2

4ξ 2 ddξ with angular displacement ϑ can be easily evaluated. The results are shown in Fig. 5.12. If the corresponding plots of the circular cylinder (Figs. 5.1 and 5.2) are examined, two dissimilarities are obvious. First, we note that the range of  is roughly halved, since at the forward stagnation point, it assumes the value of 0.5 instead of 1, 2 d2 as in the case of the cylinder. Second, the absolute magnitude of 2ξ d dξ and 4ξ dξ are approximately 1/3 to 1/5 as large. The latter fact is a sure indication that the sum of the various series associated with the MCF method will be relatively more dominated by the first term. An examination of the data given in Table 5.11 readily confirms this fact. This is the main reason that Merk [22] obtained remarkably good results when only the first term of the series was used. In physical terms, the boundary layer flow is closer to locally similar conditions. The presently computed heat transfer results, together with those evaluated from Frossling’s theory [78], are plotted in Fig. 5.13. Included are the experimental data reported by [78]. For ϑ less than about 55◦ , there is a general agreement among

The Merk–Chao–Fagbenle method for laminar boundary layer analysis

121

Figure 5.13 Heat transfer over an isothermal sphere.

the experimental data, the theoretical predictions of [78], and the MCF procedure. As ϑ increases beyond 55◦ , the expression of [78] in Eq. (5.14) above exhibits significant deviation from the data while the predictions according to the MCF series consistently follows the experiments even when ϑ attains 76.3◦ at which point  has already become negative. In making the foregoing observation, one should be mindful of the fact that the experimental data were taken at a Reynolds number of 1060, for which Eq. (5.13) might not be a good representation for the outer stream velocity distribution. Fig. 5.13 shows the heat transfer profile over an isothermal sphere, Fig. 5.14 illustrates skin friction over a sphere, Fig. 5.15 presents velocity distribution over a sphere, while Fig. 5.16 exhibits the temperature distribution over a sphere.

5.5

Forced convection over sears’ airfoils

Sears [82] analyzed the symmetrical boundary layer flow of a certain class of airfoils having external potential velocity of the form   U = a χ − χ3 , U∞

(5.15)

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 5.14 Skin friction over an isothermal sphere.

Figure 5.15 Velocity distribution in boundary layer over an isothermal sphere.

where χ = Lx , with L being the airfoil chord length. The dimensionless constant a depends, in general, on the airfoil shape. For most airfoils, Umax is approximately 1.2U∞ at zero lift and hence a = 3. Goland [83] examined the temper-

The Merk–Chao–Fagbenle method for laminar boundary layer analysis

123

Figure 5.16 Temperature distribution in boundary layer over an isothermal sphere.

ature distribution in the boundary layer of such flows for the Prandtl number of unity, when the surface was maintained at a uniform temperature and when the viscous heating was neglected. Here we compare the local velocity distribution in the boundary layer and the skin friction drag as obtained by Sears with those of the MCF procedure. The heat transfer results of [83] are also compared and discussed. Using Eq. (5.15), the following can be readily established:

χ2 χ2 1− , (5.16) ξ =a 2 2

 −2 χ2  , 1 − 3χ 2 1 − χ 2 = 1− 2

(5.17)

 −4 χ2  d 2 , = −χ 1 − 3 + χ2 1 − χ2 2ξ dξ 2

(5.18)

4ξ

2  −6 χ2  , = −2χ 1 − 13 + 3χ 2 1 − χ 2 dξ 2

2d

2

4

(5.19)

which have been numerically evaluated and the results are shown in Fig. 5.17. Results for skin friction and surface heat transfer rates are shown in Figs. 5.18 and 5.19, respectively. In both figures, the theoretical location of flow separation is designated by S; it corresponds to χ = 0.69. When the outer stream velocity distribution can be described by a simple polynomial of distance measured from the front stagnation such

124

Applications of Heat, Mass and Fluid Boundary Layers

2

2d  Figure 5.17 Variation of ξ , , 2ξ d dξ , and 4ξ dξ over Sears’ airfoil.

as Eq. (5.15), the Blasius series can be expected to outperform the MCF series. To provide the reader with “convergence” rate of the MCF series, we have included Tables 5.12 and 5.13. It may also be worthy to note that at the front stagnation point of the Sears’ airfoil,  U  √ U∞ lim (5.20) = a. 1 ξ →∞ (2ξ ) 2

5.6 Consideration in forced convection (nonisothermal) boundary layer transfer Perhaps the first extension of the methodology to the case of nonisothermal surfaces was considered in [54]. The author further considered the cases of similar and nonsimilar boundary layers applied to flow around a circular cylinder and past a flat plate. Fagbenle [26], apparently unaware of Sano’s work [54], reexamined nonisothermal boundary layer flows in a paper presented at a conference in South Africa resulting in a 3-parameter set of universal functions akin to the 3-parameter

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Figure 5.18 Skin friction over Sears’ airfoil.

Figure 5.19 Heat transfer over Sears’ airfoil with uniform surface temperature.

MCF series developed by [32] and [35]. The work in [26] later formed the basis of the doctoral thesis of [65], where nonisothermal surfaces were fully considered. The key difference between the isothermal and nonisothermal cases is in the introduction of a wall temperature distribution parameter which is represented

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Table 5.12 12 cf Re−1/2 for flow over airfoils with the external velocity given by UU = ∞   x and Re = U∞ L . 3 χ − χ 3 where χ = L ν

Table 5.13 N uRe−1/2 for flow over airfoils with the external velocity given by UU = ∞   x and Re = U∞ L . 3 χ − χ 3 where χ = L ν

as γ=

2ξ d(Tw (x) − T∞ ) . Tw (x) − T∞ dξ

(5.21)

Another alternative approach to nonisothermality is to consider T (x, 0) = Tw (x) in the boundary condition or as the condition at the wall.

The Merk–Chao–Fagbenle method for laminar boundary layer analysis

127

5.7 Consideration in mixed convection boundary layer transfer The boundary layer equations for the mixed forced and free convection laminar flow are [84]: ∂ ∂ (ru) + (rv) = 0, ∂x ∂y

(5.22)

u

∂u ∂u ∂u ∂ 2u +v =u + gγβ (T − T∞ ) + v 2 , ∂x ∂y ∂x ∂y

(5.23)

u

∂T ∂ 2T ∂T +v =α 2 , ∂x ∂y ∂y

(5.24)

∂P = 0, where P = P (x) , ∂y

(5.25)

with the associated boundary conditions u (x, 0) = v (x, 0) = 0 (noslip) , T (x, 0) = Tw (constant, Dirichlet) , u (x, ∞) = U (x) , T (x, ∞) = T∞ (constant) .

(5.26)

In the above equations r = 1 for two-dimensional flows. The velocity u is the external velocity along the outer edge of the boundary layer assumed to be known from potential flow theory or from experiment. The above equations, besides incorporating all the usual assumptions for an incompressible laminar boundary layer, have the additional assumptions of small temperature differences and small pressure increases throughout the boundary layer, while variations in density are allowable only in producing buoyancy forces. Also, γ = + cos α where, for any location on the surface of a body, α is the angle between the tangent to the surface at that point and the gravitational direction. The above equations neglect the effect of the component of the buoyancy force normal to the surface of the body. The reader is referred to [84] for more penetrating insights.

5.8 Consideration for nanofluids and other extensions of the methodology Boundary layer solutions, often suited to subsonic, attached and shock-free flows, have always been important for understanding the behavior of fluids, and they have provided very useful qualitative information. To further develop the modified and

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validated Merk series procedure of Chao and Fagbenle which is a higher-order boundary layer methodology, extensions have been made towards the coupling of several paradigms toward the study of nanofluids, which are considered a new type of heat transfer fluid that may have vast industrial applications [85]. Prior to Prof. Fagbenle’s recent demise in June 2019, and together with the second author, we were working to fully consider the MCF procedure for the study of nanofluids, porous media, electromagnetism, stretching surfaces, and with entropy generation. The outcomes of the study will be made available in the near future and in the open literature.

5.9

Conclusion

In this review, the advantages and limitations of the MCF procedure have been made clear. In principle, the MCF procedure is highly accurate, theoretically sound, and much easier to use than most other methods. There is seldom any other procedure that could match its generality and, above all, its simplicity. The procedure is a conversion of the governing PDEs into infinite ODEs. The MCF procedure of boundary layers is a disciplined method that allows the researcher to think intelligently about fluid flow problems of practical interest. The application and extension of the Merk– Chao–Fagbenle method continues to be impressive. Potential future applications and extensions of the method can be expected to be profound.

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[81] A. Fage, cited in N. Frossling, “Evaporation, heat transfer, and velocity distribution in two-dimensional and rotationally symmetrical laminar boundary-layer flow”, NACA TM 1432, 1958, Aeron. Res. Comm., Rep. and Memo, 1766, 1937. [82] W.R. Sears, The boundary layer of yawed cylinders, J.A.S. 15 (1) (1948) 49–52. [83] L. Goland, A theoretical investigation of heat transfer in the laminar flow regions of airfoils, J.A.S. (1950) 436–440. [84] S. Aliu, R.O. Fagbenle, Laminar mixed convection boundary layer flow over twodimensional and axisymmetric bodies, in: ASME 2013 IMECE, San Diego California, USA, Nov. 15–21, 2013. [85] R. Saidur, K.Y. Leong, H.A. Mohammed, A review on applications and challenges of nanofluids, Renew. Sustain. Energy Rev. 15 (3) (2011) 1646–1668. [86] S. Aliu, Laminar Mixed Convection Boundary Layer Flow Over Two-Dimensional and Axisymmetric Bodies, PhD Thesis, Department of Mechanical Engineering, University of Benin, Benin City, Nigeria, 2014.

The spectral-homotopy analysis method (SHAM) for solutions of boundary layer problems

6

S.S. Motsa, Z.G. Makukula Department of Mathematics, University of Swaziland, Kwaluseni, Swaziland

6.1 Introduction The main idea behind the spectral homotopy analysis method is to improve on some of the limitations of the homotopy analysis method (HAM). Details of the HAM, however, are not presented here as they are well documented in the literature, including [1,2] amongst others. The main focus of the SHAM was to improve the initial guess used in the algorithm since it is an important ingredient of any iterative scheme. A poor initial guess may lead to poor convergence rates or even no convergence at all. The HAM restricts the choice of the initial solution to be expressed as a sum of the base functions, which are chosen to be conveniently integrable. Good complex initial guesses end up being avoided if they do not conform to the choice guide because integration of the higher-order deformation equations may be difficult or even impossible. The spectral homotopy analysis method numerically integrates the higher-order deformation equations using the Chebyshev pseudospectral collocation method. The numerical approach overcomes the limitations of the nature of the initial guess as it is easier to integrate numerically than analytically. The use of the best possible initial solution gives rise to more accurate solutions and good convergence rates of the iterative method. The success of the spectral homotopy analysis method has been documented in the literature in the works of Motsa et al. [3–10] amongst others. The method is described in detail and later applied to two problems in the sections that follow.

6.2 The spectral-homotopy analysis method (SHAM) The details of the method are given in this section. Consider a nonlinear equation of the form N [f (x)] = g(x),

(6.1)

subject to the boundary conditions B[f (x), f  (x), . . . ] = 0,

x ∈ [a, b],

Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00014-1 Copyright © 2020 Elsevier Ltd. All rights reserved.

(6.2)

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where N represents the nonlinear equation, g(x) is a source term of the equation, f (x) is an unknown function, x signifies an independent variable, and [a, b] is the domain of the problem. Eq. (6.1) is then divided into its linear and nonlinear parts as shown below L1 [f (x)] + N1 [f (x)] = g(x),

(6.3)

where L1 and N1 represent the linear and nonlinear parts of the equation, respectively. Unlike in HAM, the initial solution used is the solution of the equation L1 [f0 (x)] = g(x),

(6.4)

with boundary conditions B[f0 (x), f0 (x), . . . ] = 0,

x ∈ [a, b].

(6.5)

Note that in the original HAM, the initial guess is chosen to satisfy the boundary conditions and must be expressed as a sum of the basis functions. The solution to Eq. (6.4) might be complex but is generally a better choice compared to that chosen to satisfy the boundary conditions only. It is only if the solution to Eq. (6.4) is zero or does not exist that we choose the initial guess to satisfy the boundary conditions. To guarantee homogeneous boundary conditions, the following transformation is introduced: u(x) = f (x) − f0 (x).

(6.6)

Substituting Eq. (6.6) in Eq. (6.3) yields L2 [u(x)] + N2 [u(x)] = ψ(x),

(6.7)

subject to B[u(x), u (x), . . . ] = 0,

x ∈ [a, b],

(6.8)

where L2 and N2 are the adapted linear and nonlinear operators respectively, while ψ(x) = g(x) − L1 [f0 (x)] − N1 [f0 (x)]. At this stage we then formulate the zeroth-order deformation equations (1 − q)L2 [U (x; q)− u0 (x)] = q (L2 [U (x; q)] + N2 [U (x; q)] − ψ(x)) , q ∈ [0, 1]. (6.9) We note again here that the auxiliary function H (x) is not necessary. It is mandatory when using HAM to make sure that the higher-order deformation equations are integrable; q and  are the embedding and convergence controlling parameters, respectively, as defined in the HAM algorithm. The initial approximation u0 (x) used here is the solution of the equation L2 [u0 (x)] = ψ(x),

(6.10)

The spectral-homotopy analysis method (SHAM) for solutions of boundary layer problems

135

with boundary conditions B[u0 (x), u0 (x), . . . ] = 0,

x ∈ [a, b],

(6.11)

which again is different from that of HAM, where the initial guess is used as the initial approximation. From the zeroth-order deformation equation (6.9), it can be shown that at q = 0 and at q = 1, U (x; 0) = u0 (x) and U (x; 1) = u(x).

(6.12)

Consequently, as q increases from 0 to 1, U (x; q) varies from the initial approximation u0 (x) to the solution ux . Using the Taylor series to expand U (x; q) about q gives  ∞  1 ∂ m U (x; q)  m U (x; q) = u0 (x) + um (x)q , um (x) = (6.13)  . m! ∂q m  m=1

q=0

As with HAM,  is chosen such that the series (6.13) converges at q = 1. Hence from Eq. (6.12) we obtain the solutions of the nonlinear problem (6.7) from u(x) = u0 (x) +

∞ 

(6.14)

um (x).

m=1

Following closely the HAM procedure, the higher-order deformation equations are formulated by differentiating the zeroth-order deformation equation m times with respect to q then dividing by m! to get L2 [um (x) − (χm + )um−1 (x)] = Rm (x),

(6.15)

with   1 ∂ m−1  Rm (x) = (N [U (x; q)] − ψ(x))  2  (m − 1)! ∂q m−1

,

(6.16)

q=0

and  χm =

0, 1,

m ≤ 1, m > 1.

In solving the higher-order deformation equations (6.15), the Chebyshev pseudospectral collocation method [11–15] is applied. With the collocation method, the unknown functions um (x) are approximated as truncated series of Chebyshev polynomials of the form um (ξ ) ≈

N  k=0

uˆ k Tk (ξj ),

j = 0, 1, 2, . . . , N,

(6.17)

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where Tk denotes the kth Chebyshev polynomial defined by Tk (ξ ) = cos[k cos−1 (ξ )],

(6.18)

uˆ k are coefficients, and ξ0 , ξ1 , ξ2 , . . . , ξN are Gauss–Lobatto points defined by ξj = cos

πj , N

j = 0, 1, . . . , N,

(6.19)

where N + 1 is the total number of collocation points. The physical domain [a, b] of the problem is mapped by an appropriate equation of the variable ξ to the domain [−1, 1] where the Chebyshev spectral method can be applied. This approach of mapping a physical domain by means of a transformation equation is known as domain truncation in [11]. The derivatives of the functions um (ξ ) are then expressed in term of the Chebyshev pseudospectral differentiation matrix D [12–15] as follows: d a um  a = Dkj um (ξj ), dξ a N

(6.20)

k=0

where a denotes the order of differentiation. The entries of the matrix D are defined as follows: ⎧ c (−1)j +k j ⎪ ⎪ ck ξj −ξk , j = k, ⎪ ⎪ ⎨ − ξk , 1 ≤ j = k ≤ N − 1, 2(1−ξk2 ) Dkj = 2 +1 ⎪ 2N ⎪ , j = k = 0, ⎪ 6 ⎪ ⎩ 2N 2 +1 j = k = N, − 6 , where  cj =

2, 1,

j = 0, N, 1, 2, . . . , N − 1.

Substituting the Chebyshev approximations (6.17)–(6.20) in the higher-order deformation equations (6.15), will result in a matrix equation of the form AUm = (χm + )AUm−1 +  Qm−1 ,

(6.21)

where A and Pm−1 are the resulting matrices after applying the Chebyshev transformations on L2 and Rm , respectively; Um = [um (ξ0 ), um (ξ1 ), um (ξ2 ), . . . , um (ξN )]T , where T stands for the transpose. The boundary conditions of the problem are then imposed on the matrix equation (6.21), and expressing Um yields ˜ m−1 + A−1 Q ˜ m−1 , Um = (χm + )A−1 AU

(6.22)

˜ and Q ˜ m−1 are the resulting matrices after applying the boundary conditions where A on the right-hand side of Eq. (6.21). Eq. (6.22) gives a recursive formula that can then

The spectral-homotopy analysis method (SHAM) for solutions of boundary layer problems

137

be used to solve for higher-order approximations um (x), m ≥ 1 starting from u0 (x) obtained from Eq. (6.10). The recursive formula obtained when using HAM involves a series of ordinary differential equations. We note here that Eq. (6.22) forms a series of algebraic equations. Therefore the results are numerical instead of analytical like in HAM.

6.3

Examples

6.3.1 Example 1: two-dimensional laminar flow between two moving porous walls The example considers a two-dimensional laminar, isothermal, and incompressible viscous fluid flow in a rectangular domain bounded by two permeable surfaces that enable the fluid to enter or exit during successive expansions or contractions. The normalized nonlinear differential equation governing the flow was derived by Majdalani et al. [16] and Dinarvand and Rashidi [17] and is given as F I V + α(yF  + 3F  ) + Re(F F  − F  F  ) = 0,

(6.23)

subject to the boundary conditions F = 0, F  = 0, at y = 0, F = 1, F  = 0, at y = 1,

(6.24) (6.25)

where α(t) is the dimensionless wall dilation rate defined to be positive for expansion and negative for contraction, and Re is the filtration Reynolds number defined positive for injection and negative for suction through the walls.

6.3.1.1

SHAM solution

To transform the domain of the problem from [0, 1] to [−1, 1] and make the governing boundary conditions homogeneous, the following transformations were used: y=

3 ξ +1 1 , U (ξ ) = F (y) − F0 (y), F0 (y) = y − y 3 . 2 2 2

(6.26)

Substituting (6.26) into the governing equation and boundary conditions (6.23)–(6.25) gives 16U I V + 8a1 U  + 4a2 U  + 2a3 U  − 3ReU + 8Re(U U  − U  U  ) = φ(y), (6.27) subject to U = 0, U  = 0, ξ = −1, U = 0, U  = 0, ξ = 1,

(6.28) (6.29)

138

Applications of Heat, Mass and Fluid Boundary Layers

where the primes denote differentiation with respect to ξ and

3 3 1 3 y − y , a2 = 3α − Re(1 − y 2 ), a1 = αy + Re 2 2 2 a3 = 3yRe, φ(y) = 12αy + 3Re y 3 .

(6.30)

The initial approximation is taken to be the solution of the nonhomogeneous linear part of the governing equations (6.27) given by 16U0I V + 8a1 U0  + 4a2 U0 + 2a3 U0 − 3ReU0 = φ(y),

(6.31)

subject to U0 = 0, U0 = 0, ξ = −1, U0 = 0, U0 = 0, ξ = 1.

(6.32) (6.33)

We use the Chebyshev spectral collocation method to solve Eqs. (6.31)–(6.33) and come up with the matrix equation AU0 = ,

(6.34)

subject to the boundary conditions U0 (ξN ) = 0, U0 (ξN ) = 0, N 

2 DN k U0 (ξk ) = 0,

k=0

N 

(6.35) D0k U0 (ξk ) = 0,

(6.36)

k=0

where A = 16D4 + 8a1 D3 + 4a2 D2 + 2a3 D − 3ReI, U0 = [U0 (ξ0 ), U0 (ξ1 ), . . . , U0 (ξN )]T ,  = [φ(y0 ), φ(y1 ), . . . , φ(yN )]T , as = diag([as (y0 ), as (y1 ), . . . , as (yN−1 ), as (yN )]), s = 1, 2, 3.

(6.37) (6.38)

In the above definitions the superscript T denotes the transpose, diag is a diagonal matrix, and I is an identity matrix of size (N + 1) × (N + 1). After implementing the boundary conditions, the values of [U0 (ξ1 ), U0 (ξ2 ), . . . , U0 (ξN−1 )] are then determined from the equation U0 = A−1 .

(6.39)

To find the SHAM solutions of (6.27), we begin by defining the following linear operator: L[U˜ (ξ ; q)] = 16

∂ 3 U˜ ∂ 2 U˜ ∂ U˜ ∂ 4 U˜ + 8a + 4a + 2a − 3ReU˜ , 1 2 3 ∂ξ ∂ξ 4 ∂ξ 3 ∂ξ 2

where U˜ (ξ ; q) is an unknown function.

(6.40)

The spectral-homotopy analysis method (SHAM) for solutions of boundary layer problems

The zeroth-order deformation equation is given by

 (1 − q)L[U˜ (ξ ; q) − U0 (ξ )] = q N [U˜ (ξ ; q)] −  ,

139

(6.41)

where  is the nonzero convergence controlling auxiliary parameter and N is a nonlinear operator given by ∂ 4 U˜ ∂ 3 U˜ ∂ 2 U˜ ∂ U˜ N [U˜ (ξ ; q)] = 16 4 + 8a1 3 + 4a2 2 + 2a3 − 3ReU˜ ∂ξ ∂ξ ∂ξ ∂ξ   ∂ 3 U˜ ∂ U˜ ∂ 2 U˜ + 8Re U 3 − . ∂ξ ∂ξ 2 ∂ξ

(6.42)

The mth order deformation equations are L[Um (ξ ) − χm Um−1 (ξ )] = Rm ,

(6.43)

subject to the boundary conditions Um (−1) = Um (1) = Um (−1) = Um (1) = 0,

(6.44)

where 1V   Rm (ξ ) = 16Um−1 + 8a1 Um−1  + 4a2 Um−1 + 2a3 Um−1 − 3ReUm−1

+ 8Re

m−1 

 (Un Um−1−n  − Un Um−1−n ) − φ(y)(1 − χm )

(6.45)

n=0

and  χm =

0, m ≤ 1, 1, m > 1.

(6.46)

Applying the Chebyshev pseudospectral transformation on Eqs. (6.43)–(6.45) gives AUm = (χm + )AUm−1 − (1 − χm ) + Pm−1

(6.47)

subject to the boundary conditions Um (ξN ) = 0, Um (ξN ) = 0, N  k=0

2 DN k Um (ξk ) = 0,

N 

D0k Um (ξk ) = 0,

(6.48) (6.49)

k=0

where A and  are as defined in (6.37), and Um = [Um (ξ0 ), Um (ξ1 ), . . . , Um (ξN )]T ,

(6.50)

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Applications of Heat, Mass and Fluid Boundary Layers

Table 6.1 Comparison of the numerical results against the SHAM approximate solutions for F (y) when α = −1 with N = 60 and  = −1. Re

y

−2

0.2 0.4 0.6 0.8

0th order 0.273828 0.532827 0.759442 0.928967

0

0.2 0.4 0.6 0.8

0.279449 0.542243 0.768950 0.933889

0.279449 0.542243 0.768950 0.933889

0.279449 0.542243 0.768950 0.933889

0.279449 0.542243 0.768950 0.933889

0.279449 0.542243 0.768950 0.933889

0.279449 0.542243 0.768950 0.933889

2

0.2 0.4 0.6 0.8

0.283996 0.549759 0.776328 0.937518

0.283983 0.549738 0.776306 0.937507

0.283983 0.549738 0.776306 0.937507

0.283983 0.549738 0.776306 0.937507

0.283983 0.549738 0.776306 0.937507

0.283983 0.549738 0.776306 0.937507

Pm−1 = 8Re

m−1 

1st order 0.273831 0.532839 0.759467 0.928990

2nd order 0.273832 0.532839 0.759468 0.928990

3rd order 0.273832 0.532839 0.759468 0.928990

Numeri-

Ref. [17]

0.273832 0.532839 0.759468 0.928990

0.273832 0.532839 0.759468 0.928990

cal

 Un (D3 Um−1−n ) − (DUn )(D2 Um−1−n ) .

(6.51)

n=0

Together with the boundary conditions, the following recursive formula for m ≥ 1 is obtained: ˜ m−1 + A−1 [Pm−1 − (1 − χm )]. Um = (χm + )A−1 AU

(6.52)

Thus, starting from the initial approximation, which is obtained from (6.39), higherorder approximations Um (ξ ) for m ≥ 1 can be obtained through the recursive formula (6.52).

6.3.1.2

Results

In this section we present a sample of the results to illustrate the performance of the method in finding solutions of the equation. The results were compared with existing results from literature. In Table 6.1 we compare the values of F (y) when α = −1 and Re = −2, 0, and 2 with the numerical and the HAM results reported in Dinarvand and Rashidi [17]. In [17] convergence up to six decimal places was achieved at the sixthorder of the HAM approximation for Re = 0 and 2. In this study the same level of convergence and accuracy was achieved at the first-order approximation for the same values of Re. For Re = −2, the convergence of the homotopy analysis method series solution was achieved at the eighth-order of approximation while with the spectral homotopy analysis method series solution gives the same level of convergence at the second order. Fig. 6.1 shows a comparison of the numerical and the SHAM solutions for the characteristic mean-flow function F (y) and F  (y) at different Reynolds numbers and α.

The spectral-homotopy analysis method (SHAM) for solutions of boundary layer problems

141

Figure 6.1 Comparison between numerical (solid line) and SHAM (figures) approximate solution of F (y) and F  (y) for different values of Re when α = −1 when  = −1.14 (for Re = −10) and  = −1 (for Re = 0, 200).

A perfect agreement between the two solutions can be observed from the figure. The strength of the method to work with very large parameters is also demonstrated in Fig. 6.1.

6.3.2 Example 2: steady von Kármán flow The classical von Kármán equations governing the boundary layer flow induced by a rotating disk. The dimensionless form of the governing equations is given by F  − F  H − F 2 + G2 = 0, 



G − G H − 2F G = 0, 





H − H H + P = 0, 

2F + H = 0,

(6.53) (6.54) (6.55) (6.56)

subject to the boundary conditions F (0) = F (∞) = 0, G(0) = 1, G(∞) = 0, H (0) = 0,

(6.57)

where F , G, and H are the velocity components in the radial, azimuthal, and axial directions, respectively, and P is the pressure. Substituting Eq. (6.56) into (6.53) and (6.54) yields 1 H  − H  H + H  H  − 2G2 = 0, 2 G − H G + H  G = 0,

(6.58) (6.59)

subject to the boundary conditions H (0) = H  (0) = H  (∞) = 0, G(0) = 1, G(∞) = 0.

(6.60)

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Applications of Heat, Mass and Fluid Boundary Layers

6.3.2.1

SHAM solution

The interval [0, L] is then transformed to the domain [−1, 1] using the algebraic mapping ξ=

2η − 1, ξ ∈ [−1, 1]. L

(6.61)

The boundary conditions are made homogeneous by applying the transformations H (η) = h(ξ ) + H0 (η),

(6.62)

G(η) = g(ξ ) + G0 (η),

(6.63)

where H0 (η) and G0 (η) are chosen so as to satisfy the boundary conditions (6.60). The chain rule gives 2  4 h (ξ ) + H0 (η), H  (η) = 2 h (ξ ) + H0 (η), L L 8    H (η) = 3 h (ξ ) + H0 (η), L 2  4  G (η) = g (ξ ) + G0 (η), G (η) = 2 g  (ξ ) + G0 (η). L L

H  (η) =

(6.64) (6.65)

Substituting (6.62)–(6.63) and (6.64)–(6.65) in the governing equations gives 4  2 h h + 2 h h − 2g 2 = φ1 (η), (6.66) 2 L L 2  2     b0 g + b1 h + b2 g + b3 h + b4 g − hg + h g = φ2 (η), (6.67) L L

a0 h + a1 h + a2 h + a3 g + a4 h −

where prime denotes derivative with respect to ξ and 8 4 2 , a1 = − 2 H0 , a2 = H0 , a3 = −4G0 , a4 = −H0 , 3 L L L 1     φ1 (η) = −H0 + H0 H0 − H0 H0 + 2G20 , 2 4 2 2 b0 = 2 , b1 = G0 , b2 = − H0 , b3 = −G0 , b4 = H0 , L L L φ2 (η) = −G0 + H0 G0 − H0 G0 . a0 =

As initial guesses we employ the exponentially decaying functions used by Yang and Liao [18], namely, H0 (η) = e−η + ηe−η − 1, G0 (η) = e

−η

.

(6.68) (6.69)

The spectral-homotopy analysis method (SHAM) for solutions of boundary layer problems

143

The initial solution is obtained by solving the linear parts of Eqs. (6.66) and (6.67), namely,   a0 h 0 + a1 h0 + a2 h0 + a3 g0 + a4 h0 = φ1 (η), b0 g0 + b1 h0 + b2 g0 + b3 h0 + b4 g0 = φ2 (η),

(6.70) (6.71)

subject to h0 (−1) =

2  2 h0 (−1) = h0 (1) = 0, g0 (−1) = 0, g0 (1) = 0. L L

(6.72)

The system (6.70)–(6.72) is solved using the Chebyshev spectral collocation method where the system is reduced into a matrix equation given by AF0 = ,

(6.73)

subject to the boundary conditions 2 D0k h0 (ξk ) = 0, L N

k=0

2 DN k h0 (ξk ) = 0, h0 (ξN ) = 0, L N

(6.74)

k=0

g0 (ξ0 ) = 0, g0 (ξN ) = 0,

(6.75)

where A=

a 0 D 3 + a 1 D 2 + a 2 D + a4 I b 1 D + b3 I

a3 I b0 D2 + b2 D + b4 I

,

F0 = [h0 (ξ0 ), h0 (ξ1 ), . . . , h0 (ξN ), g0 (ξ0 ), g0 (ξ1 ), . . . , g0 (ξN )]T ,  = [φ1 (η0 ), φ1 (η1 ), . . . , φ1 (ηN ), φ2 (η0 ), φ2 (η1 ), . . . , φ2 (ηN )]T , ai = diag([ai (η0 ), ai (η1 ), . . . , ai (ηN−1 ), ai (ηN )]), bi = diag([bi (η0 ), bi (η1 ), . . . , bi (ηN−1 ), bi (ηN )]), i = 0, 1, 2, 3, 4.

(6.76)

The superscript T denotes the transpose, diag is a diagonal matrix, and I is an identity matrix of size (N + 1) × (N + 1). The values of [F0 (ξ1 ), F0 (ξ2 ), . . . , F0 (ξN−1 )] are determined from the equation F0 = A−1 ,

(6.77)

which provides the initial approximation for the solution of Eqs. (6.66)–(6.67). We now seek the approximate solutions of (6.66)–(6.67) by first defining the following linear operators: ∂ 3 h˜ ∂ 2 h˜ ∂ h˜ ˜ + a + a2 + a3 g˜ + a4 h, 1 3 2 ∂ξ ∂ξ ∂ξ ∂ 2 g˜ ∂ h˜ ∂ g˜ ˜ ; q), g(ξ ˜ ; q)] = b0 2 + b1 ˜ + b2 + b3 h˜ + b4 g, Lg [h(ξ ∂ξ ∂ξ ∂ξ ˜ ; q), g(ξ Lh [h(ξ ˜ ; q)] = a0

(6.78) (6.79)

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Applications of Heat, Mass and Fluid Boundary Layers

˜ ; q) and g(ξ where h(ξ ˜ ; q) are unknown functions. The zeroth-order deformation equations are given by 

˜ ; q) − h0 (ξ )] = q Nh [h(ξ ˜ ; q), g(ξ (1 − q)Lh [h(ξ (6.80) ˜ ; q)] − φ1 , 

˜ ; q), g(ξ (1 − q)Lg [g(ξ (6.81) ˜ ; q) − g0 (ξ )] = q Ng [h(ξ ˜ ; q)] − φ2 . The nonlinear operators Nh and Ng are given by ∂ 3 h˜ ∂ 2 h˜ ∂ h˜ ˜ ; q), g(ξ + a3 g˜ + a4 h˜ ˜ ; q)] = a0 3 + a1 2 + a2 Nh [h(ξ ∂ξ ∂ξ ∂ξ 2 ∂ h˜ ∂ h˜ 4 ∂ 2 h˜ − 2 h˜ 2 + 2 − 2g˜ 2 , L ∂ξ L ∂ξ ∂ξ ∂ 2 g˜ ∂ h˜ ∂ g˜ ˜ ; q), g(ξ + b2 + b3 h˜ + b4 g˜ ˜ ; q)] = b0 2 + b1 Ng [h(ξ ∂ξ ∂ξ ∂ξ   ∂ g˜ 2 ∂ h˜ ˜ −h + g˜ . L ∂ξ ∂ξ

(6.82)

(6.83)

The mth order deformation equations are given by h , Lh [hm (ξ ) − χm hm−1 (ξ )] = Rm

(6.84)

g Lg [gm (ξ ) − χm gm−1 (ξ )] = Rm ,

(6.85)

subject to the boundary conditions hm (−1) = hm (−1) = hm (1) = 0, gm (−1) = gm (1) = 0,

(6.86)

where h   (ξ ) = a0 h Rm m−1 + a1 hm−1 + a2 hm−1 + a3 gm−1 + a4 hm−1

m−1  2 4    + h h − h h − 2g g n m−1−n n m−1−n L2 n m−1−n L2 n=0

− φ1 (η)(1 − χm ),

+

2 L

 + b1 hm−1 + b2 gm−1 + b3 hm−1 + b4 gm−1 m−1  (hn gm−1−n − gn hm−1−n ) − φ2 (η)(1 − χm ), n=0

(6.87)

g  Rm (ξ ) = b0 gm−1

(6.88)

and  χm =

0, m ≤ 1, 1, m > 1.

(6.89)

The spectral-homotopy analysis method (SHAM) for solutions of boundary layer problems

145

Applying the Chebyshev pseudospectral collocation transformation to Eqs. (6.84)– (6.88) gives AFm = (χm + )AFm−1 − (1 − χm ) + Qm−1 ,

(6.90)

subject to the boundary conditions N 

D0k hm (ξk ) = 0,

k=0

N 

DN k hm (ξk ) = 0, hm (ξN ) = 0,

(6.91)

k=0

gm (ξ0 ) = 0, gm (ξN ) = 0,

(6.92)

where A and  are as defined in (6.76), and (6.93) Fm = [hm (ξ0 ), hm (ξ1 ), . . . , hm (ξN ), gm (ξ0 ), gm (ξ1 ), . . . , gm (ξN )]T , ⎛  ⎞ m−1  2 4 ⎜ (Dhn )(Dhm−1−n ) − 2 hn (D2 hm−1−n ) − 2gn gm−1−n ⎟ 2 ⎟ ⎜ L L ⎟ ⎜ Qm−1 = ⎜ n=0 ⎟. m−1 ⎟ ⎜   2 ⎠ ⎝ (Dhn )gm−1−n − (Dgn )hm−1−n L n=0

(6.94) Together with the boundary conditions, this results in the following recursive formula for m ≥ 1: ˜ m−1 + A−1 [Qm−1 − (1 − χm )]. Fm = (χm + )A−1 AF

(6.95)

Thus, starting from the initial approximation, which is obtained from (6.77), higherorder approximations Fm (ξ ) for m ≥ 1 can be obtained through the recursive formula (6.95).

6.3.2.2

Results

We present sample results for the velocity distributions solved using the SHAM. The accuracy of the SHAM results is measured through comparison with the numerical solutions obtained using the Matlab bvp4c routine and with those previously published by Turkyilmazoglu [19]. The results presented in this work were generated using mostly N = 60 collocation points and L = 20 unless stated otherwise. Table 6.2 gives a comparison of the values of H (∞) obtained at different orders of the SHAM approximations against the homotopy analysis method results and the numerical results. When the same  value is used, convergence of the spectral-homotopy analysis method is achieved at the eighth order compared to the 20th order for the homotopy analysis method approximations. This shows the efficiency of the SHAM in giving the desired results quicker. Fig. 6.2 gives a comparison between the fourth-order SHAM and numerical results for the dimensionless velocity distributions H (η) and G(η), respectively. There is an

146

Applications of Heat, Mass and Fluid Boundary Layers

Table 6.2 Comparison of H (∞) at different orders of the HAM [18] and SHAM against the numerical solution when  = −1, L = 20, N = 60. Order 0 5 10 15 20

HAM [18] −1 −0.9173 −0.8747 −0.8833 −0.8845

Order 2 4 6 8 10

SHAM −0.884944 −0.884449 −0.884476 −0.884474 −0.884474

Numerical −0.884474

Figure 6.2 Comparison between the SHAM (figures) and numerical (solid line) solution of −H (η) and G(η) when  = −1, L = 20, N = 60.

excellent agreement between the two results as shown in the figure. It is worth noting that in case of the HAM in Yang and Liao [18], agreement between the numerical and the HAM results was only observed at the 30th order of approximation for H (η) and at the 10th order for G(η). As with most iterative methods, it is worth noting that the convergence rate may depend on the initial approximation used. However, since we have used the same initial approximations as Yang and Liao [18], the graphical results suggest that the SHAM converges much more rapidly than the HAM.

6.4

Pros and Cons of the SHAM

On the positive side, the way in which the initial guess is sorted in SHAM, better convergence rates are guaranteed compared to HAM. Furthermore, characteristics of the governing equation are inherited in the initial guess compared to when using the boundary conditions only. The restrictions on the initial guess are greatly reduced as long as it exists and nontrivial. Also, the auxiliary function H (x) does not have to be chosen with the SHAM algorithm. With HAM, H (x) is chosen to force all coefficients of the higher order deformation to be expressed in terms of the basis functions. This is to ensure integrability of the higher-order deformation equations. SHAM uses the

The spectral-homotopy analysis method (SHAM) for solutions of boundary layer problems

147

power of the differentiation matrix to handle complex equations. When numerical results are priority, SHAM will give fast converging solutions with improved orders of accuracy. Nonetheless, on the down side, when analytical results are a priority for a particular problem, SHAM cannot be used. Also, finding the optimal number of collocation points to be used for a particular problem is always through numerical experimentation, although once experience in using the method is acquired, it becomes easy. Otherwise, the spectral-homotopy analysis method can be used to solve nonlinear equations arising in boundary layer flows.

6.5

Conclusion

The spectral homotopy analysis method (SHAM) is described in this work. The algorithm combines ideas from a well know analytical method, the homotopy analysis method (HAM), and a numerical method well known for its accuracy, the Chebyshev pseudospectral collocation method. The SHAM algorithm decouples a nonlinear equation (system) using the same approach as that used in HAM and then differentiates the resulting equations numerically using the Chebyshev pseudospectral collocation method. Two boundary layer problems solved using SHAM have been presented to demonstrate the success and efficiency of the method. From the presented results, accuracy and good convergence rates were appreciated and thus the method can be safely used to solve boundary layer problems. The pros and cons of the method were also highlighted.

References [1] R.A. van Gorder, K. Vajravelu, On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach, Communications in Nonlinear Science and Numerical Simulation 14 (2009) 4078–4089. [2] S.J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall/CRC Press, 2003. [3] S.S. Motsa, P. Sibanda, S. Shateyi, A new spectral-homotopy analysis method for solving a nonlinear second order BVP, Communications in Nonlinear Science and Numerical Simulation 15 (2010) 2293–2302. [4] S.S. Motsa, P. Sibanda, F.G. Awad, S. Shateyi, A new spectral-homotopy analysis method for the MHD Jeffery–Hamel problem, Computers & Fluids 39 (2010) 1219–1225. [5] S.S. Motsa, P. Sibanda, On the solution of MHD flow over a nonlinear stretching sheet by an efficient semi-analytical technique, International Journal for Numerical Methods in Fluids (2011), https://doi.org/10.1002/fld.2541. [6] S.S. Motsa, S. Shateyi, A new approach for the solution of three-dimensional magnetohydrodynamic rotating flow over a shrinking sheet, Mathematical Problems in Engineering (2010), https://doi.org/10.1155/2010/586340.

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[7] Z.G. Makukula, P. Sibanda, S.S. Motsa, A novel numerical technique for two-dimensional laminar flow between two moving porous walls, Mathematical Problems in Engineering 2010 (2010) 528956, https://doi.org/10.1155/2010/528956, 15 pages. [8] Z.G. Makukula, P. Sibanda, S.S. Motsa, A note on the solution of the von Kármán equations using series and Chebyshev spectral methods, Boundary Value Problems (2010) 471793, https://doi.org/10.1155/2010/471793, 17 pages. [9] Z.G. Makukula, P. Sibanda, S. Motsa, On a linearisation method for Reiner–Rivlin swirling flow, Computational and Applied Mathematics 31 (2012) 95–125. [10] P. Sibanda, S.S. Motsa, Z.G. Makukula, A spectral-homotopy analysis method for heat transfer flow of a third grade fluid between parallel plates, International Journal of Numerical Methods for Heat & Fluid Flow 22 (1) (2012) 4–23. [11] J.P. Boyd, Chebyshev and Fourier Spectral Methods, DOVER Publications Inc., New York, 2000. [12] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988. [13] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer-Verlag, Berlin, 2007. [14] J.S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge University Press, United Kingdom, 2007. [15] L.N. Trefethen, Spectral Methods in MATLAB, SIAM, 2000. [16] J. Majdalani, C. Zhou, C.A. Dawson, Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability, Journal of Biomechanics 35 (2002) 1399–1403. [17] S. Dinarvand, M.M. Rashidi, A reliable treatment of a homotopy analysis method for twodimensional viscous flow in arectangular domain bounded by two moving porous walls, Nonlinear Analysis: Real World Applications 11 (2010) 1502–1512. [18] C. Yang, S.J. Liao, On the explicit, purely analytic solution of von Kármán swirling viscous flow, Communications in Nonlinear Science and Numerical Simulation 11 (2006) 83–93. [19] M. Turkyilmazoglu, Purely analytic solutions of magnetohydrodynamic swirling boundary layer flow over a porous rotating disk, Computers & Fluids 39 (2010) 793–799.

On a new numerical approach of MHD mixed convection flow with heat and mass transfer of a micropolar fluid over an unsteady stretching sheet in the presence of viscous dissipation and thermal radiation

7

S. Shateyia , G.T. Marewob a Department of Mathematics & Applied Mathematics, University of Venda, Thohoyandou, South Africa, b University of Limpopo, Department of Mathematics & Applied Mathematics, Sovenga, South Africa

Nomenclature A B B0 a, b, c C cp C∞ Cw Cf x D g h j u v N qr M R T T∞ Tw f Pr Ec α

Stretching parameter Micropolar parameter Uniform magnetic field strength Constants Concentration of solutes Specific heat at constant temperature Ambient concentration of the fluid Concentration of the fluid near the wall The skin friction coefficient Chemical molecular diffusivity Acceleration due to gravity Dimensionless angular velocity Microinertia per unit mass Velocity component in the x direction Velocity component in the y direction Microrotation/angular velocity Thermal radiative heat flux Magnetic parameter Thermal radiation parameter Fluid temperature Temperature away from the surface Temperature on the wall Dimensionless stream function Prandtl number Eckert number Positive constant

Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00015-3 Copyright © 2020 Elsevier Ltd. All rights reserved.

150

α0 βt βc ρ ν κ σ λ1 μ ψ γ 

Applications of Heat, Mass and Fluid Boundary Layers

Thermal conductivity Coefficient of thermal expansion Coefficient of concentration expansion Fluid density Kinematic viscosity Vortex viscosity Electric conductivity Relaxation time Dynamic viscosity Stream function Spin gradient viscosity Heat source/sink parameter

7.1 Introduction The interest on non-Newtonian fluids has considerably increased in the past few decades due to their connection with applied sciences. The motion of non-Newtonian fluids plays essential roles both in theory and many industrial processes. Flow, heat, and mass transfer behavior of non-Newtonian fluids cannot be described classically by the theory of continuum mechanics. The theory of micropolar fluids was introduced by Eringen [1] as a model for the flow of certain non-Newtonian fluids that possess internal structures. These fluids physically consist of rigid randomly oriented particles suspended in viscous media. In the micropolar fluid theory, two new variables to the velocity are added which are not present in the Navier–Stokes models. The microrotation of each particle about its centroid, as well as the translatory motion of each particle, is taken into account in the ensuing analysis (Rahman et al. [2]). Micropolar fluids mathematically represent many industrial important fluids such as paints, lubricants, polymers, human and animal blood colloidal suspensions, and liquid crystals. Details of the theory and applications of micropolar fluids can be found in the books by Eringen [3] and Bérg et al. [4]. Past studies on micropolar fluids include, among others, the boundary layer flow of a micropolar fluid over a flat plate (Crees and Bassom [5]), the flow of a micropolar fluid over a stretching sheet (Raptis, [6]), and the flow of a micropolar fluid in a porous media (Kelson and Farrell [7]; Rawat et al. [8]). Motsa et al. [9] solved the problem of a steady viscous flow of a micropolar fluid driven by injection or suction between two porous disks using the spectral modification of the homotopy analysis method. The boundary layer flow and heat transfer in a quiescent Newtonian and nonNewtonian fluid driven by a continuous stretching sheet is of paramount importance in a lot of industrial engineering processes, such as the cooling of a metallic plate in a bath, the drawing of polymer sheets or filaments extruded continuously from a die, the aerodynamics extrusion of plastic sheets, rolling, annealing and tinning of copper wires, wire and fiber cooling, etc. The mechanical properties are greatly dependent upon the cooling rate during the processes. The flow field due to a surface which is moving with a constant velocity in a quiescent fluid was first studied by Sakiadis [10]. Since then different aspects of the problem such as heat and/or mass transfer, linear

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

151

or nonlinear stretching of the surface, Newtonian and non-Newtonian fluids were considered in the literature [11–14]. Abd El-Aziz [15] performed an analysis to study the unsteady mixed convection of a viscous incompressible micropolar fluid adjacent to a heated vertical surface in the presence of viscous dissipation when the buoyancy force assists or opposes the flow. Under some circumstances, the thermal buoyancy force arising due to the heating or cooling of a continuously moving sheet alters the flow and thermal fields thus charging the heat transfer behavior in the manufacturing process. The study of buoyancy induced flows of non-Newtonian fluid continues to be a hot area of study as a result of advances in electronics, nuclear energy, and space technology. Mahmoud and Waheed [16] performed a theoretical analysis of the flow and heat transfer characteristics magnetohydrodynamic mixed convection flow of a micropolar fluid past a stretching surface with slip velocity at the surface and heat generation (absorbtion). Das [17] investigated the effects of partial slip on steady boundary layer stagnation point flow of an electrically conducting micropolar fluid impinging normally towards a shrinking sheet. Haque et al. [18] numerically studied micropolar fluid behavior on a steady MHD free convection and mass transfer though a porous medium. Rashidi et al. [19] employed the homotopy analysis method to examine free convective heat and mass transfer in a steady two-dimensional magnetohydrodynamic fluid flow over a stretching vertical surface in a porous medium. Bachok et al. [20] presented the characteristics of the flow and heat transfer caused by a stretching sheet in a micropolar fluid. Kasim et al. [21] investigated the steady magnetohydrodynamic free convection of an incompressible and electrically conduction fluid over a stretching surface under the influence of Sorec and Dufour effects with thermophoresis. Gupta et al. [22] analyzed the effect of unsteadiness on mixed convection boundary layer flow of micropolar fluid over a permeable shrinking sheet in the presence of viscous dissipation. Recently, Mohanty et al. [23] numerically investigated heat and mass transfer effect on micropolar fluid over a stretching sheet through porous media. Srinivas et al. [24] presented an analysis of the effects of a chemical reaction on an unsteady flow of a micropolar fluid over a stretching sheet embedded in a non-Darsian porous medium. Mahmood and Waheed [25] investigated the effects of slip velocity on the flow and heat transfer for an electrically conducting micropolar fluid over a permeable stretching surface with variable heat flux in the presence of heat generation absorbtion and a transverse magnetic field. The present work seeks to investigate the problem of heat and mass transfer of a magnetohydrodynamics unsteady mixed convection flow of a micropolar fluid over a stretching surface in the presence of viscous dissipation and thermal radiation. The governing partial differential equations are converted into ordinary differential equations using suitable similarity transformations. The resultant system of ordinary differential equations is then solved numerically using a recently developed Spectral Quasi-Linearization Method (SQLM).

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7.2

Applications of Heat, Mass and Fluid Boundary Layers

Mathematical formulation

We consider an unsteady two-dimensional, mixed convection flow of a viscous incompressible micropolar fluid, heat and mass transfer over an elastic, vertical impermeable stretching sheet. Following Anderson et al. [26], the sheet is assumed to emerge vertically in the upward direction from a narrow slot with velocity Uw (x, t) =

ax , 1 − αt

(7.1)

where both a and α are positive constants with dimension per unit time. We measure the positive x direction along the stretching sheet with the slot as the origin. We then measure the positive y coordinate perpendicular to the sheet in the outward direction toward the fluid flow. The surface temperature Tw and concentration Cw of the stretching sheet vary with the distance x from the sheet and time t as Tw (x, t) = T∞ +

bx , (1 − αt)2

Cw (x, t) = C∞ +

cx , (1 − αt)2

(7.2)

where b, c are constants with dimension of temperature and concentration, respectively, over length. It is noted that the expressions for Uw (x, t), Tw (x, t), and Cw (x, t) are valid only for t < α −1 . We also remark that the elastic sheet which is fixed at the origin is stretched by applying a force in the x-direction and the effective stretching rate a/(1 − αt) increases with time. Analogously, the sheet temperature and concentration increase (reduce) if b and c are positive (negative), respectively, from T∞ and C∞ at the sheet in proportion to x. We assume that the radiation effect is significant in this study. The fluid properties are taken to be constant except for density variation with temperature and concentration in the buoyancy terms. Under those assumptions and the Boussinesq approximations, the governing two dimensional boundary layer equations are given as: ∂u ∂v + = 0, ∂x ∂y

∂u ∂u μ + κ ∂ 2 u κ ∂N ∂u + +u +v = + gβt (T − T∞ ) ∂t ∂x ∂y ρ ∂y 2 ρ ∂y

∂N ∂N ∂N +u +v ∂t ∂x ∂y ∂T ∂T ∂T +u +v ∂t ∂x ∂y

σ B02 u , + gβc (C − C∞ ) − ρ

κ γ ∂ 2N ∂N − = 2N + , ρj ∂y 2 ρj ∂y

2 ∂ 2T μ+κ 1 ∂qr ∂u − = α0 2 + , ρcp ∂y ρcp ρ ∂y ∂y

∂C ∂C ∂ 2C ∂C +u +v = D 2 − κc (C − C∞ ), ∂t ∂x ∂y ∂y

(7.3)

(7.4) (7.5) (7.6) (7.7)

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

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where u and v are the velocity components along the x and y axes, respectively, T is the fluid temperature, μ is the component of the microrotation vector normal to the xy plane, γ is the spin gradient viscosity, α0 is the thermal conductivity, cp is the heat capacity at constant pressure, g is the acceleration due to gravity, βt and βc are the coefficients of thermal expansion and concentration expansion, respectively, qr is the thermal radiative heat flux, B0 is the transverse magnetic field, C is the concentration of the solutes, T∞ and C∞ denote the temperature and concentration far away from the plate, respectively, j is the microinertia density or microinertia per unit mass. The appropriate boundary conditions for the current model are: u = Uw (x, t), v = 0, N = 0, T = Tw (x, t), C = Cw (x, t) at y = 0,

(7.8)

u → 0, T → T∞ , C → C∞ as y → ∞.

(7.9)

Following the Roseland approximation, we model the radiative heat flux qr as qr =

4σ ∗ ∂T 4 , 3κ1 ∂y

(7.10)

where σ ∗ is the Stefan–Boltzman constant and κ1 is the mean absorbtion coefficient. We assume that the temperature difference within the flow is such that T 4 can be expressed as a linear combination of the temperature. We therefore expand T 4 in Taylor series about T∞ as follows: 4 3 2 + 4T∞ (T − T∞ ) + 6T∞ (T − T∞ )2 + · · · . T 4 = T∞

(7.11)

Upon neglecting higher-order terms beyond the first degree in (T − T∞ ), we obtain 4 3 − 4T∞ T. T 4 ≈ 3T∞

(7.12)

Therefore qr =

3 T + 3T 4 ) 3 ∂T 4σ ∗ ∂(−4T∞ 16σ ∗ T∞ ∞ =− , 3κ1 ∂y 3κ1 ∂y

(7.13)

then 3 ∂ 2T ∂qr −16σ ∗ T∞ , = ∂y 3κ1 ∂y 2

(7.14)

which is then substituted in the last term of the heat equation (7.6).

7.3 Similarity analysis In order to transform the governing equations (7.3)–(7.7) into a set of ordinary differential equations, we introduce the following transformation variables (Abd El-Aziz

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Applications of Heat, Mass and Fluid Boundary Layers

[15]): % $ a aν a3 η= y, ψ = xf (η), N = xh(η), ν(1 − αt 1 − αt ν(1 − αt) bx cx θ (η), C = C∞ + φ(η), T = T∞ + 2 (1 − αt) (1 − αt)2 $

(7.15)

where ψ(x, y, t) is the physical stream function which automatically satisfies the continuity equation:  A  2f + ηf  + h + λ1 θ + λ2 φ − Mf  = 0, 2 (7.16)     A (7.17) 3h + ηh − B 2h + f  = 0, λ3 h + f h − f  h − 2  A 1 4θ + ηθ  + Ec(1 + )(f  )2 = 0, (4 + 3R) θ  + f θ  − f  θ − 3Pr R 2 (7.18)   A 1  (7.19) φ + f φ  − f  φ − Kφ − 4φ + ηφ  = 0. Sc 2 (1 + ) f  + ff  − (f  )2 −

7.3.1 Boundary conditions The corresponding boundary conditions become: f (0) = 0, f  (0) = 1, h(0) = 0, θ (0) = 1, φ(0) = 1, f  (∞) = 0, h(∞) = 0, θ (∞) = 0, φ(∞) = 0.

(7.20) (7.21)

The quantities of engineering interest in the present study are the skin-friction coefficient Cf x , the local wall couple stress Mwx , the local Nusselt number N ux , and the local Sherwood number Shx . These quantities are respectively defined by:   −1 2 ∂u 2 + κN | f  (0), (7.22) (μ + κ) = 2(1 + )Re Cf x = y=0 x ∂y y=0 ρUw2

γ a ∂N Mwx = = Rex−1 h (0), (7.23) ν ∂y y=0

1 −x ∂T N ux = = −Rex2 θ  (0), (7.24) Tw − T∞ ∂y y=0

1 −x ∂C = −Rex2 φ  (0), (7.25) Shx = Cw − C∞ ∂y y=0 where A = αa is the unsteadiness parameter, θ = (T − T∞ )/(Tw − T∞ ) is the dimensionless temperature, φ = (C − C∞ )/(Cw − C∞ ) is the dimensionless concentration,

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

= κ/μ is the micropolar parameter, λ1 = gβc c a2

gβt b a2

155

is the buoyancy parameter due

to temperature gradient, λ2 = is the buoyancy parameter due to concentration gradient, λ3 = γ /μj , B = ν(1 − αt)/bj are dimensionless micropolar parameters, Ec = Uw2 /cp (Tw − T∞ ) is the Eckert number, Sc = ν/D is the Schmidt number, while K is the chemical reaction parameter.

7.4

Methods of solution

7.4.1 Spectral Quasi-Linearization Method (SQLM) 7.4.1.1

Overview

Consider the problem of finding u(x) on [a, b] satisfying a nonlinear nth order differential equation F (u, u , u , . . . , u(n) ) = G(x)

(7.26)

subject to given boundary conditions. The solution procedure consists of the following basic steps: 1. Quasi-linearization: this replaces differential equation (7.26) with its linear counterpart α0r (x)ur+1 (x) + α1r (x)ur+1 (x) + α2r (x)ur+1 (x) + · · · + αnr (x)ur+1 (x) = Rr (x) (7.27) (n)

for each r = 0, 1, 2, . . . where ur → u as r → ∞. 2. Chebyshev differentiation [27]: this reduces the problem of solving differential equation (7.27) at any point x ∈ [a, b] to the problem of solving a linear system ⎡ ⎤ u(ξ0 ) ⎢ u(ξ1 ) ⎥ ⎢ ⎥ A⎢ . ⎥ = R (7.28) ⎣ .. ⎦ u(ξN ) for u at each collocation point ξi ∈ [−1, 1] which is an image of a point xi ∈ [a, b] under a suitable transformation. 3. Interpolation: since u is now known at each xi ∈ [a, b], at any point x ∈ [a, b] we have u(x) =

N 

u(xk )Lk (x),

(7.29)

k=0

where Lk (x) is the Lagrange polynomial of degree N that is associated with node x = xk .

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Applications of Heat, Mass and Fluid Boundary Layers

7.4.1.2

Some important details

Let u = (U, U  , U  , . . . , U (n) ) and v = (V , V  , V  , . . . , V (n) ) be approximations of u and its derivatives of order up to n. If u and v are sufficiently close, it follows from Taylor’s theorem that the vector form F (u) =G(x),

(7.30)

of Eq. (7.26) may be approximated by F (v) + (u − v) · ∇F (v) =G(x),

(7.31)

upon neglecting higher-order terms. Given approximation v, one can compute an improved approximation u using Eq. (7.31). Hence we replace Eq. (7.31) with the recursive formula ur+1 · ∇F (ur ) =G + ur · ∇F (ur ) − F (ur ),

(7.32)

for generating a sequence {ur } of approximations of u whose limit we anticipate to be u itself. Upon expanding and rearranging, Eq. (7.32) takes the more familiar form of Eq. (7.27) where αir = Fu(i) (ur ) for each i = 0, 1, 2, . . . , n and Rr = G + α0r ur + α1r ur + α2r ur + · · · + αnr u(n) r − F (ur ).

(7.33)

Note that to avoid confusion with notation, we have used u in place of U . Before solving differential equation (7.27), we take a few essential steps: 1. It might happen that the problem domain [a, b] is unbounded at either end, e.g., it could be the semiinfinite interval [0, ∞). In this case it is necessary to replace it with a closed interval [0, L] where L is large enough for the boundary conditions at L to remain the same as those at ∞. Hence the new [a, b] is [0, L]. 2. It is convenient to define a change of variable x(ξ ) =

a+b b−a +ξ , 2 2

(7.34)

so that we solve (7.27) on the computational domain [−1, 1] on the ξ -axis instead of working on the physical domain [a, b] on the x-axis. 3. We form a grid of the so-called Chebyshev collocation points ξi = cos πi N on [−1, 1] with i = 0, 1, 2, . . . , N. Under the linear transformation (7.34), Eq. (7.27) takes the form α0r (ξi )β 0 ur+1 (ξi ) + α1r (ξi )β 1 ur+1 (ξi ) + · · · + αnr (ξi )β n u(n) r+1 (ξi ) = Rr (ξi ), (7.35) 2 . Chebyshev differentiation evaluates each of the at each ξi ∈ [−1, 1], where β = b−a derivatives in Eq. (7.35) using the formula (p)

ur+1 (ξi ) =

N  j =0

[D p ]i,j ur+1 (ξj ),

(7.36)

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

157

in terms of the Chebyshev differentiation matrix D so that differential equation (7.35) is replaced by a linear equation. Proceeding in a similar manner for each collocation point ξi yields the linear system Aur+1 =Rr ,

(7.37)

where ur+1 =[ur+1 (ξ0 ), ur+1 (ξ1 ), . . . , ur+1 (ξN )]T , Rr =[Rr (ξ0 ), Rr (ξ1 ), . . . , Rr (ξN )] , T

A =Aˆ (nr) Dˆ n + · · · + Aˆ (2r) Dˆ 2 + Aˆ (1r) Dˆ + I, ˆ (kr)

A

= diag {αkr (ξ0 ), αkr (ξ1 ), . . . , αkr (ξN )} ,

(7.38) (7.39) (7.40) (7.41)

Dˆ = βD, and I is the identity matrix of the same order as D.

7.4.1.3

Application

When Eq. (7.16) is written in the same form as Eq. (7.30), it becomes F (u) =0,

(7.42)

where u = (f, f  , f  , f  , h , θ, φ) and F (u) = (1 + ) f  + ff  − (f  )2 − + λ2 φ − Mf  ,

 A  2f + ηf  + h + λ1 θ 2 (7.43)

so that similar to Eq. (7.27) we obtain the linear counterpart    a0r fr+1 + a1r fr+1 + a2r fr+1 + a3r fr+1 + a4r hr+1 + a5r θr+1 + a6r φr+1 = Rr(1) ,

(7.44) of Eq. (7.42) where a0r := Ff (u) =fr ,

a1r := Ff  (u) = − 2fr − A − M, A a2r := Ff  (u) =fr − , η 2 a3r := Ff  (u) =1 + , a4r := Fθ (u) = , a5r := Fφ (u) =λ1 , a6r := Fφ (u) =λ2 , Rr(1) := ur · ∇F (ur ) − F (ur ) = − (fr )2 + fr fr .

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Applications of Heat, Mass and Fluid Boundary Layers

Similarly, quasi-linearization replaces differential equations (7.17)–(7.19) with their linear counterparts   b0r fr+1 + b1r fr+1 + b2r fr+1 + b3r hr+1 + b4r hr+1 + b5r hr+1 = Rr(2) , (7.45)     + c2r fr+1 + c3r θr+1 + c4r θr+1 + c5r θr+1 = Rr(3) , (7.46) c0r fr+1 + c1r fr+1    e0r fr+1 + e1r fr+1 + e2r φr+1 + e3r φr+1 + e4r φr+1 = Rr(4) , (7.47)

respectively, where 3 b0r = hr , b1r = −hr , b2r = − B, b3r = −fr − A − 2 B, 2 A (7.48) b4r = fr − η, b5r = λ3 , 2 Rr(2) = −fr hr + fr hr , (7.49)    c0r = θr , c1r = −θr , c2r = 2Ec (1 + )fr , c3r = −fr − 2A, A 4 + 3R c4r = fr − η, c5r = , 2 3P r R Rr(3) = (1 + )Ec(fr )2 − fr θr + f θr , A 1 e0r = φr , e1r = −φr , e2r = −fr − K − 2A, e3r = fr − η, e4r = , 2 Sc Rr(4) = −fr φr + fr φr . Upon evaluating differential equations (7.44)–(7.47) at each collocation point, and using Chebyshev differentiation to approximate derivatives, we come up with the linear system Aur+1 =Rr ,

(7.50)

subject to boundary conditions N 

D0k fr+ (ξk ) =0, hr+1 (ξ0 ) = 0, φr+1 (ξ0 ) = 0, φr+1 (ξ0 ) = 0,

(7.51)

k=0 N 

DN k fr+ (ξk ) =1, f (ξN ) = hr+1 (ξN ) = 0, θr+1 (ξN ) = φr+1 (ξN ) = 1,

k=0

(7.52) where ⎡

A(11) ⎢A(21) A =⎢ ⎣A(31) A(41)

A(12) A(22) A(32) A(42)

A(13) A(23) A(33) A(43)

⎡ (1) ⎤ ⎤ ⎤ ⎡ Rr A(14) Fr+1 ⎢ (2) ⎥ (24) ⎥ ⎥ ⎢ A ⎥ H ⎢R ⎥ ,u = ⎢ r+1 ⎥ , Rr = ⎢ r(3) ⎥ , A(34) ⎦ r+1 ⎣r+1 ⎦ ⎣ Rr ⎦ (4) r+1 A(44) Rr (7.53)

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

159

ˆ A(11) =Aˆ (0r) + Aˆ (1r) Dˆ + Aˆ (2r) Dˆ 2 + Aˆ (3r) Dˆ 3 , A(12) = Aˆ (4r) D, A(13) =Aˆ (5r) , A(14) = Aˆ (6r) , A(21) =Bˆ (0r) + Bˆ (1r) Dˆ + Bˆ (2r) Dˆ 2 , A(22) = Bˆ (3r) + Bˆ (4r) Dˆ + Bˆ (5r) Dˆ 2 , A(23) =A(24) = O, A(31) =Cˆ (0r) + Cˆ (1r) Dˆ + Cˆ (2r) Dˆ 2 , A(32) = O, A(33) =Cˆ (3r) + Cˆ (4r) Dˆ + Cˆ (5r) Dˆ 2 , A(34) = O, ˆ A(42) = A(43) = O, A(41) =Eˆ (0r) + Eˆ (1r) D, A(44) =Eˆ (2r) + Eˆ (3r) Dˆ + Eˆ (4r) Dˆ 2 , ˆ and where O is the zero matrix of the same order as D. We include boundary conditions (7.52) and (7.51) into linear system (7.50) in the same manner as Shateyi and Marewo [28]. Once this is done, we solve the resulting linear system to obtain ur+1 for each r = 0, 1, 2, . . . However, the term Rr on the right-hand side of this linear system depends on ur according to Eqs. (7.39) and (7.33). Consequently, we need to know ur first. Beginning with r = 0, we specify initial approximations f0 (η) = 1 − e−η , h0 (η) = ηe−η , θ0 (η) = e−η , φ0 (η) = e−η , which we chose to satisfy boundary conditions (7.20) and (7.21).

7.5

Results and discussion

A comprehensive numerical parametric study is conducted and the results are reported graphically and in tabular form, all presented in this section. Numerical approximations of the surface shear stresses f  (0), h (0), surface heat transfer θ  (0), and surface mass transfer φ  (0), are simulated and presented in tubular forms. The numerical results are iteratively generated by the Spectral Quasi-Linearization Method (SQLM) for the main parameters that have significant effects on the flow properties. It is remarked that the SQLM results presented in this work were obtained using N = 50 allocation points and the infinity value n∞ being 30. These values where found to give accurate results for the governing physical parameters, and beyond these values the results did not change within prescribed significant accuracy. The tolerance level was set to be  = 10−8 , which we thought to be good enough for any engineering numerical approximation. Table 7.1 displays the results generated by SQLM compared to those generated by the bvp4c routine when varying the micropolar parameter , for the skin friction coefficient and the wall couple stress. We see that the SQLM results are exactly the same to those produced by the bvp4c routine. But we remark that the SQLM converges much faster than the bvp4c routine. We observe in Table 7.1 that the skin friction coefficient |f  (0)| decreases as increases. Thus, micropolar fluids show significant drag

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Applications of Heat, Mass and Fluid Boundary Layers

reductions compared to viscous fluids. However, it is also observed in Table 7.1 that the wall couple stress h (0) increases with increasing values of the material parameter . Thus micropolar fluids experience more rotational drag resistance compared to viscous fluids. Table 7.1 Comparison of skin friction coefficient and wall couple stress for different values of . 1 3 5

−f  (0) bvp4c 0.75815723 0.53094615 0.42476812

Present 0.75815723 0.53094615 0.42476812

h (0) bvp4c 0.28501244 0.52715722 0.63699938

Present 0.28501244 0.52715722 0.63699938

Table 7.2 displays the influence of micropolar parameters on the Nusselt θ  (0) and Sherwood number φ  (0). The rate of heat transfer on the wall surface is increased with increasing values of the micropolar parameters. As the micropolar parameters increase, the fluid becomes more inviscid, thereby enhances the transfer of heat from the wall surface to fluid flow. The mass transfer rate has very large absolute values as the micropolar parameters increase. Table 7.2 Comparison of Nusselt and Sherwood numbers for different values of . 1 3 5

−θ  (0) bvp4c 0.89411422 0.82299565 0.76679375

Present 0.89411422 0.822995652 0.76679375

−φ  (0) bvp4c 2.00008779 2.01416171 2.02113863

Present 2.00008779 2.01416171 2.02113863

Table 7.3 shows the influence of the stretching parameter A, thermal buoyancy parameter λ1 , dimensionless micropolar parameters λ3 and B on the skin friction coefficient and wall couple stress. In this work we consider the accelerating cases only, λ1 > 0. Increasing the unsteadiness parameter greatly increases the drag force on the wall surface which helps to destabilize the fluid velocity. In this table we observe that the wall stresses are increasing with increasing values of the thermal buoyancy parameter λ1 . As expected, buoyancy has significant increasing effect on the flow velocities but reduces the rate of heat and mass transfer. In Table 7.3, we clearly see that the dimensionless material parameter λ3 significantly affects the couple wall stress. Increasing values of λ3 mean that the spin gradient coefficient is greatly enhanced, thereby reducing the couple wall stress as is clearly depicted in Table 7.3. We also observe in this table that the skin friction and the couple wall stress increase with increasing values of the micropolar parameter B. In Table 7.4, we display the effects of varying the unsteadiness parameter A, magnetic parameter M, the Eckert number Ec, and the chemical reaction parameter K on

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

161

Table 7.3 Values of skin friction coefficient and wall couple stress for different values of A, λ1 , λ3 , B. A 0 1 3

λ1 0.5

λ3 0.3

B 0.1

−f  (0) 0.45862358 0.75815723 1.15674408

h (0) 0.26383752 0.28501244 0.285635048

1

–1 0 1 3

0.3

0.1

1.16322365 0.90869033 0.66992922 0.22045195

0.37007346 0.30806124 0.25312018 0.15565095

1

0.5

0.1 0.3 0.5

0.65423873 0.66089865 0.66457440

0.95647649 0.51851531 0.38520719

1

0.5

0.3

0.80098707 0.79684333 0.79314915

0.05343498 0.10349781 0.15071065

0.1 0.2 0.3

Table 7.4 Values of Nusselt and Sherwood numbers for different values of A, M, Ec, K. A 0 1 3

M 1

Ec 0.1

K 0.3

−θ  (0) 0.49695274 0.89411422 1.37256946

−φ  (0) 1.51870633 2.00008779 2.73821257

0

0 1 3

0.1

0.3

0.81507017 0.60345428 0.50101120

0.92067078 0.89251991 0.78217064

1

1

0 0.2 0.4

0.92171277 0.62954370 0.22560033

0.91347151 0.91443837 0.91579397

1

1

0.1

0.77841874 0.77318449 0.76243698

0.77756028 0.91396171 1.31802332

0 1 3

the rates of heat and mass transfer on the wall surface. It is noted that increasing the values of the unsteadiness parameter A enhances the rates of both heat and mass transfer. However, increasing the values of the magnetic parameter reduces the rates of both the heat and mass transfer. Increasing the values of the magnetic parameter produces a drag force which tends to retard the fluid velocity, thus increasing the temperature and mass distributions within the fluid flow. The Eckert number has very little influence on the Sherwood number depicted in Table 7.4. But the Nusselt number is greatly reduced

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as the values of the Eckert number increase. Viscous dissipation causes the heating of the fluid thus increasing the fluid temperature. Lastly, in Table 7.4, we observe that chemical reaction has very little effect on the heat transfer on the wall surface but has a pronounced effect on the rate of mass transfer. Fig. 7.1 displays the effect of the unsteadiness parameter A on the axial velocity f  (η), angular velocity h(η), temperature θ (η), and concentration φ(η) profiles. We observe that the velocity profiles decrease as A increases. Increasing the unsteadiness

Figure 7.1 Variation of the stretching A parameter on the velocity, temperature, and concentration profiles with = 0.5, λ1 = 0.5, λ2 = 0.5, M = 0.5, λ3 = 0.5, B = 0.2, Ec = 0.1, K = 1, P r = 0.71, R = 1, Sc = 0.22.

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

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Figure 7.1 (continued)

parameter causes the velocity boundary layer thickness to decrease. Physically, we therefore conclude that stretching of the sheet can be used as a stabilizing mechanism in trying to delay the transition from laminar to turbulent fluid lows. We also observe that the angular velocity is greatly affected by the increasing values of the unsteadiness parameter. Every value of the unsteadiness causes the angular velocity profile to increase from the initial value of zero to a peak and then gradually die again to the free stream value of zero. For increasing values of the unsteadiness parameter, we also

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observe that the angular velocity profiles are greatly reduced. In the respective figures we observe that the velocity gradients at the surface are larger for larger values of the unsteadiness parameter, which in turn produces larger skin friction coefficient |f  (0)| and couple wall stress |h (0)|. In Fig. 7.1 we clearly see that increasing values of A reduce both the thermal and solutal boundary layer thicknesses, thus reducing the fluid temperature and concentration distributions.

Figure 7.2 Variation of the micropolar parameter on the velocity, angular velocity, and temperature profiles with A = 1, λ1 = 0.5, λ2 = 0.5, M = 0.5, λ3 = 0.5, B = 0.2, Ec = 0.1, K = 1, P r = 0.71, R = 1, Sc = 0.22.

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

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Figure 7.2 (continued)

The influence of the micropolar parament on the velocity components and fluid temperature is displayed in Fig. 7.2. We observe in this figure that the axial velocity f  (η) and angular velocity h(η) increase while the temperature θ (η) decreases with increasing values of the micropolar parameter . The angular velocity is greatly induced due to the vortex viscosity effect as increases. As elucidated in Table 7.1, the velocity gradients |f  (0)| and h (0) decrease as increases, thus micropolar fluids show drag reduction compared to viscous fluids. We also observe in this figure that the micropolar parameter has a more pronounced influence on the velocity distributions than on the temperature distributions, and no significant effect on the concentration (not shown here for brevity). The influence of the magnetic parameter M on the flow properties is depicted in Fig. 7.3. Applying a magnetic field perpendicular to the flow produces a drag force known as Lorentz force. This force reduces the axial velocity as can be observed in Fig. 7.3. The velocity boundary layer is reduced by increasing the magnetic strength parameter M. Fig. 7.3 shows that near the wall the angular velocity increases with increasing values of the magnetic parameter M. However, as we move away from the surface, the Lorentz force comes into effect, thus we see the angular velocity profiles being reduced with increasing values of M. Both the temperature and concentration profiles increase with increasing values of the magnetic parameter M. The reduction of the flow velocity means that more heat energy is transferred from the wall into the flow system thus increasing the temperature of the fluid. In Fig. 7.4, we display the effect of the thermal buoyancy parameter on the axial velocity f  (η), angular velocity h(η), temperature θ (η), and concentration φ(η). We observe in this figure that for opposing flows (λ1 < 0), the effect of buoyancy is to reduce the velocity compared to those for pure forced convection (λ1 = 0). Physically,

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Figure 7.3 Effect of the magnetic M parameter on the velocity, angular velocity, and temperature profiles with A = 1, λ1 = 0.5, λ2 = 0.5, = 0.5, λ3 = 0.5, B = 0.2, Ec = 0.1, K = 1, P r = 0.71, R = 1, Sc = 0.22.

a positive λ1 induces a favorable pressure gradient that enhances the fluid flow in the boundary layer, while a negative λ1 produces an adverse gradient which in turn slows down the fluid motion. It must also be remarked that for large values of the thermal buoyancy parameter (λ1 > 5), the axial velocity overshoots near the wall over the moving speed of the sheet. (This was also observed in Abd El-Aziz [15]). In Fig. 7.1 we observe that the values of the angular velocity h(η) decrease with increasing val-

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

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Figure 7.3 (continued)

ues of the thermal buoyancy parameter λ1 close to the sheet surface and at the last part of the boundary layer. However, for the middle part of the boundary layer at a fixed η position, the angular velocity h(η) increases with increasing values of the thermal buoyancy parameter λ1 . We also observe that for opposing flows (λ1 < 0), the angular velocity greatly increases near the sheet and then gradually dies to the free stream velocity. Both the thermal and solutal boundary layer thicknesses are reduced when the values of the thermal buoyancy are increased. The fluid temperature and concentration are reduced at every point other than at the wall with increasing values

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of the thermal buoyancy parameter. We remark that solutal buoyancy parameter λ2 has the same effects on the fluid properties to those of the thermal buoyancy parameter. Fig. 7.5 displays the effect of chemical reactions (left) and Eckert number (right) on the velocity profiles. We observe in this figure that both the chemical reaction and the Eckert number have little effect on the velocity profiles. It is now known that viscous dissipation produces heat due to drag force between the fluid particles and then this

Figure 7.4 Effect of the thermal buoyancy λ1 parameter on the velocity, angular velocity, and temperature

profiles with A = 1, M = 1, λ2 = 0.5, = 0.5, λ3 = 0.5, B = 0.2, Ec = 0.1, K = 1, P r = 0.71, R = 1, Sc = 0.22.

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

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Figure 7.4 (continued)

extra heating of the fluid causes an increase of the buoyancy force, thus causing fluid velocity profiles to increase as depicted in Fig. 7.5. The influence of the materials (top) and thermal radiation number (bottom) on the angular velocity profiles are depicted on Fig. 7.6. We observe in Fig. 7.6 that the angular velocity profiles are reduced with increasing values of λ3 . This means that either the spin gradient coefficient increases or the microinertia density is reduced. However, reducing the values of the microinertia density implies increasing values of B(λx/j Uw ). This in turn causes the angular velocity to increase from zero to a peak before gradually dying to the free-stream angular velocity. The an-

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Figure 7.5 Effect of the chemical reaction parameter (top) and Eckert number (bottom) on the velocity profiles with A = 1, M = 1, λ1 = λ2 = 0.5, = 0.5, λ3 = 0.5, B = 0.2, P r = 0.71, R = 1, Sc = 0.22.

gular velocity profiles are seen to be increasing with the thermal radiation parameter. Fig. 7.7 shows the effect of thermal radiation (top) and Prandtl number (bottom) on the temperature profiles. We observe in this figure that increasing the thermal radiation parameter produces a significant decrease in the thermal condition of the fluid flow. Physically a decrease in the values of R means a decrease in the Rosseland radiation absorbability k1 . Thus the divergence of the radiative heat flux decreases as k1 in-

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

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Figure 7.6 Effect of the material parameters (top) and thermal radiation number (bottom) on the angular velocity profiles with A = 1, M = 1, λ1 = λ2 = 0.5, = 0.5, λ3 = 0.5, Ec = 0.1, P r = 0.71, Sc = 0.22.

creases the rate of radiative heat transferred from the fluid, and consequently the fluid temperature decreases. It can clearly be seen in this figure that increases in P r bring about a significant decrease in the fluid temperature. By definition the Prandtl number is the ratio of momentum diffusion to thermal diffusion and therefore the thermal diffusion decreases as the Prandtl number increases, which in turn causes the thinning of thermal boundary layer. Physically, larger values of P r correspond to higher heat capacity that increases the rate of heat transfer, thus reducing the values of the dimensionless temperature.

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Figure 7.6 (continued)

Lastly, Fig. 7.8 displays the effect of the Schmidt number (top) and the chemical reaction parameter (bottom) on the concentration profiles. We observe that an increase in the Schmidt number Sc decreases the concentration profiles. The Schmidt number represents the relative ease of the occurrence of the momentum and mass transfer, and is very important in calculations of the binary mass transfer in multiphase flows. Physically, as the chemical reaction parameter increases, the concentration profiles decrease as can be seen in this figure.

7.6

Conclusion

The problem of heat and mass transfer of a magnetic hydrodynamic unsteady mixed convection flow of a micropolar fluid over a stretching surface in the presence of various dissipation and thermal radiation is numerically investigated. The partial differential governing equations were developed and transformed into a system of ordinary deferential equations by applying suitable similarity transformations. The resultant ordinary equations were solved using the Spectral Quasi-Linearization Method (SQLM). The accuracy of the SQLM is validated against the MATLAB in built bvp4c routine for solving boundary value problems. The following conclusions were drawn from the present investigation: • There was an excellent agreement between SQLM results and those obtained using the bvp4c routine giving much needed confidence to our presented results. • The velocity components are increasing functions of the increasing values of the micropolar parameter while the temperature and concentration distributions are reduced as the micropolar parameter increases.

On a new numerical approach of MHD mixed convection flow with heat and mass transfer

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Figure 7.7 The influence of thermal radiation (top) and Prandtl number (bottom) on the temperature profiles with A = 1, M = 1, λ1 = λ2 = 0.5, = 0.5, λ3 = 0.5, B = 0.2, K = 1, Ec = 0.1, Sc = 0.22.

• Presented numerical computations have demonstrated that the drag force can be reduced significantly with the proper selection of unsteadiness parameter, buoyancy parameters, and micropolar parameter. • The transfer rate of a micropolar fluid is greatly enhanced as either the values of the buoyancy parameter, unsteadiness parameter, or Prandtl number are increased. However, this rate is reduced as the values of the micropolar parameters, magnetic field parameter, Eckert number, or chemical reaction parameter are increased.

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Figure 7.8 Effect of the Schmidt number (top) and the chemical reaction parameter (bottom) on the concentration profiles with A = 1, M = 1, λ1 = λ2 = 0.5, = 0.5, λ3 = 0.5, B = 0.2, P r = 0.71, R = 1, Ec = 0.1.

• The transfer rate of a micropolar fluid is enhanced as the chemical reaction parameter, unsteadiness parameter, Schmidt number, or the buoyancy is increased. However, an opposite trend is observed when either the micropolar parameter or magnetic field parameter is increased.

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Declaration of Competing Interest The authors declare that there is no conflict of interest in the use of the above mentioned software.

Acknowledgment The first author wishes to acknowledge financial support from the University of Venda.

References [1] A.C. Eringen, Theory of micropolar fluids, Journal of Mathematics and Mechanics 16 (1966) 1–18. [2] M.M. Rahman, I.A. Eltayeb, S.M.M. Rahman, Thermo-micropolar fluid flow along a vertical permeable plate with uniform surface heat flux in the presence of heat generation, Thermal Science 13 (2009) 23–36. [3] A.C. Eringen, Microcontinuum Field Theories. II: Fluent Media, Springer, New York, 2001. [4] O.A. Bég, R. Bhargava, M.M. Rashidi, Numerical Simulation in Micropolar Fluid Dynamics, Lambert Academic Publishing, 2010. [5] D.A.S. Rees, A.P. Bassom, The Blasius boundary layer flow of a micropolar fluid, International Journal of Engineering Science 34 (1996) 113–124. [6] A. Raptis, Flow of a micropolar fluid past a continuously moving plate by the presence of radiation, International Journal of Heat and Mass Transfer 41 (1998) 2865–2866. [7] N.A. Kelson, N.A. Farrell, Micropolar flow over a porous stretching sheet with strong suction or injection, International Communications in Heat and Mass Transfer 28 (2001) 479–488. [8] S. Rawat, R. Bhargava, O.A. Bég, A finite element study of the transport phenomena in MHD micropolar flow in a Darcy–Forchheimer porous medium, in: Proc. WCECS, San Francisco, CA, 2007. [9] S.S. Motsa, S. Shateyi, P. Sibanda, A model of steady viscous flow of a micropolar fluid driven by injection or suction between a porous disk and a non-porous disk using a novel numerical technique, Canadian Journal of Chemical Engineering 88 (2010) 991–1002. [10] B.C. Sakiadis, Boundary layer behaviour on continuous solid surfaces; II. The boundary layer on continuous flat surface, AIChE Journal 7 (1961) 221–225. [11] H.A.M. El-Arabawy, Effect of suction/injection on the flow of a micropolar fluid past a continuously moving plate in the presence of radiation, International Journal of Heat and Mass Transfer 46 (8) (2003) 1471–1477. [12] R. Nazar, N. Amin, D. Filip, I. Pop, Stagnation point flow of a micropolar fluid towards a stretching sheet, International Journal of Non-Linear Mechanics 39 (7) (2004) 1227–1235. [13] Finite element analysis of combined heat and mass transfer in hydromagnetic micropolar flow along a stretching sheet, Computational Materials Science 46 (4) (2009) 841–848. [14] S. Rawat, R. Bhargava, S. Kapoor, O.A. Beg, Heat and mass transfer of a chemically reacting micropolar fluid over a linear stretching sheet in Darcy–Forheimer porous medium, International Journal of Computational and Applied Mathematics 44 (6) (2012) 40–51.

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[15] M. Abd El-Aziz, Mixed convection flow of a micropolar fluid from an unsteady stretching surface with viscous dissipation, Journal of the Egyptian Mathematical Society 21 (2013) 385–394. [16] M.A.A. Mahmoud, S.E. Waheed, MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity, Journal of the Egyptian Mathematical Society 20 (2012) 20–27. [17] K. Das, Slip effects on MHD mixed convection stagnation point flow of a micropolar fluid towards a shrinking vertical sheet, Computers and Mathematics with Applications 63 (2012) 255–267. [18] Md. Ziaul Haque, Md.M. Alam, M. Ferdows, A. Postelnicu, Micropolar fluid behaviors on steady MHD free convection and mass transfer flow with constant heat and mass fluxes, Joule heating and viscous dissipation, Journal of King Saud University. Engineering Sciences 24 (2012) 71–84. [19] M.M. Rashidi, B. Rostami, N. Freidoonimehr, S. Abbasbandy, Free convective heat and mass transfer for MHD fluid flow over a permeable vertical stretching sheet in the presence of the radiation and buoyancy effects, Ain Shams Engineering Journal 5 (2014) 901–912. [20] N. Bachok, A. Ishak, R. Nazar, Flow and heat transfer over an unsteady stretching sheet in a micropolar fluid, Meccanica 46 (2011) 935–942. [21] A.A.R.M. Kasim, N.F. Mohammad, S. Shafie, Soret and Dufour effects on unsteady MHD flow of a micropolar fluid in the presence of thermophoresis deposition particle, World Applied Sciences Journal 21 (5) (2013) 766–773. [22] D. Gupta, L. Kumar, B. Singh, Finite element solution of unsteady mixed convection flow of micropolar fluid over a porous shrinking sheet, The Scientific World Journal (2014) 362352, 11 pages. [23] B. Mohanty, S.R. Mishra, H.B. Pattanayak, Numerical investigation on heat and mass transfer effect of micropolar fluid over a stretching sheet through porous media, Alexandria Engineering Journal 54 (2015) 223–232. [24] S. Srinivas, P.B.A. Reddy, B.S.R.V. Prasad, Non-darcian unsteady flow of a micropolar fluid over a porous stretching sheet with thermal radiation and chemical reaction, Heat Transfer–Asian Research 44 (2) (2015). [25] M.A.A. Mahmoud, S.E. Waheed, Effects of slip and heat generation/absorption on MHD mixed convection flow of a micropolar fluid over a heated stretching surface, Mathematical Problems in Engineering (2010), https://doi.org/10.1155/2010/579162. [26] H.I. Andersson, J.B. Aarserth, B.S. Dandapat, Heat transfer in an liquid film on an unsteady stretching sheet, International Journal of Heat and Mass Transfer 36 (1) (2000) 69–74. [27] L.N. Trefethen, Spectral Methods in MATLAB, SIAM, 2000. [28] S. Shateyi, G.T. Marewo, On a new numerical analysis of the hall effect on MHD flow and heat transfer over an unsteady stretching permeable surface in the presence of thermal radiation and a heat source/sink, Boundary Value Problems 2014 (2014) 2180.

On the bivariate spectral quasilinearization method for nonlinear boundary layer partial differential equations

8

Vusi M. Magagulaa,b,c , Sandile S. Motsaa,b , Precious Sibandaa of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Pietermaritzburg, South Africa, b Faculty of Science and Engineering, Department of Mathematics, University of Swaziland, Kwaluseni, Swaziland, c DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Pretoria, South Africa a School

8.1 Introduction Nearly all physical processes are described by systems of complex nonlinear differential equations. Finding solutions to such equations poses significant challenges since in most instances these equations have no closed-form solutions. Currently there exist a wide range of both numerical and analytical techniques to approximate solutions to such equations. However, these methods are not universally applicable, may suffer from instabilities, or may not be sufficiently accurate or computational efficient for certain uses. Devising new and more efficient algorithms is always a challenge and an interesting research objective. Various numerical techniques have been developed to solve a class of nonsimilar boundary layer equations. The problem considered in this paper relates to the flow of a viscous incompressible magnetohydrodynamic fluid that finds applications in many engineering processes. The nonlinear equations that describe this fluid flow model have been solved previously by Yih [1] using the Keller-box method. The Kellerbox method has been widely used to solve nonsimilar boundary layer equations. For example, Cebeci [2], Yih [1,3], Aydin [4], Prasad [5], and Afify [6] have all used the Keller-box method and its variants to solve nonsimilar boundary layer equations. The Keller-box method is an implicit finite difference scheme that is second-order accurate both in space and time. It permits the step sizes in time and space variables to be nonuniform. This makes it efficient and appropriate for the solution of nonsimilar boundary layer equations. The main disadvantage of the method is that the computational effort per time step is expensive due to the need to replace higher-order derivatives by first-order derivatives, so that an nth order nonsimilar boundary layer equation is expressed as a system of n first-order equations [7]. The size of the coefficient matrix from discretization increases due to introduction of the derivatives as unknowns which increases the computational cost and time. This in turn implies that a lot of grid points are needed to achieve accurate results. Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00016-5 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Other forms of finite differences have been used to solve a class of nonsimilar boundary layer equations. Watanabe [8,9] used the backward finite differences, Ariel [10] used a combination of finite differences with quasilinearization, and Watabane [11] used a Gregory–Newton finite difference method to solve a class of nonsimilar boundary layer equations. Chamka [12,13] used an implicit iterative tridiagonal finite difference method. Implicit methods are unconditionally stable, whereas the explicit methods break down if the time step is too large. The extra accuracy and efficiency of an implicit method may offset the computing time overhead of solving the linear system using an iterative method and hence the extra accuracy is a drawback for the method. The Runge–Kutta–Fehlberg scheme with a shooting technique has been utilized to find numerical approximation of nonsimilar boundary layer equations [14–17]. However, for stiff problems, explicit Runge–Kutta methods are very inefficient. Spectral methods have been used successfully in many different fields in the sciences because of they give accurate solutions of differential equations. Chebyshev spectral methods are defined everywhere in the computational domain. Thus, it is easy to obtain an accurate value of the function under consideration at any point of the domain. Spectral collocation methods are simple to implement and are adaptable to various problems, including variable coefficient and nonlinear differential equations. The interest in using Chebyshev spectral methods in solving nonlinear differential equation stems from the fact that these methods require fewer grid points to achieve accurate results and are computationally efficient when solving problems with smooth solutions. Spectral methods coupled with finite differences have been used to solve nonlinear partial differential equations [18]. The spectral methods were used to solve the partial differential equations in space and finite differences in time. However, applying finite differences in time compromises the accuracy and computationally efficiency of the spectral methods. We can solve this problem by applying spectral methods in both space and time. This idea was successfully used to solve nonlinear evolution equations by Motsa [19]. The method is termed the bivariate interpolation spectral quasilinearization method (BI-SQLM). Motsa [19] observed that the method achieved accurate results with relatively fewer spatial grid points. It was observed that the method converged fast to the exact solution and approximated the solution of the problem in a computationally efficient manner with simulations completed in fractions of a second. The objective of this paper is to use for the first time the bivariate interpolation spectral quasilinearization method (BI-SQLM) to solve a system of nonsimilar boundary layer equations that model magnetohydrodynamic forced convection flow adjacent to a nonisothermal wedge. This problem was previously solved numerically using the Keller-box method by Yih [1]. We show that the BI-SQLM method can be used to solve nonlinear differential equations, with particular reference to nonsimilar boundary layer equations. We propose to show that the method is more accurate than some traditional numerical methods, being both computationally efficient and robust [24–26].

On the bivariate spectral quasilinearization method for nonlinear boundary layer partial differential

179

8.2 Bivariate interpolated spectral quasilinearization method In this section we consider nonsimilar boundary layer equations that model the problem of magnetohydrodynamic forced convection flow adjacent to a nonisothermal wedge. The equations are given in dimensionless form as [1]  2 

∂ 2f ∂f ∂ 3f ∂f + βf + m 1 − + ξ 1 − ∂η ∂η ∂η3 ∂η2

∂f ∂ 2 f ∂ 2 f ∂f = (1 − m)ξ − 2 , (8.1) ∂η ∂η∂ξ ∂η ∂ξ  2 

2 ∂θ ∂f ∂f ∂ 2f ∂f 1 ∂ 2θ + βf − (m + ξ ) − 2mθ + Ec +ξ P r ∂η2 ∂η ∂η ∂η ∂η ∂η2

∂f ∂θ ∂θ ∂f = (1 − m)ξ − , (8.2) ∂η ∂ξ ∂η ∂ξ subject to the boundary conditions f (0, ξ ) = 0,

∂f (0, ξ ) = 0, ∂η

∂f (∞, ξ ) = 1, ∂η

θ (0, ξ ) = 1,

θ (∞, ξ ) = 0, (8.3)

where m, P r, Ec, ξ , η, θ , and f are the dimensionless pressure gradient parameter, Prandtl number, Eckert number, magnetic parameter, pseudosimilarity variable, dimensionless temperature function, dimensionless stream function, respectively, and β = (1 + m)/2. In the analysis of boundary layer flow problems, quantities of physical interest are the skin friction, and the local Nusselt number given in dimensionless form (see Yih [1]) as ∂ 2f (ξ, 0), ∂η2 1/2 ∂θ N ux = −Rex (ξ, 0), ∂η 1/2

Cf Rex

=2

(8.4) (8.5)

respectively, where Rex is the local Reynolds number and Cf is the local friction coefficient. We now find the approximate solutions of Eqs. (8.1)–(8.3). We apply the quasilinearization method by assuming that the difference fr+1 − fr , and all its derivatives, are small. The quasilinearization method is the Taylor series expansion about a point of the first two linear terms. It was introduced by Bellman and Kalaba [22]. Applying the quasilinearization method to Eq. (8.1) and rearranging terms in Eq. (8.2)

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gives    fr+1 + a1,r (η, ξ )fr+1 + a2,r (η, ξ )fr+1 + a3,r (η, ξ )fr+1 + a4,r (η, ξ )

 ∂fr+1

∂ξ

∂fr+1 + a5,r (η, ξ ) = a6,r (η, ξ ), ∂ξ

(8.6)

  θr+1 + α1,r (η, ξ )θr+1 + α2,r (η, ξ )θr+1 + α3,r (η, ξ )

∂θr+1 = α4,r (η, ξ ), ∂ξ

(8.7)

where the primes denote differentiation with respect to η and ∂fr ∂f  , a2,r (η, ξ ) = −2mfr − ξ − (1 − m)ξ r , ∂ξ ∂ξ    a3,r (η, ξ ) = βfr , a4,r (η, ξ ) = −(1 − m)ξfr , a5,r (η, ξ ) = (1 − m)ξfr ,

∂f  ∂fr a6,r (η, ξ ) = βfr fr − m(fr )2 − m − ξ − (1 − m)ξ f  r − f  , ∂ξ ∂ξ ∂fr , α2,r (η, ξ ) = −2mP rfr , α1,r (η, ξ ) = βP r fr + P r(1 − m)ξ ∂ξ α3,r (η, ξ ) = −P r(1 − m)ξfr ,   α4,r (η, ξ ) = −EcP r (fr )2 − (m + ξ )fr + ξ(fr )2 . a1,r (η, ξ ) = βfr + (1 − m)ξ

The solution of the now linear differential equation (8.6) is obtained by approximating the solution f (η, ξ ) using a Lagrange polynomial F (η, ξ ) which interpolates f (η, ξ ) at selected collocation points, 0 = ξ 0 < ξ 1 < ξ 2 < · · · < ξ Nξ = L ξ . Thus, the approximation for f (η, ξ ) has the form f (η, ξ ) ≈

Nξ  j =0

F (η, ξj )Lj (ξ ) =

Nξ 

(8.8)

Fj (η)Lj (ξ ),

j =0

where Fj (η) ≡ F (η, ξj ) and Lj (ξ ) is the characteristic Lagrange cardinal polynomial defined as Lj (ξ ) =

 Nξ , ξ − ξk 0 , with Lj (ξk ) = δj k = 1 ξ j − ξk

j =0 j =k

if j = k, if j = k.

(8.9)

The solutions Fj (η) are obtained by substituting (8.8) into (8.6) and requiring the equation to be satisfied exactly at the points ξi , i = 0, 1, 2, . . . , Nξ . For the derivatives of the Lagrange polynomial with respect to ξ to be computable analytically, it is convenient to transform the interval of ξ ∈ [0, Lξ ] to that of ζ ∈ [−1, 1] then choose

On the bivariate spectral quasilinearization method for nonlinear boundary layer partial differential

181



iπ as collocation points. After using Nξ a linear transformation to transform ξ to the new variable ζ , the derivative of f  with respect to ξ at the collocation points ζj is computed as

Chebyshev–Gauss–Lobatto points ζi = cos

 Nξ Nξ   dLj ∂f    =2 Fj (η) di,j Fj (η), i = 0, 1, 2, . . . , Nξ , (8.10) (ζi ) = ∂ξ ξ =ξi dζ j =0

j =0

where di,j =

dLj (ζi ), i = 0, 1, . . . , Nξ , dζ

(8.11)

are entries of the standard Chebyshev differentiation matrix (see, for example, [20, 21]), and d = (2/Lξ )d. Applying the collocation at ξi in (8.6) and (8.7) gives    Fr+1,i (η) + a1,r Fr+1,i (η) + a2,r Fr+1,i (η) + a3,r Fr+1,i (η) (i)

(i) + a4,r

Nξ 

(i)

(i)  di,j Fr+1,j (η) + a5,r

j =0

(i)

Nξ 

(i) di,j Fr+1,j (η) = a6,r ,

(8.12)

j =0

(i)  (i) (i) r+1,i (η) + α1,r r+1,i (η) + α2,r r+1,i (η) + α3,r

Nξ 

(i) di,j r+1,j = α4,r ,

j =0

(8.13) (i)

(i)

where ak,r ≡ ak,r (η, ξi ) (k = 1, 2, 3, 4, 5, 6) and αs,r ≡ αs,r (η, ξi ) (s = 1, 2, 3, 4). Consequently, for each ξi , Eqs. (8.12) and (8.13) form a system of linear ordinary equations with variable coefficients. To solve Eqs. (8.12) and (8.13), we apply the Chebyshev spectral collocation independently in the η direction by choosing Nη + 1 Chebyshev–Gauss–Lobatto points 0 = η0 < η1 < · · · < ηNη = ηe , where ηe is a finite value that is chosen to be sufficiently large to approximate the conditions at infinity. Again, before implementing the collocation, the interval of η ∈ [0, ηe ] is transformed into that of τ ∈ [−1, 1] using

a linear transformation. Thus, the collocation points are jπ . chosen as τj = cos Nη The derivatives with respect to η are defined is terms of the Chebyshev differentiation matrix as  p  Nη 2 d p Fr+1,i  p = Dj,k Fr+1,i (τk ) = Dp Fr+1,i , (8.14) dηp η=ηj ηe k=0

where p is the order of the derivative, D = (2/ηe )[Dj,k ] (j, k = 0, 1, 2, . . . , Nη ), with [Dj,k ] being an (Nη + 1) × (Nη + 1) Chebyshev derivative matrix, and the vector Fr+1,i is defined as Fr+1,i = [Fr+1,i (τ0 ), Fr+1,i (τ1 ), . . . , Fr+1,i (τNη )]T .

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Applications of Heat, Mass and Fluid Boundary Layers

Thus applying Eq. (8.14) in Eqs. (8.12) and (8.13) gives A Fr+1,i − 2ξi (1 − ξi ) (i)

Nξ 

(i)

di,j DFr+1,j = R1 ,

(8.15)

(i) 1 ,

(8.16)

j =0

B(i) r+1,i + α (i) 3,r

Nξ 

di,j  r+1,j =

j =0

where (i)

(i)

(i)

(i)

A(i) = a0,r D3 + a1,r D2 + a2,r D + a3,r , (i) (i) 2 B(i) = α (i) 0,r D + α 1,r D + α 2,r , (i)

(i)

R1 = a6,r , (i) 1

(i)

= α 4,r .

(i) (i) ak,r (k = 0, 1, 2, 3, 4, 5), α s,r (s = 0, 1, 2, 3) are diagonal matrices with vectors (i)

(i)

(i)

(i)

(i)

(i)

[ak,r (τ0 ), ak,r (τ1 ), . . . , ak,r (τNx )]T , [αs,r (τ0 ), αs,r (τ1 ), . . . , αs,r (τNx )]T placed on the (i) (i) (i) (i) (i) (i) main diagonal, a6,r = [a6,r (τ0 ), a6,r (τ1 ), . . . , a6,r (τNx )]T , and α 4,r = [α4,r (τ0 ), (i)

(i)

α4,r (τ1 ), . . . , α4,r (τNx )]T . After imposing the boundary conditions, for each i = 0, 1, . . . , Nξ , Eqs. (8.15) and (8.16) can be written in matrix form as ⎡

A0,0 ⎢ A1,0 ⎢ ⎢ .. ⎣ .

A0,1 A1,1 .. .

... ... .. .

A0,Nξ A1,Nξ .. .

ANξ ,0

ANξ ,1

...

ANξ ,Nξ

B0,0 ⎢ B1,0 ⎢ ⎢ .. ⎣ .

B0,1 B1,1 .. .

... ... .. .

B0,Nξ B1,Nξ .. .

BNξ ,0

BNξ ,1

...

BNξ ,Nξ

⎡

⎤⎡

Fr+1,0 ⎥ ⎢ Fr+1,1 ⎥⎢ ⎥ ⎢ .. ⎦⎣ .

⎤

 r+1,0 ⎥ ⎢ r+1,1 ⎥⎢ ⎥⎢ .. ⎦⎣ .  r+1,Nξ

(0)

R1 (1) ⎥ ⎢ R1 ⎥ ⎢ ⎢ ⎥=⎢ . ⎦ ⎣ ..

(Nξ )

Fr+1,Nξ ⎤⎡

⎡

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

(8.17)

R1 ⎤

⎡

(0) 1 (1) 1

⎤

⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ , ⎥=⎢ . ⎥ . ⎦ ⎣ . ⎥ ⎦

(8.18)

(Nξ ) 1

where (i)

(i)

Ai,i = A(i) + a4,r di,i D + a5,r di,i , i = 0, 1, . . . , Nξ , (i)

(8.19)

Ai,j = a4,r di,j D + a5,r di,j , when i = j ,

(i)

(8.20)

(i) Bi,i = B(i) + α 3,r di,i , (i) Bi,j = α 3,r di,i .

(8.21) (8.22)

On the bivariate spectral quasilinearization method for nonlinear boundary layer partial differential

183

Eqs. (8.17) and (8.18) can be expressed in the form AF = R, = . B

(8.23) (8.24)

The solutions are found by multiplying by the inverses of A and B, respectively, to get

8.3

F = A−1 R,

(8.25)

 = B−1 .

(8.26)

Results and discussion

In this section we present the numerical solutions obtained using the BI-SQLM algorithm. The results presented in this section were generated using MATLAB . The number of collocation points in the space η variable used to generate the results is Nη = 60 in Tables 8.1 through 8.4. The number of collocation points in the magnetic parameter ξ variable used is 10 ≤ Nξ ≤ 30 in all cases. It was found that sufficient accuracy was achieved using these values in all numerical simulations. In Tables 8.1 through 8.3, we compare our results with those of Yih [1]. The results are in excellent agreement. The time to compute the approximate solutions by the Bivariate spectral quasilinearization method is displayed to give an indication of the computational speed. The time to compute the approximate solutions using the Bivariate spectral quasilinearization method is less than three seconds for all cases which is remarkable since traditional numerical methods require more computational time. We remark that in Yih’s article [1], the stopping criteria and time steps were not provided, and hence we were not able to compare the time to compute the same results using the Keller-box method. Table 8.1 Comparison of the numerical values of the skin friction f  (0, 0) for various values of m. m \ Nξ −0.05 0.0 1/3 1.0

10 0.213484 0.332057 0.757448 1.232588

15 0.213484 0.332057 0.757448 1.232588

20 0.213484 0.332057 0.757448 1.232588

25 0.213484 0.332057 0.757448 1.232588

Time (s)

0.181133

0.469017

0.908688

1.532474

Yih [1] 0.213484 0.332057 0.757448 1.232588

Table 8.1 shows the numerical values of the skin friction f  (0, 0) for various values of m. The numerical values obtained using the BI-SQLM method agree exactly with those obtained by Yih [1] using the finite differences based method, the Kellerbox method. We also note that the BI-SQLM approximates the skin friction within a fraction of a second.

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Table 8.2 Numerical values of the skin friction f  (0, ξ ) at different values of ξ when m = 1. ξ \ Nξ 0 1 4 9 25

10 1.232588 1.585336 2.346667 3.240943 5.147966

15 1.232588 1.585329 2.346663 3.240950 5.147966

20 1.232588 1.585331 2.346663 3.240950 5.147966

25 1.232588 1.585331 2.346663 3.240950 5.147966

CPU Time (s)

0.448792

0.83394

1.410784

2.250584

Yih [1] 1.232588 1.585331 2.346663 3.240950 5.147964

Table 8.2 shows the numerical values of the skin friction f  (0, ξ ) at different values of ξ when m = 1. The numerical values obtained using the BI-SQLM method are in agreement with those obtained by Yih [1] using the Keller-box method. It should be noted that the BI-SQLM approximates the skin friction within a fraction of a second. Table 8.3 Numerical values of the Nusselt number −θ  (0, ξ ) at different values of ξ when m = 0, P r = Ec = 1. ξ \ Nξ 0.0 0.5 1.0 1.5 2.0

10 0.166029 0.201426 0.216787 0.226296 0.232981

15 0.166029 0.201411 0.216786 0.226302 0.232981

20 0.166029 0.201411 0.216786 0.226302 0.232981

25 0.166029 0.201411 0.216786 0.226302 0.232981

CPU Time (s)

0.15710

0.350761

0.834859

1.496277

Yih [1] 0.166029 0.201452 0.216814 0.226323 0.232998

Table 8.3 shows the numerical values of the Nusselt number −θ  (0, ξ ) at different values of ξ when m = 0, P r = Ec = 1. The numerical values obtained using the BISQLM method are in agreement with those obtained by Yih [1] using the Keller-box method. The BI-SQLM approximates the skin friction within a fraction of a second. Table 8.4 Numerical values of the Nusselt number −θ  (0, ξ ) at different values of ξ when m = 1/2, and P r = Ec = 1. ξ \ Nξ 0 0.5 1 1.5 2

10 0.720626 0.759480 0.786931 0.808036 0.825096

15 0.720626 0.759480 0.786931 0.808036 0.825096

20 0.720626 0.759480 0.786931 0.808036 0.825096

25 0.720626 0.759480 0.786931 0.808036 0.825096

30 0.720626 0.759480 0.786931 0.808036 0.825096

CPU Time (s)

0.239541

0.468491

0.865754

1.488947

2.17121

Table 8.4 shows the numerical values of the Nusselt number −θ  (0, ξ ) at different values of ξ when m = 0.5, P r = Ec = 1. In this case, we use the grid independence

On the bivariate spectral quasilinearization method for nonlinear boundary layer partial differential

185

test to show that our method can generate approximate values of the Nusselt number with any m in the range of the values of m. We also note that the BI-SQLM approximates the skin friction within a fraction of a second. Theorem 8.3.1 (Order of Convergence). A method is convergent with convergence rate c if ||f (k+1) − f ∗ ||∞ = V, k→∞ ||f (k) − f ∗ ||c∞ lim

(8.27)

where V is a finite positive constant (see [23]). The convergence rate c can be calculated from the ratio

ek+2 ek+1 c = , (8.28) ek+1 ek and hence the convergence rate c is given by c=

ln(ek+2 /ek+1 ) , ln(ek+1 /ek )

(8.29)

where ek+2 , ek+1 , and ek are the norms computed using three consecutive iterations. Bellman and Kalaba [22] observed that the quasilinearization method converges quadratically, that is, c = 2. We prove numerically that the BI-SQLM converges quadratically. Table 8.5 shows the convergence rate values of f (η, ξ ) at different values of Nη , when Nξ = 8, m = 0, and P r = Ec = 1. Table 8.6 shows the convergence rate values of θ (η, ξ ) at different values of Nη , when Nξ = 8, m = 0, and P r = Ec = 1. The values in Tables 8.5 and 8.6 prove numerically that BI-SQLM converges quadratically. Table 8.5 Convergence rate values of f (η, ξ ) at different values of Nη when Nξ = 8, m = 0, and P r = Ec = 1. Iterations \Nη 3 4 5 6 7 8

Convergence rate 15 2.0100 2.0000 1.9800 2.0000 2.0000 2.0000

30 2.0000 2.0000 2.0000 2.0000 1.9900 2.0000

45 2.0000 2.0000 2.0000 2.0000 2.0000 1.9800

60 2.0000 2.0000 2.0000 2.0000 2.0000 1.9800

Table 8.7 shows the skin friction and Nusselt number results obtained by solving analytically the governing equations. These results were obtained by directly solving the differential equations sequences obtained, for the large ξ limiting cases, using Mathematica’s DSolve command. The number of terms of the series required to give

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Applications of Heat, Mass and Fluid Boundary Layers

Table 8.6 Convergence rate values of θ(η, ξ ) at different values of Nη when Nξ = 8, m = 0, and P r = Ec = 1. Iterations \ Nη

Convergence rate 15 1.9500 1.9500 2.0500 2.0300 2.0000 1.9800

3 4 5 6 7 8

30 1.9600 1.9900 2.0300 2.0100 2.0000 1.9900

45 1.9600 1.9900 2.0300 2.0100 2.0000 1.9900

60 1.9600 1.9900 2.0300 2.0100 2.0000 1.9900

Table 8.7 Comparison of the BI-SQLM for f  (0, ξ ) and θ  (0, ξ ) against the exact solution. ξ 5 10 15 20 25 30 35 40

BI-SQLM f  (0, ξ )

θ  (0, ξ )

0.2842719 0.1428115 0.09523208 0.07142715 0.05714239 0.04761886 0.04081624 0.03571424

−3.5066677 −7.0008399 −10.5002490 −14.0001050 −17.5000538 −21.0000311 −24.5000196 −28.0000131

it

Exact solution f  (0, ξ )

5 4 4 4 3 3 3 3

0.2842719 0.1428115 0.09523208 0.07142714 0.05714239 0.04761886 0.04081624 0.03571424

θ  (0, ξ ) −3.5066677 −7.0008399 −10.5002490 −14.0001050 −17.5000538 −21.0000311 −24.5000196 −28.0000131

k 3 2 1 1 1 1 1 1

results that are accurate up to at least 8 decimal digits are indicated in the table. The analytic results are compared with those obtained by using the BI-SQLM method. In order to determine the level of accuracy of the BI-SQLM approximate solution at a particular time level, we report the maximum error which is defined by  - . Ef = max |Fr+1,i − Fr,i  |∞ , : 0 ≤ i ≤ Nξ , i  - . r+1,i −  r,i  |∞ , : 0 ≤ i ≤ Nξ , Eθ = max | i

(8.30) (8.31)

where Fr+1,i and Fr,i are the approximate solutions obtained by Eq. (8.25) at the current and previous time steps, respectively. Figs. 8.1 and 8.2 show the convergence graphs of f and θ , respectively. The BI-SQLM method converges fully after eight iterations. In Fig. 8.1, the BI-SQLM method converges to 10−94 after eight iterations. We note that it converges quadratically and hence the parabolic shape of the curve. This is in agreement with the results in Tables 8.5 and 8.6 and the theory that the quasilinearization method converges quadratically as observed by Bellman and Kalaba [22]. In Fig. 8.2, the BI-SQLM method converges quadratically to 10−95 after less than eight iterations. This in turn implies that the method needs few iterations to

On the bivariate spectral quasilinearization method for nonlinear boundary layer partial differential

187

Figure 8.1 Ef vs iterations for all η and ξ .

Figure 8.2 Eθ vs iterations for all η and ξ .

converge fully and hence takes less computational time. This is in agreement with the computational times displayed in Tables 8.1, 8.2, 8.3, and 8.4. Figs. 8.3 and 8.4 show the velocity and temperature profiles of the governing equations for different values of ξ . Increasing the values of ξ reduces the values of the velocity and temperature profiles. We also observe that the BI-SQLM method is capable of solving system of equations with large values of ξ .

Figure 8.3 Velocity profile.

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 8.4 Temperature profile.

8.4

Conclusion

In this paper we have presented an innovative BI-SQLM method with quadratic convergence. We used the method to solve a system of nonlinear differential equations that model magnetohydrodynamic forced convection flow adjacent to a nonisothermal wedge. The main goal of the study was to develop the BI-SQLM algorithm for solving nonsimilar boundary layer equations and to assess the accuracy, robustness, and effectiveness of the method. Residual error analysis, convergence analysis, and grid independence tests were conducted to establish the accuracy, convergence, and validity of the algorithm. Computational order of convergence was proved numerically, and this was found to be in agreement with theory of quasilinearization methods. The results were validated against published results. It was observed that the method converges quadratically while using only a few discretization points. We further observed that the BI-SQLM algorithm minimizes computation time and that its accuracy is independent of the size of the dependant variables in the differential equation. The study confirms that BI-SQLM gives accurate results in a computationally efficient manner. The success of the method can be attributed to the use of the Chebyshev spectral collocation method with bivariate Lagrange interpolation both in space and time. This work contributes to the existing body of literature by providing a new tool for solving systems of nonlinear nonsimilar boundary layer equations. Further work needs to be done to establish whether BI-SQLM can be equally successful in solving higher-order nonlinear systems of differential equations arising in fluid mechanics.

On the bivariate spectral quasilinearization method for nonlinear boundary layer partial differential

189

Declaration of Competing Interest The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgments The authors would like to acknowledge funding from the School of Mathematics, Statistics, and Computer Science at the University of KwaZulu-Natal. The first author would also like to acknowledge funding from the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoEMaSS). However, the opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the CoE-MaSS.

References [1] K.A. Yih, MHD forced convection flow adjacent to a non-isothermal wedge, International Communications in Heat and Mass Transfer 26 (1999) 819–827. [2] T. Cebeci, P. Bradshaw, Physical and Computational Aspects of Convective Heat Transfer, Springer-Verlag, New York, 1984, p. 385. [3] K.A. Yih, Uniform suction or blowing effect on forced convection about a wedge, uniform heat flux, Acta Mechanica 128 (1998) 173–181. [4] O. Aydin, A. Kaya, MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate, Applied Mathematical Modelling 33 (2009) 4086–4096. [5] K.V. Prasad, P.S. Datti, K. Vajravelu, MHD mixed convection flow over a permeable nonisothermal wedge, Journal of King Saud University, Science 25 (2013) 313–324. [6] A.A. Afify, M.A.A. Bazid, MHD Falkner–Skan flow and heat transfer characteristics of nanofluids over a wedge with heat source/sink effects, Journal of Computational and Theoretical Nanoscience 11 (8) (2014) 1844–1852, (9). [7] J.C. Tannehill, D.A. Anderson, R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, second edition, Taylor and Francis, 1997. [8] T. Watanabe, Magneto hydrodynamic stability of boundary layers along a flat plate with pressure gradient, Acta Mechanica 65 (1986) 41–50. [9] T. Watanabe, I. Pop, Thermal boundary layers in magneto hydrodynamic flow over a flat plate in the presence of a transverse magnetic field, Acta Mechanica 105 (1994) 233–238. [10] P.D. Ariel, Hiemenz flow in hydromagnetics, Acta Mechanica 103 (1994) 31–43. [11] T. Watanabe, I. Pop, Hall effects on magnetohydrodynamic boundary layer flow over a continuous moving plate, Acta Mechanica 108 (1995) 35–47. [12] A.J. Chamkha, M. Mujtaba, A. Quadri, I. Camille, Thermal radiation effects on MHD forced convection flow adjacent to a non-isothermal wedge in the presence of a heat source or sink, Heat and Mass Transfer 39 (2003) 305–312. [13] C.J. Chamkha, A.M. Rashad, MHD forced convection flow of a nanofluid adjacent to a non-isothermal wedge, Computational Thermal Sciences: An International Journal (2014), https://doi.org/10.1615/ComputThermalScien.2014005800, or Computational Thermal Sciences 27–39.

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[14] D. Pal, H. Mondal, Influence of temperature-dependent viscosity and thermal radiation on MHD forced convection over a non-isothermal wedge, Applied Mathematics and Computation 212 (2009) 194–208. [15] D. Pal, H. Mondal, Influence of thermophoresis and Soret–Dufour on magnetohydrodynamic heat and mass transfer over a non-isothermal wedge with thermal radiation and Ohmic dissipation, Journal of Magnetism and Magnetic Materials 331 (2013) 250–255. [16] S. Mukhopadhyay, I.C. Mandal, Boundary layer flow and heat transfer of a Casson fluid past a symmetric porous wedge with surface heat flux, Chinese Physics 23 (2014). [17] Z. Uddin, M. Kumar, S. Harmand, Influence of thermal radiation and heat generation or absorption on MHD heat transfer flow of a micropolar fluid past a wedge with hall and ion slip currents, Thermal Science 18 (2014) 489–502. [18] S.S. Motsa, P.G. Dlamini, M. Khumalo, Spectral relaxation method and spectral quasilinearization method for solving unsteady boundary layer flow problems, Advances in Mathematical Physics 2014 (2014) 341964, https://doi.org/10.1155/2014/341964, 12 pages. [19] S.S. Motsa, V.M. Magagula, P. Sibanda, A bivariate Chebyshev spectral collocation quasilinearization method for nonlinear evolution parabolic equations, The Scientific World Journal 2014 (2014) 581987, https://doi.org/10.1155/2014/581987, 13 pages. [20] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988. [21] L.N. Trefethen, Spectral Methods in MATLAB, SIAM, 2000. [22] R.E. Bellman, R.E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems, Elsevier, New York, NY, USA, 1965. [23] S.S. Motsa, P. Sibanda, Some modifications of the quasilinearization method with higherorder convergence for solving nonlinear BVPs, Numerical Algorithms 63 (2013) 399–417. [24] S. Liao, An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate, Communications in Nonlinear Science and Numerical Simulation 11 (2006) 326–339. [25] R. Nazar, N. Amin, I. Pop, Unsteady boundary layer flow due to stretching surface in a rotating fluid, Mechanics Research Communications 31 (2004) 121–128. [26] R. Nazar, N. Amin, D. Filip, Ioan Pop, Unsteady boundary layer flow in the region of the stagnation point on a stretching sheet, International Journal of Engineering Science 42 (2004) 1241–1253.

Mixed convection heat transfer in rotating elliptic coolant channels

9

Olumuyiwa Ajani Lasode Department of Mechanical Engineering, University of Ilorin, Ilorin, Nigeria

9.1 Introduction Holzworth [1] appears to be one of the first investigators to suggest that this problem could be solved by the use of blind radial holes situated in the rotor blade filled with a suitable fluid. This form of arrangement is often referred to as thermosiphon. Owing to the intense centripetal acceleration present, free convection currents occur in the fluid causing the warmer fluid to move towards the axis of rotation. This stream of warm fluid adjacent to the wall is simultaneously replaced by a central core of relatively cool fluid located in the rotor motor shaft, resulting in the blade material being maintained at a high temperature level compatible with its mechanical strength. The motion of the fluid in the thermosiphon tubes described above is entirely due to free convection. In many nonrotating heat transfer systems, the influence of free convection (due to the Earth’s gravitation) is often neglected in comparison with (forced convection due to) pressure gradients. However, in a gas turbine, the centripetal acceleration may be as high as 104 g, the free convection velocities encountered are as high as those usually associated with forced convection in nonrotating devices. Cooling of turbine rotor blades has also been effected by the forced circulation of suitable coolant through internal passages, the circulation of the fluid being maintained by externally generated pressure gradients. Owing to the radial and tangential components of acceleration caused by rotation, modifications to this otherwise forced convection process occur. This class of problems, where forced cooling of rotating components occur, is not only restricted to the field of gas turbines, but also to large electrical machines. The power output from electrical machines is to some extent governed by the permissible temperature rise in the insulation surrounding the rotor conductors. Although cooling of these conductors is commonly achieved by the forced convection of air over the rotor periphery, there are advantages to be gained if the heat transfer is effected through a suitable coolant flowing inside the conductors themselves especially for large electrical generating machines such as those found in hydroelectric power stations. It is thus evident that the problem of forced flow through heated rotating channels is interesting both academically and practically. Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00017-7 Copyright © 2020 Elsevier Ltd. All rights reserved.

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9.1.1 Some historical background and literature review Extensive research works [1–8,12,14,15,17] have been carried out to study heat transfer and fluid flow in rotating and nonrotating coolant channels especially of the circular-type geometry, while researches [9–11,13] carried out on elliptic geometry are limited to nonrotating systems. In his monograph, Morris [2] has demonstrated a number of instances in practice where the effect of rotation on the hydrodynamic and thermal characteristics of channel-type flows may have important consequences on the performance of cooling systems of prime movers. He also critically reviewed the assorted literature available on this generalized topic with a view to present it in a form which bridges the gap between the academic researchers, on the one hand, and the eventual industrial user, on the other hand. A particular work closely related to this research project was cited [2]. Heated flow in circular-sectioned duct was studied using perturbation analysis adopting a power series solution up to first order for a horizontal duct rotating about a parallel axis. The results obtained are used to validate the present work especially for the case of tube eccentricity due to a secondary flow driven by the Coriolis force. It was also shown by analyzing the two wall temperature conditions, that is, the constant wall temperature gradient and the uniform wall temperature, that the Nusselt number is almost the same for both these conditions. Siegel [8] analyzed laminar heat transfer in a tube rotating about an axis perpendicular to the tube axis for a flow that is either radially outward from the axis of rotation or radially inwards to the axis of rotation. The flow considered is fully developed, with uniform heat addition at the tube wall. Power series solution up to second order using Taylor number as a perturbation parameter was constructed. It was shown that the secondary flow induced by the Coriolis term always tends to increase the heat transfer coefficient; this effect can dominate for small heating. For radial inflow, buoyancy tends to increase heat transfer, while for radial outflow, however, buoyancy tends to reduce heat transfer. For large heating, this effect can dominate, and there can be a net reduction in heat transfer coefficient. Bello-Ochende [9] has conducted a numerical study of natural convection in horizontal elliptic cylinders. The method of discretization he proposed always lets mesh points fall on the cylinder boundary so that the problem of irregular boundary is avoided. He presented results of nonuniform heat flux applications at the cylinder periphery in graphical forms for heat transfer and flow regimes for some values of eccentricity and a range of Rayleigh numbers. In another research work, Bello-Ochende [10] studied the thermal problem of transposition-point heat transfer for forced laminar convection in heated horizontal elliptic ducts, using the concept of scale analysis. The obtained results indicate that in the neighborhood of the eccentricity, e = 0.866, optimum results are predicted for the generalized transition-point Nusselt number based on the major diameter and the corresponding generalized thermal entrance length for the parameter space, 0.75 ≤ e < 1.0. Abdel-Wahed et al. [11] have done an extensive experimental investigation in the in the area of laminar developing and fully developed flows and heat transfer in a horizontal elliptic duct. The working fluid was air, and two thermal situations were considered; the first was when the wall temperature was linear. They presented hydrodynamic and thermal results. Morton [12] did a wonderful job along the same lines but with a radial temperature distribution, axial velocity, and stream function all in power series.

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193

Also worthy of note is the work of Adegun [13] who investigated the laminar forced convective heat transfer in an inclined elliptic duct using scale and perturbation techniques. Thermal and hydrodynamic entrance problems were investigated using the scale approach while the perturbation approach was used to analyze the fully developed region of the duct. Useful results were obtained, in particular, for optimum heat transfer, a critical aspect ratio of 0.5 (e = 0.866) is predicted and perturbation results indicate a considerable effect of inclination on circular ducts and elliptic geometry of e = 0.433, while the effect is negligible for e = 0.866. Faris and Viskanta [14] studied laminar combined forced and free convection heat transfer in a horizontal tube using a perturbation method. They presented approximate analytical solutions, as well as average Nusselt number graphically for a range of Prandtl and Grashof numbers. Tormcej and Nandakumar [15] studied mixed convective flow of a power-law fluid in horizontal ducts. Rabadi [16] studied heat transfer in coils under a boundary condition of uniform heat flux. Siegwarth et al. [17] studied the effect of a secondary flow on the temperature field and primary flow in a heated horizontal tube while Van Dyke [18] in his mathematical models solved a lot of perturbation problems in fluid mechanics for both singular and regular perturbation cases. All the same, these investigations served not only as a means to success in this research work but also in comparison and validation.

9.1.2 Objectives of the study The present study is an investigation of laminar combined free and forced convective heat transfer in horizontal elliptic ducts rotating about a parallel axis. A physical model for a solution of the problem is shown in Fig. 9.1. Particular attention is paid to the fully developed flow regime in this study where the temperature and axial velocity are far ahead of axial locations along the ducts where entry effects could be felt. The governing equations of continuity, momentum and energy transfer were solved using single parameter perturbation technique. The technique is an approximate analytic method in which the normalized axial velocity, temperature, and crossflow stream function were expanded in power series using the rotational Rayleigh number, Raτ . For the elliptic tubes concerned, a boundary coordinate, ξ , is developed from the parametric equations of an ellipse for the solution of the normalized governing equations to be valid at any boundary location, for the Dirichlet problem. The normalized equations for the stream function and axial velocity are derived from the momentum equations while a dimensionless form of temperature equation is generated from the energy transport equation. These equations are nondimensionalized by the transformation parameters presented in Sect. 9.2. The effects of the perturbation parameter on the temperature and axial velocity are investigated. The effects of varying eccentricity on the temperature map, axial velocity profile, and flow map have also been investigated. The effect of the perturbation parameter on the mean Nusselt number is studied and the highest value is found to occur for the eccentricity e = 0.433. The effect of varying fluid property – Prandtl number – on the established flow conditions is studied, and remarkable effects are reported for the temperature field, axial velocity profile, peripheral local Nusselt number, and mean Nusselt number. The formulation of the

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governing equation is contained in Sect. 9.2 of this report while Sect. 9.3 presents the single parameter perturbation technique used in the solution of the normalized governing equations. The actual perturbation solution was carried out manually, while a computer program was developed to generate the results and plot the graphs for the sets for the parameter spaces.

9.2 Governing equations for horizontal elliptic duct rotating in parallel mode The physical model and the cylindrical polar coordinate (r, Θ, z) systems are shown in Fig. 9.1. The elliptic duct is horizontal and rotates in the parallel mode. The major diameter is A, and the minor diameter is B. For the flow condition, the following assumptions should be noted. Fully developed flow is treated, hence no axial velocity gradient is included in the formulation. A uniformly heated tube is considered. The uniformity of the thermal boundary condition means uniform heat transfer per unit length of duct considered. This is coupled with the assumption that the thermal conductivity of the tube wall material is high enough to smooth out circumferential variations in wall temperature. The combined assumption of fully developed flow and uniform axial heating allows the fluid temperature distribution to be expressed in the mathematical form, T = τ z + F (r, θ ) ,

(9.1)

where τ is the axial temperature gradient, F (r, Θ) is a function of the crossstream coordinates r and Θ. Eq. (9.1) is applicable at the tube wall, meaning that the wall temperature will increase uniformly in the direction of the flow and will also be functionally related to the crossstream coordinates. The flow is assumed laminar and, with the exception of density, the fluid properties are taken to be constant with temperature. If the velocity components at a point p in the coordinate directions shown in Fig. 9.1 are u, v, and w, then since these velocities refer to a moving frame of reference, the respective acceleration components at the point p are given by the following: (In the radial direction) fr = u

∂u v ∂u v 2 + − − 2Ωv − (r + H cos θ ) Ω 2 , ∂r r ∂θ r

(9.2)

(In the azimuthal direction) fr = u

∂v v ∂v uv + − + 2Ωu − Ω 2 H sin θ, ∂r r ∂θ r

(9.3)

Mixed convection heat transfer in rotating elliptic coolant channels

195

Figure 9.1 Physical model and the cylindrical polar coordinate (r, Θ, z) system for horizontal elliptic duct rotating about a parallel axis.

(In the axial direction) fz = u

∂w v ∂w + , ∂r r ∂θ

(9.4)

where Ω is the angular velocity of the tube, H is the distance between the axis of rotation and the tube axis (see Fig. 9.1). With the exception of the terms involving Ω, the acceleration components above are the same as those in nonrotating tubes. The additional terms may be thought of as body forces, similar to gravity, producing free convection currents. In order to include the effect of free convection in the basic conservation laws, density must vary with temperature. This variation is sufficiently small to be ignored in all terms except the buoyancy force in the momentum equations.

9.2.1 Equations for the working fluid The following are the equations that govern the fluid flow in the elliptic duct. The continuity equation is essentially a mathematical model for the conservation of mass. For a steady incompressible flow, the continuity equation in the dimensional form is ∂u u 1 ∂v + + = 0, ∂r r r ∂θ

(9.5)

where u, v, and w are velocities in the radial, azimuthal, and axial directions, respectively. The momentum equation is derived from Newton’s second law of motion which states that “for a constant mass, the product of mass and acceleration of a body is

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equal to the sum of the external forces acting on the body”. The momentum equations in their dimensional forms in the r, Θ, z directions, respectively, are expressed as follows: (In the radial direction) u

∂u v ∂u v 2 1 ∂P + − =− + Ω 2 β (r + H cos θ ) (Tw − T ) ∂r r ∂θ r ρ ∂r

u 2 ∂v + 2Ωβv (Tw − T ) + ϑ ∇ 2 u − 2 − 2 , r r ∂θ

(9.6)

(In the azimuthal direction) u

v ∂v uv 1 ∂P ∂V + + =− − Ω 2 Hβ (Tw − T ) sin θ − 2Ωβu (Tw − T ) ∂r r ∂θ r ρ ∂θ

v 2 ∂u , (9.7) + ϑ ∇ 2v − 2 − 2 r r ∂θ

(In the axial direction) u

∂w v ∂w 1 ∂P + =− + ϑ∇ 2 w, ∂r r ∂θ ρ ∂z

(9.8)

∂2 1 ∂ ∂2 1 ∂2 + + . + ∂r 2 r ∂r r 2 ∂θ 2 ∂z2

(9.9)

where ∇2 =

The left-hand side (LHS) terms in Eqs. (9.6)–(9.8) represent the inertia terms. The first terms on the right-hand side (RHS) are the pressure gradient terms, in Eqs. (9.6) and (9.7) the two middle terms represent the body forces due to rotation while the last terms represent the viscous or friction terms. For the energy transport equation, it is assumed that there are no chemical reactions, no heat sources within the fluid, and radiation is neglected. Therefore, for a steady incompressible flow without dissipation, the differential form of the energy transport equation is given as u

∂T v ∂T ∂T + +w = α∇ 2 T , ∂r r ∂θ ∂z

(9.10)

where α=

k , ρCp

α is thermal diffusivity, k is thermal conductivity, ρ is fluid density, and Cp is the specific heat at constant pressure.

(9.11)

Mixed convection heat transfer in rotating elliptic coolant channels

197

9.2.2 Normalization parameters For distances along the tube large enough to ignore entry effects, the continuity equation may be satisfied by specifying a dimensionless stream function, ψ, to replace components of velocity in the crossstream directions as

∂ψ v = −υ (9.12) ∂r and u=

υ r



∂ψ ∂θ

(9.13)

.

Because distances far away from the inlet effects are being considered, the pressure distribution must be of the form P = σ z + P (r, θ ) ,

(9.14)

where σ is the axial pressure gradient and P (r, Θ) is a function specifying the distribution of pressure in the (r, Θ)-plane. The nondimensionalization scheme adopted for the dependent and independent variables following Morton [12] and Morris [2] was

r z wa Tw − T R = ;Z = ;W = ;η = , (9.15) a a υ τ aP r where P r is the Prandtl number of the fluid, a is the semimajor diameter of the tube, Tw is the local temperature of the wall, T is the local fluid temperature, and υ is the kinematic viscosity of the fluid. For the derivation of the equation for normalized stream function, ψ, the momentum equations (9.6) and (9.7) are required. The pressure gradient terms in the radial and azimuthal momentum equations (9.6) and (9.7) are eliminated by differentiating equation (9.6) with respect to Θ and Eq. (9.7) with respect to r, and subtracting the result obtained from that of Eq. (9.7). Then, using the aforementioned transformation parameters, that is, (i)

(iii)

u=

υ ∂ψ , r ∂θ

r R= , a

(ii)

v = −υ

(iv)

η=

∂ψ , ∂r

(Tw − T ) , τ aP r

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Applications of Heat, Mass and Fluid Boundary Layers

into the result and collecting like terms yields  

1 ∂ ψ, ∇ 2 ψ ∂η Raτ ∂η 1 ∂η 4 ∇ ψ+ + Raτ cos θ + sin θ + R ∂ (R, θ ) R ∂θ ∂R εa ∂θ Raτ .Ro∗ 1 ∂ (η, ψ) = 0, + Rem R ∂ (R, Θ) where

  ∇ 4ψ = ∇ 2 ∇ 2ψ .

(9.16)

(9.17)

The expanded form of the biharmonic operator is ∇4 =

∂4 ∂4 2 ∂3 1 ∂2 1 ∂ 2 ∂3 2 . (9.18) + + + − + ∂R 4 R ∂R 3 R 2 ∂R 2 R 3 ∂R R 2 ∂θ 2 ∂R 2 R 3 ∂θ 2 ∂R

The Jacobian operator is defined as ∂ (A, B) ∂A ∂B ∂B ∂A = . − . . ∂ (R, θ ) ∂R ∂θ ∂R ∂θ

(9.19)

Eq. (9.16) is the normalized equation for the stream function. The normalization procedure adopted highlights the following dimensionless groups which parametrically govern this problem: Raτ =

Ω 2 Hβτ a 4 , αυ

(9.20)

where Raτ is the rotational Rayleigh number, β is the volumetric coefficient of thermal expansion, Ro∗ =

−a 2 ∂P 2H Ωρυ ∂Z

(9.21)

is the Rossby number, Rem =

−a 3 ∂P 4ρυ 2 ∂z

(9.22)

is the modified Reynolds number, εa =

a H

is the axis displacement parameter, and υ Pr = α is the Prandtl number.

(9.23)

(9.24)

Mixed convection heat transfer in rotating elliptic coolant channels

199

The rotational Rayleigh number, Raτ , emerges from the centripetal buoyancy terms in the momentum equations. This is similar to the Rayleigh number encountered by Morton [12] in the study of buoyancy force due to the Earth’s gravitational acceleration replaced by the centripetal acceleration measured at the centerline. The Rossby number, Ro∗ , has its origin in the Coriolis terms. The modified Reynold’s number, Rem , is defined in an identical mathematical form to the usual through-flow Reynolds number, Re, when the buoyancy effects are not included in the analysis. For the normalized axial velocity, we recall Eq. (9.8) and use the following transformations: W=

wa υ ∂ψ ∂ψ r (Tw − T ) ,u = , v = −υ ,η = ,R = , υ r ∂θ ∂r τ aP r a

∇ 2W +

1 ∂ (ψ, W ) + 4Rem = 0. R ∂ (R, θ )

(9.25)

Eq. (9.25) is the normalized axial velocity equation. For the normalized energy transport equation, substituting u=

υ ∂ψ ∂ψ , v = −υ , r ∂θ ∂r

employing the normalized variables in Eq. (9.10), and rearranging the result gives ∇ 2η +

9.3

P r ∂ (ψ, η) + W = 0. R ∂ (R, Θ)

(9.26)

Governing equations for vertical elliptic duct rotating in parallel mode

The physical model and the cylindrical polar coordinate (r,Θ, z) system are shown in Fig. 9.2. For the flow condition, the following assumptions should be noted: • The flow is laminar and fully developed. • The elliptic duct is vertical and rotates in the parallel mode. • The heated tube is treated and the thermal conductivity of the tube material is high enough to smooth out circumferential variation in wall temperature. • The fluid temperature distribution can be mathematically stated as Tw = T0 + τ z.

(9.27)

Due to the combined assumption of fully developed flow and uniform axial heating, Eq. (9.27) is applicable at the tube wall, meaning that the wall temperature will increase uniformly in the direction of flow. At any axial location, the difference in the wall temperature, Tw , and any local value of temperature in the flow will also be functionally related to the axial temperature gradient.

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 9.2 Physical model, coordinate axes, and regions of the vertical ducts rotating about a parallel axis.

• With the exception of density, the fluid properties are taken to be constant with temperature. Because distances well away from inlet influences are being considered, the pressure distribution is constrained to be of the form P = γ z + p (r, θ ) .

(9.28)

• It is assumed that there are no chemical reactions, no heat sources within the fluid, radiation is neglected, and viscous dissipation is ignored. The following nondimesionalization parameters are adopted for the dependent and independent variables:

r z w Tw − T R = ; ;W = ;η = a a

υ τ aP

r (9.29) μ δψ δψ ,μ = ν = −υ δr r δr The normalized governing equations are as follows. The normalized stream function becomes  

1 ∂η ∂η 1 ∂ ψ, ∇ 2 ψ 4 ∇ ψ+ + Raτ Cos θ + sin θ R ∂ (R, θ ) R ∂θ ∂R ∂η Raτ Ro∗ 1 ∂ (η, ψ) + =0 (9.30) + Raτ ∈a ∂θ Rem R ∂ (R, θ )

Mixed convection heat transfer in rotating elliptic coolant channels

201

where   ∇ 4ψ = ∇ 2 ∇ 2ψ .

(9.31)

Eq. (9.18) above is a biharmonic operator. The normalized axial velocity equation becomes ∇ 2W +

1 ∂ (ψ, W ) + 4 Rem −η = 0. R ∂ (R, θ )

(9.32)

The normalized energy transport equation becomes ∇ 2η +

Pr ∂ (ψ, η) + W = 0. R ∂ (R, θ )

(9.33)

The normalization procedure adopted highlights the following dimensionless groups, which parametrically govern this problem: 2

4

a Raτ = Ω Hβτ (Rotational Rayleigh number), αυ a3 ∂ρ Rem = − 4ρv 2 ∂z (Modified Reynolds number),

Ro∗ =

∂ρ a2 2H Ωρv ∂z (Rossby number), 4 βgτ a αυ (Gravitational Rayleigh

Gaτ = number), εa = Ha (Axis displacement parameter), Pr = υα (Prandtl number). The rotational Rayleigh number, Raτ , emerges from the centripetal buoyancy terms in the momentum equations. This is similar to the Rayleigh number encountered by Morton (1959) in the study of buoyancy force due to the Earth’s gravitational field but with the gravitational acceleration replaced by the centripetal acceleration measured at the centerline. The Rossby number, Ro∗ , has its origin in the Coriolis acceleration terms. The modified Reynolds number, Rem , approximates to the usual through-flow Reynolds number when the buoyancy effects are not included.

9.4 Parameter perturbation analysis for horizontal elliptic ducts in parallel mode rotation This section presents a set of parameter perturbations for the normalized equations for temperature distribution, η, streamfunction, ψ, and axial velocity, W . The conservation equations are solved using a series expansion in ascending powers of the rotational Rayleigh number, Raτ , which governs the free convection due to centrifugal buoyancy. The value of this parameter also determines the rates of heating and rotation in practical terms. This asymptotic series expansion represents an approximate solution.

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Applications of Heat, Mass and Fluid Boundary Layers

9.4.1 Boundary conditions The need for the satisfaction of the boundary conditions for different polar coordinates at the boundary of the ellipse needs the consideration of the eccentricity, e, and angular position, Θ. The derivation of the boundary coordinate, ξ , the following parametric equations of an ellipse are therefore needed: x2 y2 + = 1, a 2 b2

(9.34)

a 2 − b2 = a 2 e2

(9.35)

x = rb cos θ,

(9.36)

y = rb sin θ,

(9.37)

where

and

with a and b being the major and minor radii, respectively. Upon substituting Eqs. (9.28)–(9.30) into Eq. (9.27), we have   1 − e2  , ξ= (9.38) 1 − e2 cos 2 θ where ξ= or

 r 2

r  b

a

b

a

=

,

(9.39)

/ ξ,

(9.40)

where ξ is the dimensionless boundary coordinate. The angle, Θ, is measured from the horizontal position of the circumference of the ellipse (see Fig. 9.3) using the cylindrical polar coordinate system, while z is measured from the point of fully developed flow. The boundary coordinate as expressed in Eq. (9.38) is the radial distance measured from the center of the duct to the boundary where the numerical values of η, ψ, and W are known a priori for the Dirichlet problem. For evaluation of the normalized governing equations (9.16), (9.25), and (9.26), the following boundary conditions are imposed: √ (i) Ψ , W , and η are zero when R = ξ , that is, at the boundary; (ii) Ψ , W , and η are finite at R = 0, that √ is, at the core of the duct; 1 ∂ψ (iii) ∂ψ and are also zero at R = ξ but finite at the center of the elliptic duct. ∂R R ∂θ

Mixed convection heat transfer in rotating elliptic coolant channels

203

Figure 9.3 Coordinate system for the analysis of the boundary coordinate.

We note that all the solutions of Eqs. (9.16), (9.25), and (9.26) satisfy the boundary conditions, and the approximate solutions can be expanded as power series which are presented in what follows.

9.4.2 Power series Although exact solutions of Eqs. (9.16), (9.25), and (9.26) would be extremely difficult to find, if at all possible, an approximate solution may readily be obtained using a technique suggested by Morris [2]. Applied to the present problem, the technique involves the expansion of the dimensionless velocity, stream function, and temperature fields in ascending powers of a suitably small parameter. It is necessary that the parameter selected be small in magnitude for the solutions obtained to be valid. This technique has been successfully used by Morton [12] and Morris [2]. The method of analysis presented in this work is similar to that used by Morton [12]. However, owing to the influence of tangential and radial acceleration components in the present problem, the resulting Eqs. (9.16), (9.25), and (9.26) for ψ, W , and η are expanded in terms of a single parameter, the rotational Rayleigh number, Raτ , which governs the free convection due to rotation: (i) Streamfunction ψ=

n 

Raτi ψi = ψ0 + Raτ ψ1 + Raτ2 ψ2 + . . . ;

(9.41)

i=0

(ii) Axial velocity W=

n  i=0

Raτi Wi = W + Raτ W1 + Raτ2 W2 + . . . ;

(9.42)

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Applications of Heat, Mass and Fluid Boundary Layers

(iii) Temperature Field η=

n 

Raτi ηi = η + Raτ η1 + Raτ2 η2 + . . . .

(9.43)

i=0

Substituting into Eqs. (9.16), (9.25), and (9.26), respectively, yields the following. The stream function becomes   ∇ 4 ψ0 + Raτ ψ1 + Raτ2 + . . .     1 ∂ ψ0 + Raτ ψ1 + Raτ2 ψ2 + . . . , ∇ 2 ψ0 + Raτ ψ1 + Raτ2 ψ2 + . . . + R ∂ [R, θ ]   1 ∂  + Raτ η0 + Raτ η1 + Raτ2 η2 + . . . cos θ R ∂θ    ∂ 2 η0 + Raτ η1 + Raτ η2 + . . . sin θ + ∂R  Raτ ∂  + η0 + Raτ η1 + Raτ2 η2 + . . . εa ∂θ ∂ Raτ .Ro 1 . . + Rem R ∂ [R, θ ]     × η0 + Raτ η1 + Raτ2 η2 + . . . , ψ0 + Raτ ψ1 + Raτ2 ψ2 + . . . = 0. (9.44) The axial velocity becomes   ∇ 2 W0 + Raτ W1 + Raτ2 W2 + . . . 1 ∂ R ∂ [R, θ ]     × ψ0 + Raτ ψ1 + Raτ2 ψ2 + . . . , W0 + Raτ W1 + Raτ2 W2 + . . . +

+ 4Rem = 0.

(9.45)

The temperature field becomes  Pr  ∂ ∇ 2 η0 + Raτ η1 + Raτ2 η2 + . . . + R ∂ [R, θ ]     × ψ0 + Raτ ψ1 + Raτ2 ψ2 + . . . , η0 + Raτ η1 + Raτ2 η2 + . . .   (9.46) × W0 + Raτ W1 + Raτ2 W2 + . . . = 0.

Mixed convection heat transfer in rotating elliptic coolant channels

205

9.4.3 Zeroth-order solutions The zeroth-order equation for the streamfunction is obtained by equating coefficients Raτ0 to zero. Thus,   1 ∂ ψ 0 , ∇ 2 ψ0 4 ∇ ψ0 + . = 0. (9.47) R ∂ (R, θ ) Therefore, the leading term of ψ is zero, that is, Ψ0 = 0,

(9.48)

since there can be no flow in the (r, Θ)-plane when Raτ = 0, due to the absence of circulation or secondary flow. This corresponds to no heating and no rotation condition. The basic equation for the zeroth-order axial velocity is ∇ 2 W0 +

1 ∂ (ψ0 , W0 ) . + 4Rem = 0. R ∂ (R, θ )

(9.49)

Substituting Ψ0 = 0 into the above equation reduces it to ∇ 2 W0 + 4Rem = 0,

(9.50)

where ∇2 =

∂2 1 ∂ 1 ∂2 + . . . + ∂R 2 R ∂R R 2 ∂θ 2

(9.51)

The radial direction is to be considered since the integration is performed along the radial lines for any angular position. The angular position is fixed and frozen while integrating along any particular radius. Then Eq. (9.51) reduces to ∇2 =

∂2 1 ∂ + . . 2 R ∂R ∂R

This can be written in a more compact form as

1 ∂ ∂ ∇2 = R . R ∂R ∂R Substituting this into Eq. (9.50) and rearranging yields

1 ∂ ∂W0 R = −4Rem . R ∂R ∂R

(9.52)

(9.53)

(9.54)

Multiplying both sides by R and integrating yields R

∂W0 = −2R 2 Rem + A1 . ∂R

(9.55)

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Applications of Heat, Mass and Fluid Boundary Layers

Dividing both sides of the above by R and integrating yields W0 = −2R 2 Rem + A1 ln R + B1 ,

(9.56)

where A1 and B1 are unknown constants of integration. They can be determined from the boundary conditions which are stated below: √ At R = ξ , W0 = 0; At R = 0, W0 is infinite. From these equations and for the solution to be unique or valid, A1 must be zero. Upon substituting these conditions into Eq. (9.56), we have B1 = Rem ξ Putting the values of A1 and B1 into Eq. (9.56) and simplifying gives   W0 = Rem ξ − R 2 .

(9.57)

The equation for the zeroth-order form of the temperature distribution is retrieved as ∇ 2 η0 +

P r ∂ (ψ0 , η0 ) . + W0 = 0. R ∂ (R, θ )

(9.58)

Substituting the values of ψ0 and W0 into Eq. (9.58) yields, upon rearranging,   ∇ 2 η0 = −Rem ξ − R 2 . (9.59) Following the same steps for the zeroth-order axial velocity distribution yields, after integration,

2 R4 R η0 = −Rem ξ (9.60) − + A2 ln R + B2 , 4 16 where A2 and B2 are constants of integration to be determined by considering the following boundary conditions: √ At R = ξ , : η0 = 0, At R = 0, η0 is finite. Substituting the boundary conditions into Eq. (9.60) gives A2 = 0, for validity of the resulting equation, and also   3Rem ξ 2 B2 = . 16 Substituting the values of A2 and B2 into Eq. (9.60) gives η0 =

 Rem  2 3ξ − 4ξ R 2 + R 4 , 16

(9.61)

Mixed convection heat transfer in rotating elliptic coolant channels

207

or η0 =

  Rem  ξ − R 2 3ξ − R 2 . 16

(9.62)

Proceeding further to the first-order solutions is what we do in the following.

9.4.4 First-order solutions The equation that governs the first-order stream function can be recalled from Eq. (9.44) as        ∂ ψ 0 , ∇ 2 ψ1 1 ∂ ψ1 , ∇ 2 ψ 0 1 ∂η0 ∂η0 4 ∇ ψ1 + + + cos θ + sin θ R ∂ (R, θ ) ∂ (R, θ ) R ∂θ ∂R +

1 ∂η0 Ro 1 ∂ (η0 , ψ0 ) . + = 0. εa ∂θ Rem R ∂ (R, θ )

Substituting the value of ψ0 = 0 into Eq. (9.63) and taking ∇ 4 ψ1 + But

∂η0 sin θ = 0. ∂R

(9.63) ∂η0 ∂θ

= 0 gives (9.64)

  ∇4 = ∇2 ∇2

and

1 ∂ ∂ R . ∇ = R ∂R ∂R 2

Hence, ∇4 =







1 ∂ ∂ ∂ 1 ∂ R R . R ∂R ∂R R ∂R ∂R

(9.65)

Substituting the above expression for η0 into Eq. (9.64) gives





 1 ∂ Rem  3 ∂ 1 ∂ ∂ R R ψ1 + 4R − 8ξ R sin θ = 0. (9.66) R ∂R ∂R R ∂R ∂R 16 For uniqueness of the solution, R1 in Eq. (9.66) is taken as a variable. Hence, it is used to multiply through the equation after it has been integrated. Considering the first step of integration gives



  ∂ Rem 1 ∂ ∂ R (9.67) R ψ1 + sin θ R 5 − 4ξ R 3 + A3 R = 0. ∂R R ∂R ∂R 16

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Applications of Heat, Mass and Fluid Boundary Layers

Here R in the first term of Eq. (9.67) is again taken as a variable, the equation is integrated and divided through by R to give

 5  Rem ∂ R A3 R 3 B3 1 ∂ R ψ1 + sin θ − ξ R3 + + = 0. (9.68) R ∂R ∂R 16 6 4 R This same procedure is adopted for Eq. (9.68) and gives   R5 A3 3 ∂ψ1 Rem sin θ R 7 + −ξ + R + B3 R ln R + C3 R = 0. R ∂R 16 36 4 4 And finally, adopting the same technique gives   R3 R Rem sin θ R 7 R5 ψ1 + −ξ + A3 + C3 = 0, 16 288 24 16 2

(9.69)

(9.70)

where A3 and C3 are constants to be determined by applying the boundary conditions stated below to Eqs. (9.69) and (9.70): At R =

/

ξ , : ψ1 =

∂ψ1 = 0, ∂R

and At R = 0, ψ1 and

∂ψ1 ∂R

are finite.

The results upon substituting the boundary conditions are A3 =

21 2 ξ Rem sin θ 288

and C3 = −

5 2 ξ Rem sin θ. 1122

Putting A3 and C3 into Eq. (9.70) and simplifying gives ψ1 =

 Rem sin θ  .R 10ξ 2 − 21ξ 2 R 2 + 12ξ R 4 − R 6 , 4608

(9.71)

ψ1 =

2   Rem sin θ  .R ξ − R 2 10ξ − R 2 . 4608

(9.72)

or

The first-order axial velocity in the basic form can be recalled from Eq. (9.45) as   1 ∂ (ψ0 , W1 ) ∂ (psx1, W0 ) + = 0. (9.73) ∇ 2 W1 + R ∂ (R, θ ) ∂ (R, θ )

Mixed convection heat transfer in rotating elliptic coolant channels

Substituting ψ0 = ∇ 2 W1 −

∂W0 ∂θ

209

= 0 into Eq. (9.73) and expanding the Jacobian operators gives

1 ∂W0 ∂ψ1 . = 0. R ∂R ∂θ

(9.74)

Substituting for ψ1 and W1 in Eqs. (9.71) and (9.57), then differentiating with respect to Θ and R, respectively, yields

 ∂W1 Rem cos θ  1 ∂ R + 20ξ 3 R − 42ξ 2 R 3 + 24ξ R 5 − 2R 7 = 0. (9.75) R ∂R ∂R 4608 Adopting the same method of integration as used for the first-order streamfunction and simplifying we obtain Re2 cos θ W1 + m 4608



 R B4 1 9 5 2 3 7 2 5 1 7 ξ R − ξ R + ξ R − R + A4 + = 0. (9.76) 2 4 2 40 2 R

If the boundary conditions are imposed as follows: / At R = ξ , W1 = 0, and At R = 0, W1 is finite, then B4 = 0, for validity of resulting equation, and 49 A4 = − 20



2 cos θ ξ 4 Rem . 4608

Substituting for A4 and B4 in Eq. (9.76) gives W1 =

  2 cos θ Rem .R 49ξ 4 − 100ξ 3 R 2 + 70ξ 2 R 4 − 20ξ R 6 + R 8 , 184320

(9.77)

W1 =

  Rem cos θ  .R ξ − R 2 49ξ 3 − 51ξ 2 R 2 + 19ξ R 4 − R 6 . 184320

(9.78)

or

The first-order temperature distribution in the basic form can be recovered as   P r ∂ (ψ0 , η1 ) ∂ (ψ1 , η0 ) ∇ 2 η1 + (9.79) + + W1 = 0. R ∂ (R, θ ) ∂ (R, θ ) Substituting for ψ0 = 0 in Eq. (9.79) gives ∇ 2 η1 −

P r ∂η0 ∂ψ1 . + W1 = 0. R ∂R ∂θ

(9.80)

210

Applications of Heat, Mass and Fluid Boundary Layers

We now substitute for ψ1 and η0 in the above equation, and then differentiate these terms with respect to Θ and R, respectively. The result is then multiplied by PRr , and subtracted from W1. Integrating the result while adopting the earlier approach for the first-order streamfunction and axial velocity, after rearranging, gives  2 cos θ  P rRem 26 3 5 15 2 7 7 1 11 4 3 9 −10ξ R + ξ R − ξ R + ξ R − R η1 = 4608 × 16 3 4 10 30  2 cos θ  49 26 35 1 1 Rem ξ 4R3 − ξ 3R5 + ξ 2R7 − ξ R9 + R 11 − 184320 8 6 24 4 120 B5 A5 R+ , (9.81) + 2 R where A5 and B5 are the constants of integration that can be determined upon the application of the following boundary conditions: At R =

/

ξ , η1 = 0; while at R = 0, : η1 is finite.

Substituting these boundary conditions into Eq. (9.81), we have A5 =

2 cos θ 2ξ 5 Rem [381 + 1325P r] , 22118400

where B5 = 0 is needed for the validity of equation. Upon substituting these constants into Eq. (9.81) and after simplification, we have η1 =

2 cos Θ P rRem 4608 × 16 × 300   × −3000ξ 4 R 3 + 2600ξ 3 R 5 − 1125ξ 2 R 7 + 210ξ R 9 − 10R 11

 2 cos Θ  Rem 735ξ 4 R 3 − 500ξ 3 R 5 + 175ξ 2 R 7 − 30ξ R 9 + R 11 184320 × 120 2 cos θ ξ 5 Rem + (9.82) [381 + 1325P r] . 22118400 −

Further simplification gives a more compact form as η1 =

2 cos θ  Rem (381 + 1325P r) ξ 5 R − (735 + 3000P r) ξ 4 R 3 22118400 + (500 + 2600P r) ξ 3 R 5 − (175 + 1125P r) ξ 2 R 7  + (30 + 210P r) ξ R 9 − (1 + 10P r) R 11 .

(9.83)

Mixed convection heat transfer in rotating elliptic coolant channels

211

9.4.5 Second-order solutions The equation for the second-order streamfunction can be retrieved from Eq. (9.44) as        ∂ ψ0 , ∇ 2 ψ 2 ∂ ψ 1 , ∇ 2 ψ1 ∂ ψ2 , ∇ 2 ψ0 + + ∂ (R, Θ) ∂ (R, Θ) ∂ (R, Θ)   ∗ 1 ∂η1 ∂η1 1 ∂η1 Ro 1 ∂ (η0 , ψ1 ) ∂ (η1 , ψ0 ) + . cos θ + sin θ + + + R ∂θ ∂R εa ∂θ Rem R ∂ (R, Θ) ∂ (R, θ ) = 0. (9.84)

1 ∇ ψ2 + R 4

Substituting ψ0 = 0 into Eq. (9.84) and expanding the Jacobian operators yields      ∂ ∇ 2 ψ1 ∂ψ1 ∂ψ1 ∂ ∇ 2 ψ1 . − . ∂R ∂θ ∂R ∂θ   1 ∂η1 ∂η1 − cos θ + sin θ R ∂θ ∂R   1 ∂η1 Ro∗ 1 ∂η0 ∂ψ1 − . + . . εa ∂θ Rem R ∂R ∂θ

1 ∇ ψ2 = R 4

(9.85)

Since the values of η0 , η1 , and ψ1 have been determined in earlier sections, we can differentiate η1 and ψ1 with respect to R and Θ, and substitute the result into parts of Eq. (9.85). Similarly, we differentiate η0 with respect to R only. Upon rearranging, ∇ ψ2 = 4

2 sin 2θ Rem

(4608)2

 (288 + 14400P r) ξ 4

R2 + (2112 − 4992P r) ξ 3 R 4 5

R8 − (2448 − 3240P r) ξ 2 R 6 + (2304 − 4032P r) ξ 5  2 sin θ  R Rem R 10 + − (216 − 240P r) (9144 + 31800P r) ξ 5 2 5 25 (4608) εa 3 R + (480 + 2496P r) ξ 3 R 5 − (3528 + 14400P r) ξ 4 5 R9 − (168 + 1080P r) ξ 2 R 7 + (144 + 1008P r) ξ 5  R 11 − (24 + 240P r) 25  Ro∗ .Rem cos θ  −20ξ 4 R + 52ξ 3 R 3 − 95ξ 2 R 5 + 14ξ R 7 − R 9 . − 4608 × 4 (9.86) The value of ∇ 4 as determined previously for the first-order streamfunction is adopted along with the technique of integration. The following boundary conditions are em-

212

Applications of Heat, Mass and Fluid Boundary Layers

ployed to obtain the solution: /

At R =

ξ , : ψ2 =

∂ψ2 = 0; while at R = 0, ψ2 ∂R

and

∂ψ2 are finite. ∂R

The solution is presented within summation signs and the actual value is tabulated in Table 9.1. ψ2 =

 2 2 sin 2θ   Rem (4608)2 +

7  

 C(2r−1) + D(2r−1) P r ξ

r=1



C2r ) + D(2r) P r ξ



15 2 −r



R (2r−1)

 (7−r)

R

(2r)

r=3

 7  2 sin θ   (7−s) (2s+1) Rem + C(2s+1) + D(2s+1) P r ξ R 22118400εa s=0  6  ∗ Ro .Rem cos θ  + C(2t+1) ξ (6−t) R (2t+1) . 18432

(9.87)

t=0

The coefficients C(2r−1) , D(2r−1) , C2r , D2r , C(2s+1) , D(2s+1) and C(2t+1) are tabulated in Table 9.1. The equation for the second-order axial velocity can be recovered from Eq. (9.45) as   1 ∂ (ψ0 , W2 ) ∂ (ψ1 , W1 ) ∂ (ψ2 , W0 ) ∇ W2 + + = 0. R ∂ (R, θ ) ∂ (R, θ ) ∂ (R, θ ) 2

(9.88)

The knowledge of ψ0 = 0 and expanding the Jacobian operators helps reduce Eq. (9.88) to ∇ 2 W2 =

  1 ∂W0 ∂ψ2 ∂W1 ∂ψ1 ∂ψ1 ∂W1 . + . − . . R ∂R ∂θ ∂R ∂θ ∂R ∂θ

(9.89)

We now use the values of W0 , W1 , ψ1 , and ψ2 , with their appropriate differentials, then integrate the result, and apply the following boundary conditions to arrive at the final solution: At R =

/

ξ , : W2 = 0; while at R = 0, : W2 is finite.

The final solution contains numerical coefficients of an unwieldy nature, therefore they have been grouped within summation signs and the values are tabulated in Ta-

Mixed convection heat transfer in rotating elliptic coolant channels

213

Table 9.1 ψ2 coefficients. r 1 2 3 4 5 6 7

C(2r−1) 1.3235 −2.0295 C2r 0.1097 0.9578 −0.3925 0.0325 −0.0015

D(2r−1) 4.1401 −7.8261 D2r 5.4857 −2.2639 0.5195 −0.057 0.0017

s 0 1 2 3 4 5 6 7

C(2s−1) 1.0332 −2.4931 1.9844 −0.6380 0.1302 −0.0182 0.0015 −0.0000

D(2s+1) 3.3059 −8.1727 6.9010 −2.6042 0.6771 −0.1172 0.0104 −0.0003

t 0 1 2 3 4 5 6

C(2t+1) 0.0436 −0.1129 0.1042 −0.0451 0.0117 −0.0015 0.0000

ble 9.2. W2 =

 3 3 cos 2θ   Rem (4608)2 +

10 



E(2r−1) − F(2r−1) P r ξ



17 2 −r

r=1





E2(r−2) + F2(r−2) P r ξ

r=4



R (2r−1)

 (10−r)

R

2(r−2)

 8  3 cos θ    (8−s) (2s+1) Rem − E(2s+1) + F(2s+1) P r ξ R 11059200 s=0  7  2 sin θ  Ro∗ Rem (7−t) (2t+1) + E(2t+1) ξ R 9216 t=0

214

Applications of Heat, Mass and Fluid Boundary Layers

Table 9.2 W2 coefficients. r 1 2 3 4 5 6 7 8 9 10

+

3 Rem

(4608)2

E(2r−1) 0.3036 −0.6618 0.3383 E2(r−2) 0.0483 0.16 −0.2939 0.1330 −0.02 0.0024 −0.0001

F(2r−1) 1.0358 −2.0701 1.3044 F2(r−2) – – −0.3483 0.0915 −0.0145 0.0012 −0.0000

s 0 1 2 3 4 5 6 7 8

E(2s+1) −0.0596 0.1292 −0.1039 0.0413 −0.008 0.0011 −0.0001 0.0000 −0.0000

F(2s+1) −0.1888 0.4132 −0.3405 0.1438 −0.0326 0.0056 −0.0007 0.0000 −0.0000

t 0 1 2 3 4 5 6 7

E(2t+1) −0.0025 0.0055 −0.0047 0.0022 −0.0006 0.0001 −0.0000 0.0000

x 1 2 3 4 5 6 7 8

 15 2

−1.3572ξ R +

8 

E2x 4.0833 −6.7633 7.26 −4.6333 1.7298 −0.3472 0.0287 −0.0008

 E2x ξ

(8−x)

R

2x

.

(9.90)

x=1

The coefficients E(2r−1) , F(2r−1) , E2(r−2) , F2(r−2) , E(2s+1) , F(2s+1) , E(2t+1) and E2x are tabulated in Table 9.2.

Mixed convection heat transfer in rotating elliptic coolant channels

215

The equation for the second-order temperature distribution solution can be recovered as   P r ∂ (ψ0 , η2 ) ∂ (ψ1 , η1 ) ∂ (ψ2 , η0 ) 2 (9.91) + + + W0 = 0. ∇ η2 + R ∂ (R, θ ) ∂ (R, θ ) ∂ (R, θ ) Substituting ψ0 = 0 and expanding the Jacobian operators reduce Eq. (9.91) to Pr ∇ η2 = R



2

 ∂η1 ∂ψ1 ∂ψ1 ∂η1 ∂η0 ∂ψ2 . − + . ∂R ∂θ ∂R ∂θ ∂R ∂θ

(9.92)

We now use the values of η0 , η1 , ψ1 , ψ2 , and W2 with their differentials with respect to R and Θ as appropriate, then integrate the result, and apply the following boundary conditions to arrive at the final solution: At R =

/ ξ , : η2 = 0;

At R = 0, : η2 is infinite. The final solution has been grouped within summation signs and the actual coefficients of the solution have been tabulated in Table 9.3 so that for the second-order streamfunction and axial velocity solutions we have: η2 = +

3 Rem

(4608)2 12  



4    19 −r   2 R (2r−1) L(2r−1) + M(2r−1) P r + N(2r−1) P r ξ 2 r=1





L2(r−3) + M2(r−3) P r + N2(r−3) P r 2 ξ (12−r) R 2(r−3)

r=5

 9   3 cos θ  Rem + L(2s+1) + M(2s+1) P r + N(2s+1) P r 2 ξ (9−s) R (2s+1) 22118400εa s=0  2    19 −t  3  Rem 2 + R (2t−1) L(2t−1) + M (2t − 1) P r + M(2t−1) P r ξ 2 (4608)2 t=1  11    2 (11−t) 2(t−2) + R L2(t−2) + M2(t−2) P r + N2(t−2) P r ξ t=3

 8  2 sin θ    (8−x) (2x+1) Ro∗ .Rem − L(2x+1) + M (2x + 1) P r ξ R . (9.93) 73728 x=0

The coefficients L(2r−1) , M(2r−1) , N(2r−1) , L2(r−3) , M2(r−3) , N2(r−3) , L(2s+1) , M(2s+1) , N(2s+1) , L(2t−1) , M(2t−1) , N(2t−1) , L2(t−2) , M2(t−2) , N2(t−2) , L(2x+1) and M(2x+1) are tabulated in Table 9.3.

216

Applications of Heat, Mass and Fluid Boundary Layers

Table 9.3 η2 coefficients. r 1 2 3 4 5 6 7 8 9 10 11 12 s 0 1 2 3 4 5 6 7 8 9 t 1 2 3 4 5 6 7 8 9 10 11

L(2r−1) 0.019 −0.0380 0.0276 −0.0070 L2(r−3) – −0.0014 −0.0025 0.0030 −0.0009 0.0002 −0.0000 0.0000 L(2s+1) 0.0075 −0.0149 0.0108 −0.0043 0.0010 −0.0001 0.0000 −0.0000 0.0000 −0.0000 L(2t−1) −0.0119 0.1697 L2(t−2) – −0.2722 0.1932 −0.1152 0.0468 −0.0121 0.0018 −0.0001 0.0000

M(2r−1) 0.1422 −0.2949 0.1984 −0.0483 M2(r−3) 0.0090 0.0051 −0.0209 0.0093 0.0003 −0.0002 0.0000 −0.0000 M(2s+1) 0.0508 −0.1118 0.0971 −0.0478 0.0138 −0.0024 0.0003 −0.0000 0.0000 −0.0000 M(2t−1) −0.0894 – M2(t−2) 0.2646 −0.4264 0.4466 −0.2837 0.1127 −0.0283 0.0042 −0.0003 0.0000

N(2r−1) 0.2842 −0.5175 0.4123 −0.0815 N2(r−3) −0.0302 0.1202 −0.2470 0.0996 −0.0474 0.0079 −0.0006 0.0000 N(2s+1) 0.0856 −0.2066 0.2047 −0.1145 0.0378 −0.0082 0.0014 −0.0002 0.0000 −0.0000 N(2t−1) −0.2993 – N2(t−2) 0.9201 −1.6063 1.8732 −0.13647 0.6305 −0.1802 0.0288 −0.0022 0.0001

Mixed convection heat transfer in rotating elliptic coolant channels

x 0 1 2 3 4 5 6 7 8

L(2x+1) 0.0013 −0.0025 0.0018 −0.0008 0.0002 −0.0000 0.0000 −0.0000 0.0000

217

M(2x+1) 0.0045 −0.0109 0.0112 − 0.0067 0.0024 −0.0006 0.0001 −0.0000 0.0000

9.4.6 Solutions of power series approximations The power series approximations in rotational Rayleigh number, Raτ , for the solution of the formulation up to the second order are expressed in what follows. For the stream function ψ = ψ0 + Raτ ψ1 + Raτ2 ψ22 + . . . , 

 sin θ Rem  .R 10ξ 3 − 21ξ 2 R 2 + 12ξ R 4 − R 6 4608    2 2    152 −r (2r−1) Rem + Raτ2 C + D P r ξ R (2r−1) (2r−1) (4608)2 r=1

ψ = Raτ

+

7 

(C2r + D2r P r) ξ (7−r) R 2r

r=3

 7  2 sin θ   (7−s) (2s+1) Rem C(2s+1) + D(2s+1) P r ξ − R 22118400εa s=0   6 Ro∗ .Rem cos θ  (6−t) (2t+1) + C(2t−1) ξ R . 18432

(9.94)

t=0

For the axial velocity W = W0 + Raτ W1 + Raτ2 W2 + . . . ,   W = Rem ξ − R 2  2  Rem cos θ  + Raτ .R 49ξ 4 − 100ξ 3 R 2 + 70ξ 2 R 4 − 20ξ R 6 + R 8 184320

218

Applications of Heat, Mass and Fluid Boundary Layers



 3   3 cos 2θ    172 −r (2r−1) Rem E + F P r ξ R (2r−1) (2r−1) (4608)2 r=1  10    (10−r) 2(r−2) E2(r−2) + F2(r−2) P r ξ + R

+ Raτ2

r=4

 8  3 cos θ   (8−s) (2s+1) Rem E(2s+1) + F(2s+1) P r ξ R − 1159200εa s=0   7 Ro∗ .Rem sin θ  (7−t) (2t+1) + E(2t+1) ξ R 9216 t=0   8 3  15 Rem 2x (8−x) + E2x ξ R −1.3572ξ 2 R + . (4608)2 x=1

(9.95)

For the temperature η = η0 + Raτ η1 + Raτ2 η2 + . . . ,  Rem  2 3ξ − 4ξ R 2 + R 4 16 2 cos θ  Raτ Rem + (381 + 1325P r) ξ 5 R − (735 + 3000P r) ξ 4 R 3 22118400 + (500 + 2600P r) ξ 3 R 5 − (175 + 1125P r) ξ 2 R 7  + (30 + 210P r) ξ R 9 − (1 + 10P r) R 11   4   19 −r  3  Re m 2 2 L(2r−1) + M(2r−1) P r + N(2r−1) P r ξ 2 + Raτ R (2r−1) (4608)2 r=1  12    L2(r−3) + M2(r−3) P r + N2(r−3) P r 2 ξ (12−r) R 2(r−3) +

η=

r=5

 9   3 cos θ  Rem 2 (9−s) (2s+1) + L(2s+1) + M(2s+1) P r + N(2s+1) P r ξ R 22118400εa s=0  2   19 −t  2  Rem 2 L(2t−1) + M(2t−1) P r + N(2t−1) P r ξ 2 + R (2t−1) (4608)2 t=1  11    2 (11−t) 2(t−2) L2(t−2) + M2(t−2) P r + N2(t−2) P r ξ + R t=3

  8 2 sin θ    (8−x) (2x+1) Ro∗ .Rem − L(2x+1) + M(2x+1) P r ξ R . 73728 x=0

(9.96)

Mixed convection heat transfer in rotating elliptic coolant channels

219

9.4.7 Peripheral Nusselt number Nusselt number is a dimensionless quantity indicative of the rate of energy conversion from the surface. For the estimation of the peripheral local Nusselt number, N u(Θ), we define the temperature difference motivating heat transfer to be that between the wall temperature, Tb , and the local convective heat transfer coefficient based on the bulk temperature. At the wall of the tube, heat transfer to the tube is solely due to conduction in the radial direction. Thus for a small angular element of the tube periphery, the locally prevailing heat transfer rate, q(Θ), may be expressed as  ∂T  q (θ) = hf (θ) (Tw − Tb ) = −k . (9.97) ∂r r=rb We now express Eq. (9.97) in terms of the dimensionless dependent and independent variables r rb / (Tw − T ) (Tw − Tb ) R = , x (θ ) = = ξ , η = , ηb = . a a τ aP r τ aP r Substituting the above transformation variables into Eq. (9.97) gives  kτ aP r ∂η  hf (θ ) τ aP rηb = . a ∂R R=x(θ) Simplifying and rearranging Eq. (9.98) gives  hf (θ) a 1 ∂η  . = k ηb ∂R R=x(θ)

(9.98)

(9.99)

The local peripheral Nusselt number, N u(Θ), based on the major diameter of the tube can be defined as N u (θ ) =

hf (θ) 2a . k

Hence,  2 ∂η  . N u (θ) = ηb ∂R R=x(θ)

(9.100)

However, the heat transfer from the surrounding space to the elliptic tube is influenced by the components of the radial temperature gradients, say, due to the tangential temperature gradient and the normal temperature gradient. While the normal temperature gradient is the agent of heat transfer across the tube periphery into the fluid, the tangential temperature gradient maintains the wall at constant temperature. Hence of more importance and of useful consideration is the normal temperature gradient at the wall of the elliptic tube. The slope of the normal to the tangent at any point (x1 , y1 ) on the

220

Applications of Heat, Mass and Fluid Boundary Layers

Figure 9.4 Resolution of temperature gradients.

elliptic tube can be derived from the equation of an ellipse x2 y2 + = 1, a 2 b2

(9.101)

subject to the orthogonality condition for any two perpendicular lines m1 m2 = −1. Thus, the slope of the normal is   m= 

y1 b2

x1 a2

.

(9.102)

Using the relation, a 2 − b2 = a 2 e2 in the above gives tan λ =

y1  , x1 1 − e 2

(9.103)

where λ is the angle which the normal to the tangent makes with the horizontal. Hence,   y1 −1   . λ = tan (9.104) x1 1 − e 2 Resolving the radial temperature gradient, ∂η/∂R, into its normal component gives ∂η ∂η = cos (λ − θ ) , ∂n ∂R

(9.105)

where ∂η/∂n is the normal temperature gradient at any point on the elliptic tube. Hence, this peripheral local Nusselt number, N u(Θ), based on the normal temperature gradient and the major diameter of the elliptic tube is given by  2 ∂η  N u (θ ) = cos (λ − θ ) . (9.106) ηb ∂R R=x(θ)

Mixed convection heat transfer in rotating elliptic coolant channels

221

It is more convenient to use the bulk temperature, Tb , also referred to as weighted mean temperature upon which the solution is based. For the solution of Eq. (9.106), we define 2π x(θ ) η (R, θ ) W (R, θ ) RdRdθ ηb = 0 02π x(θ ) , (9.107) W (R, θ ) RdRdθ 0 0 where ηb is the normalized form of Tb , the bulk temperature. The mean Nusselt number is obtained from  2π 1 N um = N u (θ) dθ. (9.108) 2π 0 The result obtained from the above equation by a computer algorithm is just the summation of the peripheral local Nusselt number averaged over the total angular coordinate of the elliptic duct. Following Morris [3], the influence of rotation on the established resistance to flow may be examined by evaluating a Blasius friction factor, Cf r , defined as Cf r = −

a ∂p , 2 ∂z ρwm

where wm is the mean axial velocity defined as  2π  a 1 wrdrdθ, wm = Ar 0 0

(9.109)

(9.110)

and Ar is the cross-sectional area of the tube. For the elliptic tube, when using the earlier established normalization parameters, the normalized mean axial velocity is  2π  x(θ) υ wm = wRdRdθ. (9.111) 1  0 πa 1 − e2 2 0 Adopting the normalization parameters, the normalized form of the friction coefficient can be shown to be   4π 2 1 − e2 Rem Cf r =  (9.112) 2 . 2π x(θ) wRdRdθ 0 0

9.5

Parameter perturbation analysis for vertical elliptic ducts in parallel mode rotation

The normalized governing equations (9.30), (9.32), and (9.33) are solved using a series expansion in ascending powers of the rotational Rayleigh number, Raτ . This

222

Applications of Heat, Mass and Fluid Boundary Layers

asymptotic series expansion is truncated at the second order, and therefore presents an approximate solution. This technique was successfully used by Morton [12], Morris [3], as well as Bello-Ochende and Lasode [19]. The need for the satisfaction of the boundary conditions for different polar coordinates at the boundary of the ellipse requires the consideration of the eccentricity, e, and the angular position, θ . For the derivation of the boundary coordinate, ξ , the parametric equations of an ellipse are invoked (see Fig. 9.3) and this variable is then given as   1 − e2 . ξ= (9.113) 1 − e2 cos2 θ

9.5.1 Boundary conditions The normalized boundary constrains are as follows: √ (i) ψ, W and η are zero when R = ξ , that is, at the boundary; (ii) ψ, W and η are finite at R√ = 0, that is, at the core of the duct; 1 ∂ψ (iii) ∂ψ ∂R , R ∂θ are zero at R = ξ , but finite at the center of the elliptic duct. The parameter perturbation technique adopted in the solution of the problem gave rise to the power series representation of the normalized governing equations, which is expanded with rotational Rayleigh number, Raτ , as follows. The stream function is n 

ψ=

Raτi ψi = ψ0 + Raτ ψ1 + Raτ2 ψ2 + ....

(9.114)

i=0

The axial velocity is W=

n 

Raτi Wi = W0 + Raτ W1 + Raτ2 W2 + ....

(9.115)

i=0

The temperature field is η=

n 

Raτi ηi = η0 + Raτ η1 + Raτ2 η2 + ....

(9.116)

i=0

Substituting Eqs. (9.114), (9.115), and (9.116) into Eqs. (9.30), (9.32), and (9.33), respectively, it is possible, upon integrating the resulting cascade of differential equations and applying the boundary constraints, to arrive at the solutions as in the following sections.

9.5.2 Zeroth-order solutions The zeroth-order stream function is ψ0 = 0.

(9.117)

Mixed convection heat transfer in rotating elliptic coolant channels

223

There can be no flow in the (r,θ)-plane when Raτ = 0 due to the absence of circulation or secondary flow. This corresponds to a no-heating condition. The zeroth-order axial velocity is   W0 = ξ − R 2 Rem . (9.118) The zeroth order temperature is η0 =

 Rem  2 3ξ − 4ξ R 2 + R 4 . 16

(9.119)

9.5.3 First-order solutions The first-order stream function is  Rem sin θ  ψ1 = R 10ξ 3 − 21ξ 2 R 2 + 12ξ R 4 − R 6 . 4608

(9.120)

The first-order axial velocity is    Re2m cos θ ξ − R 2 49ξ 3 − 51ξ 2 R 2 + 19ξ R 4 − R 6 W1 = 1.843×10 5R   Rem + 576Ra R 6 − 9ξ R 4 + 27ξ 2 R 2 − 19ξ 3 τ

(9.121)

The first-order temperature ⎡

⎤ (381 + 1325 Pr) ξ 5 R − (735 + 3000 Pr) ξ 4 R 3 cos θ ⎦ η1 = 22118400 ⎣ + (500 + 2600 Pr) ξ 2 R 5 − (175 + 1125 Pr) ξ 2 R 7 + (30 + 210 Pr) ξ R 9 − (1 + 10 Pr) R 11   Gaτ Rem 211ξ 4 − 304ξ 3 R 2 + 108ξ 2 R 4 − 16ξ R 6 + R 8 − 3.686×10 4 Ra Re2m

τ

(9.122)

9.5.4 Second-order solutions For the second-order stream function, the final solution contains numerical coefficients of an unwieldy nature, therefore they have been grouped within summations signs and actual values have been tabulated in Table 9.4:     7 12 −r (2r−1) sin 2θ 02  ψ2 = Rem4608 C + D Pr ξ R (2r−1) (2r−1) r=1  0 + 7r=3 (C2r + D2r Pr) ξ 7−r R 2r    7−s 2s+1  Re2m εa sin θ 07  (9.123) + 2.21x10 C Pr ξ R + D (2s+1) (2s+1) 7 s=0     ∗ Re cos θ 06 m + Ro1.843x10 C(2t+1) ξ 6−t R 2t+1 4 0t=0   6 Gaτ Rem sin θ 5−u R 2u+1 − 4.424x10 C ξ (2u+1) 7 Ra u=0 τ

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Applications of Heat, Mass and Fluid Boundary Layers

The second-order axial velocity (see Table 9.5 for actual values of coefficients) is 0  Re3m 8 8−r R 2r E ξ W2 = r 2 (4608)  r=0    172 −s (2s−1) Re3m cos 2θ 03  E + F Pr ξ R + 2.123x107 (2s−1) (2s−1) s−1  0 (10−s) R 2 (s − 2) + 10 s−4 (E2 (s − 2) + F2 (s − 2) Pr) ξ   (6−1) (2t+1)  Gaτ Re2m cos θ 06  (9.124) E + F Pr ξ R + 2.123x10 (2t+1) (2t+1) 7 Ra t=0 τ     (8−u) (2u+1) Re3m εa cos θ 08 R − 1.106x10 7 Ra u=0 E(2u+1) + F(2u+1) Pr ξ τ   (7−v) (2v+1)  Ro∗ Re2m sin θ 07 + E(2v+1) ξ R 9216 0v=0   2  Gaτ Rem 5 (5−x) R (2x) E ξ − 3.686x10 (2x) 4 Ra x=0 τ

The second-order temperature (see Table 9.6 for actual values of coefficients) is 0   9  Rem 3 2 ξ 9−r R 2r η2 = 2.123×10 J + K + L Pr 2r 2r 2r 7    r=0   19 04  3 2 ξ 2 −s R (s−3) Rem cos 2θ J + K Pr +L Pr (2s−1) (2s−1) s=1 + 2.123×107   012(2s−1) J + + K Pr +L2(s−3) Pr2 ξ (12−s) R 2(s−3) 2(s−3) 2(s−3) s=5 0 2 2   Gaτ Rem cos θ 7 (7−t) R (2t+1) + 2.123×10 7 Ra 2 1=0 J(2t+1) + K(2t+1) Pr ξ τ   3   0 Rem εa cos θ 9 2 ξ (9−u) R (2u+1) J + K Pr +L Pr − 2.123×10 (2u+1) (2u+1) (2u+1) 7  u=0  (8−v) (2v+1)  Ro∗ Re2m sin θ 08 J(2v+1) + K(2v+1) Pr ξ R 7.373×104 0v=0  Gaτ2 Re2m 6 (6−x) R 2x x=0 J2x ξ 3.686×104 Ra 2 τ

(9.125) The overall validity range for the solution presented is obtained by the continuation procedure suggested by Tormcej and Nandakumar [15] in which the bisection method is incorporated.

9.5.5 Peripheral local Nusselt number The Nusselt number is a dimensionless quantity indicative of the rate of energy convection from the surface. For the conduction referenced heat transfer with respect to the bulk temperature and considering the normal temperature gradient (see Fig. 9.4), we have the peripheral local Nusselt number, N u(θ), as N u (θ ) = where

2 ∂η  R=x(θ) cos (λ − θ ) , ηb ∂R

2πχ (θ) ηb = 02πχ 0(θ) 0

0

η (R, θ ) W (R, θ ) RdRdθ η (R, θ ) W (R, θ ) RdRdθ

(9.126)

(9.127)

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Table 9.4 ψ2 coefficients. s 0 1 2 3 4 5 6 7

C(2s+1) 1.0315 −2.4913 1.9842 −0.6380 0.1302 −0.0182 0.0015 0.0000 t 0 1 2 3 4 5 6

D(2s+1) 3.3052 −8.1719 6.9006 −2.6040 0.6770 −0.01172 0.0104 0.0003

r 1 2 3 4 5 6 7

C(2r−1) 1.3223 −2.0294 C 2r 0.1097 0.9581 −0.3925 0.0312 −0.0015

C(2t+1) 0.0432 −0.1125 0.1042 −0.0451 0.0117 −0.0015 0.0000

u 0 1 2 3 4 5

C(2u+1) 2986.08 −6365.03 3800 −450 30 −1

D(2r−1) 4.1400 −7.826 D 2r 5.3864 −2.2647 0.5195 −0.0545 0.0017

and χ (θ ) =

/

ξ.

The mean Nusselt number is obtained from  2π 1 N u (θ) dθ. N um = 2π 0

(9.128)

(9.129)

The trapezoidal rule was used to evaluate Eq. (9.129) above. The normalized form of the friction coefficient (the parameter indicating the influence of rotation on the established resistance to flow using the Blasius friction factor) is given by   4π 2 1 − e2 Rem (9.130) Cf r =  2 . 2πχ (θ) W RdRdθ 0 0

9.6 Discussion and conclusions of the effects of the variables on fluid flow and heat transfer Figs. 9.5(A) and (B) respectively present typical illustrations of perturbation components of the temperature and axial velocity profiles along the major diameter for e = 0.433 for a rotating horizontal elliptic duct. The zeroth-order component is

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Table 9.5 W2 coefficients. r 0 1 2 3 4 5 6 7 8

t 0 1 2 3 4 5 6

Er −0.6148 3.0625 −6.3406 7.0583 −4.5609 1.7125 −0.3448 0.0286 −0.0008

E(2t+1) −81.4559 277.39 −272.7988 178 −57.9 7.05 −0.2857

s 1 2 3 4 5 6 7 8 9 10

F(2t+1) 79.1229 159.7988 −120.000 52 13.5 1.68 −0.0571

v 0 1 2 3 4 5 6 7

E(2v+1) −0.0024 0.0054 −0.0047 0.0022 −0.0006 0.0001 0.0000 0.0000

E(2s−1) 0.3029 −0.661 0.3382 E 2(s-2) 0.0483 0.16 −0.2939 0.133 −0.0298 0.0024 −0.0001 u 0 1 2 3 4 5 6 7 8 x 0 1 2 3 4 5

F(2s−1) 1.0359 −2.07 1.3043 F 2(s-2) – – −0.3483 0.915 −0.0145 0.0011 0.0000

E(2u+1) −0.0594 0.1289 −0.1038 0.0413 −0.008 0.0012 −0.0001 0.0000 0.0000

F( 2u + 1) −0.1888 0.4131 −0.3405 0.1438 −0.0326 0.0056 −0.0007 0.0000 0.0000

E2x −36.51 52.75 −19.0 3.000 −0.25 0.001

parabolic, and it is the one usually encountered in pure forced convective flows. The first- and second-order components are sinusoidal in nature. They are the ones that perturbed to the zeroth-order bring about the shift in the maximum local values of temperature and axial velocity profiles. These results also show the typical effect of the rotational Rayleigh number, Raτ , on the temperature and axial velocity distributions. The tendency for the warmer and less dense fluid to move towards the outer region of tube cross-section under the influence of radial component of acceleration is clearly shown. This centrifugal buoyancy is what is really responsible for the distor-

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Table 9.6 η2 coefficients. s 1 2 3 4 5 6 7 8 9 10 11 10 r 0 1 2 3 4 5 6 7 8 9

J(2s−1) 0.0316 −0.0505 0.0275 0.007 J 2(s−3) – −0.0014 −0.0025 0.003 −0.0009 0.0002 −0.0000 0.0000

J2r −0.0635 0.1537 −0.1914 0.1761 −0.1103 0.0456 −0.0119 0.0018 −0.0001 0.0000

K(2s−1) 0.1999 −0.3930 0.1984 −0.0483 K 2(s−3) 0.009 0.0051 −0.0006 0.0349 −0.0054 0.0002 −0.0002 0.0000

K2r 0.0301 0.1985 −0.3998 0.4342 −0.2514 0.0806 −0.028 0.0041 −0.0001 0/0000 v 0 1 2 3 4 5 6 7 8

L(2s−1) 0.2548 −0.6900 0.4123 0.1087 L2(s−3) −0.0002 0.1827 −0.1168 0.1352 −0.0466 0.0079 −0.0006 0.0000

L2r 0.1826 0.6901 −1.5059 1.760 −1.215 0.6045 −0.1781 0.0281 −0.0021 0.0000

J(2v+1) −0.0028 0.0024 −0.0002 0.0008 −0.0002 0.0000 −0.0000 0.0000 −0.0000

u 0 1 2 3 4 5 6 7 8 9

K(2v+1) 0.0021 0.0108 0.0177 0.0067 0.0024 0.0006 −0.0001 0.0000 −0.0000

t 0 1 2 3 4 5 6 7

J(2t+1) −2.5242 10.182 −11.5579 5.6833 −2.225 0.4825 −0.042 0.0013

J(2u+1) −0.0073 0.0143 −0.0103 0.0042 −0.001 0.0001 −0.000 0.000 −0.000 0.000 x 0 1 2 3 4 5 6

K(2t+1) −6.4092 15.3173 −17.036 12.7152 −5.924 1.4275 −0.0943 0.0042

K(2u+1) −0.0485 0.107 −0.0932 0.0459 −0.0132 0.0023 −0.0003 0.0000 −0.0000 0.0000

L(2u+1) −0.0818 0.1983 −0.1965 0.1096 −0.0363 0.0079 −0.0013 0.0001 −0.0000 0.0000

J2x 6.2614 −9.0375 3.2594 −0.5278 0.0469 −0.0025 0.0001

tion of the temperature and axial velocity profiles. However, the first- and second-order perturbations are insignificantly small along the minor diameter, and the zeroth-order component contributes to the axial velocity and temperature distributions.

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Figure 9.5 Typical illustration of temperature and axial velocity perturbation components along the major diameter for a horizontal elliptic duct (Rem = 50, P r = 1, Ro∗ =1, a =1/48, e = 0.433). (A) Temperature, Raτ = 10; (B) Axial velocity, Raτ = 20.

Figs. 9.6(A) and (B) present typical illustrations of temperature and axial velocity perturbation components, respectively, along the major diameter for e = 0.433 for a rotating vertical elliptic duct. The zeroth-order component is parabolic, and it is the one usually encountered in pure forced convective flows. The first- and second-order perturbation components are sinusoidal in nature. The results show the effects of the first- and second-order components on the zeroth-order component. The sinusoidal nature of the first- and second-order perturbation components account for the shift in the maximum local values of the temperature and axial velocity profiles. The effects of rotational Rayleigh number, Raτ , on the temperature and axial velocity distributions can also be noticed. The tendency for the warmer and less dense fluid to move towards the outer region of the tube’s cross-section under the influence of centripetal buoyancy is clearly shown. The centrifugal buoyancy is what is really responsible for the distortion of the temperature and axial velocity profiles. At the maximum local temperature value, the contributions of the zeroth-, first-, and second-order perturbations are 99.23%, 0.41%, and 0.36%, respectively, while at the maximum axial velocity value, the contributions are 99.72%, 0.25%, and 0.03%, respectively. Considering the rigor involved in obtaining the second-order coefficients and their percentage contribution to the entire solution, obtaining higher-order terms may not significantly alter the accuracy of the results. Fig. 9.7 shows the plot of mean Nusselt number against duct eccentricity at rotational Rayleigh number Raτ = 10 for a rotating horizontal elliptic duct. The plot shows also a maximum mean Nusselt number at e = 0.433. The values of mean Nusselt number is higher for Raτ = 10 than that for Raτ = 0 for each elliptic geometry considered. In terms of the mean Nusselt number, Figs. 9.8(A) and (B) show the plots of the mean Nusselt number against duct eccentricity at rotational Rayleigh number Raτ = 5 and Raτ = 10, respectively, for a rotating vertical elliptic duct. Fig. 9.8(A) indicates

Mixed convection heat transfer in rotating elliptic coolant channels

229

Figure 9.6 Typical illustration of temperature and axial velocity perturbation components along the major diameter for a vertical elliptic duct (Raτ =10, Rem =50, P r = 1, Ro∗ =1, a =1/48, e = 0.433, Gaτ =1). (A) Temperature; (B) Axial velocity.

Figure 9.7 Mean Nusselt number versus eccentricity (Raτ =10, Rem =50, P r = 1, Ro∗ =1, a =1/48).

Figure 9.8 Effect of eccentricity on the mean Nusselt number (Rem =50, P r = 1, Ro∗ =1, a =1/48, Gaτ =1). (A) Rotational Rayleigh number Raτ = 5; (B) Rotational Rayleigh number Raτ = 10.

that the mean Nusselt number is invariant with eccentricity up to e = 0.433, beyond which it drops sharply. However, for Raτ = 10 (Fig. 9.8(B)), the mean Nusselt number monotonically decreases with eccentricity with a change to higher gradient of decrease noticeable at e = 0.6. The optimum heat transfer seems to be at e = 0. The values of the mean Nusselt number are higher for Raτ = 10 than for Raτ = 5 for each

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elliptic geometry considered. This result is different from that of Bello-Ochende and Lasode [19] in which optimum heat transfer was indicated at e = 0.433 for the horizontal configuration they considered. The difference can be attributed to the effect of gravitational buoyancy included in this analysis.

References [1] H. Holzworth, Die Entwicklung der Holzworth – Gas Turbine, Holzworth – Gas turbine, G.m.b.H. Muehlmein, Ruhr, 1938. [2] W.D. Morris, Heat Transfer and Fluid Flow in Rotating Coolant Channels, Research Studies Press, John Wiley and Sons, 1981. [3] W.D. Morris, Laminar convection in a heated vertical tube rotating about a parallel axis, Journal of Fluid Mechanics 21 (Part 3) (1965) 453–464. [4] W.D. Morris, Heat Transfer Characteristics of a Rotating Thermosyphon, PhD Thesis, University of Wales, Swansea, 1964. [5] T.H. Davies, W.D. Morris, Heat transfer characteristics of a closed loop thermosyphon, in: Proceedings, Third International Heat Transfer Conference, vol. 2, American Institute of Chemical Engineers, Chicago, USA, 1966, p. 172. [6] W.D. Morris, Terminal laminar convection in a uniformly heated rectangular duct, in: Thermo and Fluid Mechanics Convention, I. Mech E, Bristol, 1968, Paper No. 4. [7] Y. Mori, W. Nakayama, Convective heat transfer in rotating radial circular pipes, International Journal of Heat Transfer 11 (1968) 1027–1040. [8] R. Siegel, Analysis of buoyancy effect on fully developed laminar heat transfer in a rotating tube, Transactions of the ASME 107 (1985) 338–344. [9] F.L. Bello-Ochende, A numerical study of natural convection in horizontal elliptic cylinders, Revista Brasileira de Ciencias Mecanicas VII (4) (1985) 353–371. [10] F.L. Bello-Ochende, Scale analysis of entrance region heat transfer for forced convection in elliptic cylinders, in: Proceedings of the 11th ACBM Mechanical Engineering Conference, Sao Paulo, SP – Brazil, 1991. [11] R.M. Abdel-Wahed, A.E. Attia, M.A. Hifini, Experiments on laminar flow and heat transfer in an elliptic duct, International Journal of Heat and Mass Transfer 27 (12) (1984) 2397–2411. [12] B.R. Morton, Laminar convection in uniformly heated horizontal pipes at low Rayleigh number, Quarterly Journal of Mechanics and Applied Mathematics XII (1959) 411–420. [13] I.K. Adegun, Analytical Study of Convective Heat Transfer in Inclined Elliptic Ducts, M.Eng. Thesis, University of Ilorin, Ilorin, 1992. [14] G.N. Faris, R. Viskanta, An analysis of laminar combined forced and free convective heat transfer in a horizontal tube, International Journal of Heat and Mass Transfer 12 (1969) 1295–1309. [15] R. Tormcej, K. Nandakumar, Mixed convective flow of a power law fluid in horizontal duct, The Canadian Journal of Chemical Engineering 64 (1986) 743. [16] N.J. Rabadi, Heat transfer in coils, uniform heat flux, International Journal of Heat and Technology 6 (1) (1990) 93–113. [17] D.P. Siegwarth, R.D. Mikesell, T.C. Readal, T.J. Hanratty, Effect of secondary flow in a heated horizontal tube, International Journal of Heat and Mass Transfer 12 (1969) 1535–1552. [18] M. Van Dyke, Perturbation Method in Fluid Mechanics, Academic Press, 1964.

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[19] F.L. Bello-Ochende, O.A. Lasode, Convective heat transfer in horizontal elliptic ducts in parallel mode rotation, International Journal of Heat and Technology 13 (1) (1995) 105–122.

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Numerical techniques for the solution of the laminar boundary layer equations

10

O.M. Amoo, A. Falana Department of Mechanical Engineering, University of Ibadan, Ibadan, Nigeria

10.1

Introduction – laminar boundary layer equations

The types of problems engineers are called upon to solve often have no exact closedform, or rather analytical solutions seldom exist. Engineers and scientists model fluid flow problems using various numerical techniques to aid fundamental understanding of their complex physics and enhance the quality and performance of various products of technological implications. No singular numerical technique is superior, instead, the choice of any numerical technique employed depends on the problem. Two-dimensional laminar boundary layers in incompressible flows can be described by the following set of equations [1]: ∂U ∞ ∂u ∂u ∂U ∞ ∂ 2u ∂u +u +v = + U∞ +ν 2, ∂t ∂x ∂y ∂t ∂x ∂y ∂u ∂v + = 0. ∂x ∂y

(10.1) (10.2)

Here u is the local velocity in the streamwise direction, v is the local velocity in the perpendicular to the streamwise direction, ν is the fluid viscosity, and U ∞ is the free-stream velocity of the fluid (see Fig. 10.1). Various numerical methods can solve this set of momentum and continuity equations. This paper presents a review of the numerical techniques essential for the solution of the laminar boundary layer equations. Solving Eqs. (10.1) and (10.2) in their full form requires the development of a relatively complex numerical code. We will illus-

Figure 10.1 Schematic representation of a laminar boundary layer on a flat plate. Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00018-9 Copyright © 2020 Elsevier Ltd. All rights reserved.

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trate the most important numerical techniques using simplified versions of Eq. (10.1). Two fundamental and very important versions were selected: the Blasius equation and the unsteady one-dimensional heat equation. The Blasius equation is one of the earliest and most celebrated similarity solutions of the laminar boundary layer equations. This equation was selected to show the techniques typically used for the cases when a similarity solution allows for the transformation of the partial differential equation (PDE) for momentum and continuity into an ordinary differential equation (ODE). In another regard, the unsteady one-dimensional heat equation is usually associated with the temperature distribution but can also be considered as one of the simplified cases of Eq. (10.1). This equation was selected to illustrate the solution techniques applied in cases when the momentum equation cannot be transformed into an ODE.

10.1.1 Blasius equation Blasius equation is the self-similar form of Eqs. (10.1)–(10.2) that represents the case of a laminar zero pressure gradient boundary layer flow on a flat plate. In this case ∞ ∞ U ∞ = const, ∂U∂t + U ∞ ∂U ∂x = 0. A complete derivation of the Blasius equation can be found in numerous references, such as [2], [1]. Blasius equation has the following form: f  + ff  = 0

(10.3)

with f = f (η) being the dimensionless streamfunction and η being the dimensionless similarity variable. Here η ∼ y/δ (x) and f  = u/U ∞ , where δ (x) is the boundary layer thickness (see Fig. 10.1). The physical boundary conditions for a flat plate boundary layer are u = 0 at the wall and u = U ∞ at the free-stream boundary. For the Blasius equation, these conditions can be formulated as:   f  η=0 = f |η=0 = 0, f  η=∞ = 1. (10.4) The exact analytic solution for the Blasius (or its more generalized form, the Falkner–Skan equation [1]) has never been obtained. A considerable amount of literature is devoted to the numerical solution of these equations. The most important numerical methods, as well as current research trends, will be outlined in this review.

10.1.2 One-dimensional unsteady heat equation A one-dimensional unsteady heat equation is a second-order in space, first-order in time partial differential equation that has the following formulation: ∂ 2u ∂u − ν 2 = 0. ∂t ∂x

(10.5)

This equation represents a simplified form of Eq. (10.1) when only the diffusion effects are considered. In its most common interpretation, Eq. (10.5) can be used to

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235

describe the distribution of heat in a given region over time. In this case, u is the temperature, and ν is the diffusion coefficient. We will consider Eq. (10.5) on the domain x ∈ [0, 1]. The Dirichlet boundary conditions are applied u|x=0,t = u|x=1,t = 0,

(10.6)

and the initial condition is u|x,t=0 = sin (πx) .

(10.7)

This boundary value problem has an exact solution u = sin (πx) e−νπ t . 2

(10.8)

The possibility to compare the numerical solution with the exact solution is extremely practical to assess the performance of the numerical schemes. Because of its relative simplicity, a wide range of application, and the existence of analytical solutions, the heat equation is generally popular in numerical engineering mathematics as a basic test problem [3], [4]. For all these reasons, it was selected in this review paper as well.

10.2

Numerical methods – general background and the most important techniques in the context of the laminar boundary layer ODEs

The Blasius equation introduced in the first section is a third-order nonlinear ordinary differential equation (ODE). Together with its three boundary conditions, it forms a classical two-point boundary value problem. This section provides the background needed to understand the numerical techniques commonly used for the solution of the Blasius or Falkner–Scan problems. The emphasis will be first given to the numerical solution of initial value problems. Typically, finding the numerical solution of an initial value problem is a simpler task than finding the numerical solution of a boundary value problem. However, as it will be shown later in Sect. 10.3, the most popular methods used for the solution of the Blasius or Falkner–Scan equations rely on initial value problem algorithms. Thus, this section develops the techniques that will be used later to implement a MATLAB based solution algorithm. Also provided here is a discussion on the stability, convergence, and error analysis of different methods. The general theory of this overview is given based on [5], [4], [6], [3].

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10.2.1 General overview – Euler scheme example A simple example of a numerical solution of an initial value problem will be discussed first. We consider a first-order differential equation with a single boundary condition: dy = f (y, t) , y|0 = y0 . dt

(10.9)

This equation has to be solved with regards to y (t) in the interval 0 < t ≤ tf . The boundary condition is formulated at the initial point of the solution interval. To find a numerical solution to this problem, the solution interval has to be discretized into N points t0 , t1 , . . . tn , . . . , tN where t0 = 0, tN = tf , and tn+1 = tn + t. Here t is the discretization interval, also called the “step size.” It can be either constant or vary over the solution interval. We will consider here and later only the case of the constant t. The task of any numerical method is to find a discrete approximation of the continuous function y on the entire solution interval. It means that at each tn we have to find yn that would approximate the real y (t) with the desired accuracy. After the discretization is completed, we can start constructing a simple numerical solution for the given problem. Every yn+1 can be represented using yn information via a Taylor series yn+1 = yn + tyn +

( t)2  ( t)3  yn + yn + . . . . 2 6

(10.10)

Eq. (10.9), which we are solving numerically, has the form y  = f (y, t). Thus, we can set yn = f (yn , tn ). With that, leaving only the first two terms on the right-hand side in the above series expansion given by Eq. (10.10), one can obtain yn+1 = yn + tf (yn , tn ) .

(10.11)

At t0 we have the boundary condition for y0 . Using these two values, we now can obtain y1 at t1 , and then every consecutive value of the numerically approximated function yn can be determined by repeatedly applying Eq. (10.11) until yN is reached. This numerical solution process is schematically shown in Fig. 10.2. The formula derived here represents the Euler method – one of the first and most known numerical methods for the solution of initial value problems. In what follows, we will discuss some of the properties of the Euler method. It will give us a relatively simple framework to illustrate the general procedure of characterization of the other, more complex, numerical techniques. First, let us consider the numerical errors and method accuracy. Each numerical approximation represents the real function only up to a certain degree of accuracy. The difference between the actual y (tn ) and the approximation yn obtained by a numerical method is called the numerical error εn = y (tn ) − yn (see Fig. 10.2). This error can be divided into two parts: the truncation error and the round-off error. The truncation error is the error that is caused by the accuracy deficiencies of the numerical method. The global truncation error of a method consists of a local truncation

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237

Figure 10.2 Schematic representation of the Euler method.

error and a propagated truncation error. The local truncation error is the error arising at each discretization point. The propagated truncation error results from the approximation errors on the previous discretization steps. Finally, the round-off error is caused by the forced round-off of each value produced by a numerical solution to a certain number of significant digits. The computer memory limitations entirely determine this error. In the examples considered here, the truncation errors make the main impact into the global numerical error, and thus the round-off errors won’t be considered further. Comparing the Taylor series approximation for yn+1 (Eq. (10.10)) with the formula of the Euler method (Eq. (10.11)), we can determine the local truncation error εnlocal =

( t)2  ( t)3  yn + yn + . . . . 2 6

(10.12) 2

 The leading term in this formula is ( t) 2 yn , and it is usually assumed that the leading term is the main contributor to the magnitude of   the truncation error. Thus, the local truncation error for the Euler method is O ( t)2 . It can also be said that locally, the Euler method is second-order accurate. That is, the error will scale quadratically with the discretization interval t. Practically, the global truncation error is a much more relevant measure than the local truncation error because a calculation never consists of just one step. The global truncation error of the Euler method is more complex to determine (see [6], [4]). It can be shown that the global truncation error has the order O ( t) and thus globally, the Euler method is only first-order accurate. The order of accuracy of a numerical method is a very useful performance metric of an algorithm. It indicates how rapidly the accuracy can be improved with the reduction of the discretization interval t. For example, for a first-order scheme, the reduction of the discretization interval by a factor of 2 will lead to the reduction of the

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truncation error also by approximately a factor of 2. For the second-order method, the same reduction of the discretization interval will lead to the reduction of the truncation error by a factor of 4. As discretization becomes increasingly fine, every useful numerical method should give the numerical solution that tends to the exact solution. This is called the convergence of the numerical method. It can be shown analytically that the Euler method is convergent. The next characteristic that we will consider is the numerical stability of the method. A numerical solution of a differential equation can start producing absolute numerical values that are unjustifiably high and grow uncontrollably (“blow up”) even if the exact solution of this equation doesn’t demonstrate such behavior. In this case, the numerical solution is called unstable. Stability analysis is used to find the conditions at which the numerical scheme can remain stable. There are three different stability categories of the numerical techniques: stable, unstable, and conditionally stable. A stable scheme provides numerical solutions that will not grow unbounded at any value of the parameters of the scheme (for example, any t). An unstable scheme means that the numerical solution will “blow up” at any value of the scheme parameters. A conditionally stable scheme is a scheme that leads to a stable numerical solution with a certain choice of parameters such as t. Numerical method stability analysis is typically performed on the linear model equation y  = λy with a complex coefficient λ. While this equation is simple in appearance, interpreting λ as the eigenvalue of the right-hand side operator allows for the “real-world” behavior of the numerical schemes typical for a range of problems. More details and stability analysis examples can be found in [6], [4]. The stability analysis of the Euler scheme shows that this method is conditionally stable. For example, applied to the model problem y  = λy with the complex coefficient λ the discretization interval t has to satisfy the condition t ≤ 2/|λ|. In practice, this result means that when using the Euler scheme given by Eq. (10.11), the user must choose a sufficiently small discretization step based on the coefficients of the equation that is being solved. The Euler scheme of Eq. (10.11) is explicit. It means that the solution on the step n + 1 can be found with the formula that involves only f (yn , yn−1 , . . . , y0 , tn , tn−1 , . . . , t0 ) but does not involve f (yn+1 , tn+1 ). Any numerical method that includes f (yn+1 , tn+1 ) to obtain the solution for yn+1 is called implicit. Implicit methods are usually unconditionally stable. The Euler scheme considered here also exists in its implicit and unconditionally stable form: yn+1 = yn + tf (yn+1 , tn+1 ) .

(10.13)

Unconditional stability typically comes at an increased computational cost. In the cases when f is nonlinear, implicit methods require a solution of a nonlinear system of algebraic equations that use either costly iterative procedures or linearization techniques. Additionally, numerical stability doesn’t imply accuracy. A method can be stable but not accurate.

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10.2.2 Runge–Kutta and predictor–corrector methods As pointed out in the previous section, the explicit Euler method globally is only first-order accurate and has very limited stability characteristics. For that reason, in practice, more advanced methods became popular for the solution of the initial value problems. In this section, two classes of higher-order methods will be introduced – Runge–Kutta and multistep methods. Runge–Kutta (RK) methods are probably the most popular techniques to solve the Blasius or Falkner–Skan equations practically.

10.2.2.1

Runge–Kutta methods

Runge–Kutta methods are the class of methods that use several Euler-style intermediate substeps on each numerical integration step to obtain a higher order of accuracy. In this section, second- and fourth-order RK methods will be introduced. Consider the first-order ODE y  = f (y, t) ,

(10.14)

then the general form of the two-stage second-order RK method is yn+1 = yn + γ1 k1 + γ2 k2 ,

(10.15)

where k1 and k2 are the intermediate step functions defined as k1 = tf (yn , tn ) ,

(10.16)

k2 = tf (yn + βk1 , tn + α t) . Here α, β, γ1 , γ2 are the constants of the method. These constants are selected such that the global truncation error of the method becomes second-order. We will not reproduce here the constant derivation process, but it can be found, for example, in [6], [4]. The resulting set of constants is γ2 = 1/2α, β = α, γ1 = 1 − 1/2α with α being a free parameter that is often chosen to be α = 1/2. Thus, the final form of the RK second-order formulas is yn+1 = yn + k2 ,

(10.17)

k1 = tf (yn , tn ) ,

1 1 k2 = tf yn + k1 , tn + t . 2 2 Accuracy analysis shows that the presented formula indeed represents a method of second-order accuracy. Stability analysis leads to the conclusion that although the second-order RK method is also only conditionally stable, the stability region is larger than for the explicit Euler method. More details on the accuracy and stability analysis can be found in [4], [6].

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Furthermore, we will present the formulation of the fourth-order RK method, which is arguably the most widely used method of this class: 1 1 1 yn+1 = yn + k1 + (k2 + k3 ) + k4 , 6 3 6

(10.18)

k1 = tf (yn , tn ) ,

1 1 k2 = tf yn + k1 , tn + t , 2 2

1 1 k3 = tf yn + k2 , tn + t , 2 2 k4 = tf (yn + k3 , tn + t) . The constants are chosen such that the method becomes fourth-order accurate. The stability analysis of the fourth-order RK shows a much larger stability region than for the second-order RK (see [6], [4]). Consequently, the popularity of the fourth-order RK method can be explained by considering its enhanced stability and accuracy characteristics, on the one hand, and the relative simplicity of its formulation, on the other hand.

10.2.2.2

Multistep methods – Adams–Bashforth

Another class of methods that achieves higher order accuracy are multistep or predictor–corrector methods. These methods use not only yn and f (yn , tn ) to calculate yn+1 , but also involve information from the previous steps: yn−1 , yn−2 , f (yn−1 , tn−1 ), f (yn−2 , tn−2 ), etc. The price for higher accuracy is that more memory must be used to store information from previous time steps. Furthermore, to start a solution by a multistep method, the first steps have to be made by a method that doesn’t involve the information on the previous steps, like Euler or Runge–Kutta. One of the most popular methods of this class is the second-order Adams–Bashforth method: yn+1 = yn +

3 t t f (yn , tn ) − f (yn−1 , tn−1 ) . 2 2

(10.19)

Stability analysis of the second-order Adams–Bashforth method shows that this method is less stable than even the second-order Runge–Kutta. However, this method leads to smaller computational cost than RK since only one equation is solved per each time step. For many practical tasks, the reduction in stability is offset by the simplicity of coding and potentially quicker execution time. The Adams–Bashforth method can be used in application to the Blasius equation.

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241

Application of different numerical methods for the solution of the Blasius equation

10.3.1 Blasius equation as a numerical boundary value problem Having completed an overview of the terminology and methods that are typically used for the numerical solution of initial value problems, we can return to the Blasius equation introduced in Sect. 10.1. As stated above, it is a third-order nonlinear ODE of the form: f  + ff  = 0,

(10.20)   with the boundary conditions f  η=0 = f |η=0 = 0, f  η=∞ = 1. This is not an initial value problem as considered in the simple first-order equations in the examples of Sects. 10.2.1 and 10.2.2. Instead of having three boundary conditions at η = 0, we have two at η = 0 and one at η = ∞. This situation, when the values for ODE variables are prescribed at more than one point, is called the boundary value problem. For the initial value problems, the solution is usually fully determined by the ODE and its initial conditions, commonly at the zero value of the independent variable. Thus, the solution can be started at the initial point and marched forward by a numerical method until the desired endpoint. In contrast to that, for boundary value problems, the number of prescribed boundary values at the initial point is smaller than the order of ODE. This is the case for Eq. (10.20) at η = 0. The system will not have a unique solution by considering only the η = 0 data. A “random guess” for the absent boundary condition at η = 0 will almost certainly not lead to the solution that would satisfy the prescribed value at η = ∞. A numerical integration technique is needed that would provide a global solution satisfying all the prescribed conditions. These techniques are usually more complex than those implemented for the initial value problems, but some boundary value problem methods use the initial value problem methods described in Sect. 10.2 as an important building block. Two classes of methods are typically used for the boundary value problems: shooting methods and finite-differencing methods. Shooting methods are an iterative technique developed to find a unique global solution of a boundary value problem while using, by and large, the same ODE integration methods as for the initial value problems. First, it involves choosing the values for all the dependent variables at one boundary. These values need to satisfy all the prescribed conditions for this boundary and be a “free guess” for the parameters that cannot be uniquely determined. Then, the equations are solved numerically by one of the initial value methods (like Euler method or one of the Runge–Kutta methods). When the solution arrives at the other boundary, obtained values are compared with the required boundary condition of the problem. This almost always results in a discrepancy. Now we have to adjust the value of the free boundary condition so that the discrepancy is zeroed out. The methods for the solution of this problem will be discussed later. The finite-differencing methods represent an extremely important numerical technique that is used in many applications far beyond the problem considered here. The

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general background information on these methods can be found in [5], [4], [6], [3]. Here we will give only a very brief overview of this technique and discuss the challenges of its application to the Blasius or Falkner–Skan equations. In finite-differencing methods, the differential equations are first rewritten as discrete finite differences of the dependent and independent variable. At every discretizay −y tion point j , y  can be approximated with different “stencils.” Formula y  = j +1 t j y −y is referred to as the first-order forward difference. Formulas y  = j tj −1 and y −y y  = j +12 t j −1 represent the first-order backward-difference and the second-order central difference, respectively. Higher order derivatives are approximated similarly. Noncentered stencils introduce numerical dissipation into the calculation. Using the central difference approximation for the model equation y  = λy, we would generate a system of algebraic equations yj +1 − yj −1 = λyj , j = 1, 2, . . . , N, 2 t

(10.21)

with N being the number of the discretization points. The resulting system of algebraic equations (linear or nonlinear, depending on the initial ODEs) has to be solved on the discretization grid that covers the entire range of integration. Boundary conditions are also rewritten in the finite-differencing form and are included in the system. This results in a matrix that has to be inverted using various methods, either direct or iterative, depending on the system characteristics. The number of variables in the algebraic set of equations depends directly on the number of discretization points, and so does the computational cost. There are many boundary value problems where finite differencing methods work better than shooting methods. However, it is not a popular technique for the solution of the Blasius or Falkner–Skan equations. This situation at least partly can be attributed to the fact the equation is nonlinear and that one of the boundary conditions is prescribed at η = ∞. Obtaining a numerical solution on a semiinfinite interval using a finite-difference method or even a shooting method is impossible from the practical point of view. The most widespread approach to overcome this obstacle is truncating  the initial semiinfinite interval η ∈ [0, ∞) to a finite computational interval η ∈ 0, η∞ , choosing η∞ to be large enough in magnitude not to affect the accuracy of the solution [7]. This approach is mostly used in conjunction with the shooting methods. It has been found that with shooting methods the solution converges very quickly for large η∞ and that η∞ = 10 is large enough in order to obtain an accurate solution to the Blasius problem [4]. Shooting methods generally have an advantage here because of their highly variable integration step size that easily adapts to the solution behavior. For the finite difference approach, choosing a relatively large η∞ might result in a high computational cost. If a uniform discretization is used ( η is the same on the entire interval and is small enough for the accurate approximation in the near-wall region), the computational cost might even be prohibitive. This problem can be partly solved by using the nonuniform discretization. In this case, the discretization points can be clustered near η = 0 where the solution changes rapidly and can have larger η spacing near η = η∞ . However, due to the requirement of the inversion of a potentially

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very large matrix (even in the case of the nonuniform discretization), application of the finite-difference approach requires more coding and, potentially, more computational cost than the shooting method. Finally, we will discuss the application of the finite-element method to the boundary value problem of the Blasius equation. For the general description of the finiteelement method, we can consider a one-dimensional model boundary value problem F (u) = f (x)

(10.22)

with the boundary conditions B (u) = b(x). Here F denotes the general differential operator that can be of various order and linear or nonlinear with regards to u. The solution domain for u(x) is first divided into N finite elements by placing N -1 nodes on the interior. Each element has the width i = xi − xi−1 where xi denotes the ith node. In the next step, the functions approximating the solution for each element are developed. The simplest choice here are the piecewise linear functions: ⎧ x < xi−1 or x ≥ xi+1 , ⎪ ⎨ 0, x−xi−1 , xi−1 ≤ x < xi , i = 1, 2, . . . , N − 1. (10.23) φi (x) = xi −xi−1 ⎪ ⎩ x−xi+1 , xi ≤ x < xi+1 , xi −xi+1 Higher-order polynomials can also be used for the approximation of φi (x). The critical requirement for the approximation functions is that they should be continuous and differentiable within each element. If Eq. (10.23) is used for the basis functions, then the numerical solution for u(x) can be formulated as: u (x) ≈ u˜ =

N 

ui φi (x) .

(10.24)

i=0

After substituting this into the original equation (10.22) and taking the inner product of that equation with the basis functions, we can obtain a system of algebraic equations (linear or nonlinear), similar to those that appear in the finite-difference methods. The resulting matrices, like for the finite-difference methods, have to be inverted using various techniques, either direct or iterative, depending on the system characteristics. The example shown here in Eqs. (10.22), (10.23), and (10.24) doesn’t go into detail and doesn’t provide the full description of the finite-element method. Tools from functional analysis are needed to fully elaborate the method, including the weak form of differential equations, inner products, and minimizing a residual with respect to a norm. The reader should consult [3], [5], [4] for more detailed and general information. The challenges for applying a finite-element method to the Blasius or Falkner–Scan equation are similar to those that arise with the finite-difference method. The first and foremost one is the semiinfinite solution interval η ∈ [0, ∞). Like with the finitedifference methods, this interval can be truncated to the finite computational interval  η ∈ [0 , η∞ with η∞ being large enough in magnitude not to affect the accuracy of the

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solution. However, like with finite-differences, large η∞ might result in an extremely high computational cost. Additionally, the matrices that result from the finite-element method have a different structure than in the finite-difference methods, and this structure might not allow for an efficient inversion method. Furthermore, the procedure of the basis function selection and application of the inner product represents additional mathematical challenges. Later, in the concluding section, we will provide more discussion on the application of the finite differences and finite-element methods (as well as some other techniques, like the spectral methods) for the Blasius or Falkner–Scan equation. We will also consider some of the current research trends including the methods that avoid the ne cessity of truncating the semiinfinite interval η ∈ [0 , ∞) to η ∈ [0 , η∞ . However, those methods are a subject of active research and often pose their own unique set of challenges. Thus, the finite difference and finite-element methods won’t be included into the practical section devoted to the numerical solution of the Blasius equation, and we will concentrate on the application of the shooting method technique, which is by far the most popular practical solution implementation for this problem. The illustration of the practical application of the finite element method will be provided in Sect. 10.5, where unsteady heat equation PDE will be solved. This equation is better suited for the solution using finite-differences and finite-element methods. The finiteelement solution provided in Sect. 10.5 will demonstrate the numerical procedures typically used in a broad range of cases when the boundary layer momentum equation (10.1) cannot be transformed into a Blasius-type ODE.

10.3.2 Practical application of the shooting method for the solution of the Blasius equation 10.3.2.1

Change of variables and first steps

To solve the Blasius equation numerically using the shooting method, this third-order nonlinear ODE is usually transformed into a nonlinear system of 3 ODEs of first-order. For that, a change of variables is performed: F = f  , G = f  , H = f . The resulting system is: F  = −F H,

(10.25)

G = F, H  = G. The boundary conditions on the wall are G|η=0 = H |η=0 = 0. The third boundary condition is formulated at η = ∞: G|η=∞ = 1. As discussed in Sect. 10.3.1, the semiinfinite integration interval is  typically truncated, and the solution is being obtained on the interval η ∈ [0 , η∞ with η∞ = 10. Thus, the modified set of the boundary conditions is: G|η=0 = H |η=0 = 0, and G|η=10 = 1.

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Now the solution procedure can be started. As described in Sect. 10.3.1, we have to specify a “guess” value for F |η=0 (“absent” initial condition). This first guess can be, for example, F |η=0 = F0 = 0. After that the system can be marched along the discretized interval [η0 , .., ηN ] using Euler, or Runge–Kutta, or any other method of choice until it reaches ηN = 10. For the explicit Euler method (see Eq. (10.11)), the solution on any step ηn+1 can be calculated as: Fn+1 = Fn − ηFn Hn ,

(10.26)

Gn+1 = Gn + ηFn , Hn+1 = Hn + ηGn . For the Runge–Kutta or Adams–Bashforth methods, the solution algorithm can be defined using the general formulas from Sects. 10.2.2.1 and 10.2.2.2. These formulas will be presented later in the section devoted to the practical code development. After the ODE system is solved for over the entire discretization interval and the value GN is obtained, it can be compared with the desired G|η=10 = 1. Here a measure for the error in GN (numerically obtained G at η = 10) can be introduced as εG = G|η=10 − GN .

(10.27)

In the ideal case, the absolute value of εG has to be smaller than a certain, userspecified, error threshold that is close to zero (for example, |εG | < 10−6 ). This implies that GN → G|η=10 , and that the numerical solution with the guess for the absent boundary condition F |η=0 = F0 was able to satisfy the endpoint boundary condition G|η=10 . Unless we are extremely lucky, the first guess won’t lead to a solution with εG within the error threshold. Thus, a second iteration is needed. We cannot continue randomly guessing the value of F0 . A method is needed to systematically iterate towards the value of F0 that would lead to εG → 0, and thus to the accurate global solution of the Blasius equation system. Here GN and εG can be considered as a function of F0 and our task can be formulated as finding the root F0 of εG (F0 ) = G|η=10 − GN (F0 ). This can be done using a variety of methods existing in the literature. A detailed description of these methods, such as the bisection method, secant method, or Newton– Raphson method, can be found in [5], [6], [3]. Here we will consider the secant method and the bisection method.

10.3.2.2

Secant method for the iteration towards finding the accurate F |η=0

To start the secant method iterations, two initial guesses have to be made for F0 . We will label them F01 and F02 . The corresponding GN values are G1N and G2N . The first approximation for the function GN (F0 ) is based on a straight line, and this approximation can be made using G1N (F01 ) and G2N (F02 ) as schematically shown in Fig. 10.3. Every point on this straight line can be then found using the line equation. After that,

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Figure 10.3 Illustration of the secant method iterations. Note that the curve shown here is given just for illustrative reasons and its shape doesn’t represent the real dependency given by the Blasius system of equations.

we can find the next guess for F0 as the value at which the straight-line approximation of the symbolic curve shown in Fig. 10.3 intersects the value G|η=10 . This is F03 that can be seen in Fig. 10.3. The equation for F03 is   F03 = F02 + m G|η=10 − G2N . (10.28) Here m is the slope of the straight line that approximates the relationship between G1N (F01 ) and G2N (F02 ): m = (F02 − F01 )/(G2N − G1N ).

(10.29)

This F03 is the new value of the initial condition F0 that will be used in the third iteration of the ODE solver. With F03 the new G3N can be obtained and εG calculated according to Eq. (10.27). If |εG | is still not within a user-specified error threshold, the secant method iteration is repeated to obtain the next initial value F04 , using F02 , F03 , G20 , and G30 . On any successive iteration k, the general formula for the next F0k+1 is   F0k+1 = F0k + m G|η=10 − GkN with m = (F0k − F0k−1 )/(GkN − Gk−1 N ). (10.30) Usually, less than ten secant iterations are needed to obtain a converged solution that would lead to the small enough εG . At this point, the numerical solution of the equation system (10.25) is finished, and the global solution that satisfies all the boundary conditions is found. In summary, the iterative procedure of the numerical solution for the Blasius equation via shooting method in combination with the secant method is as follows: 1) Make two initial guesses for the absent boundary condition F |η=0 , F01 and F02 . 2) Run two initial iterations of the ODE solver using Eqs. (10.26) or equations of any other method of choice. Obtain G1N and G2N . 3) Calculate |εG | for both G1N and G2N using Eq. (10.27). If |εG | for G1N or G2N are more than the chosen error threshold start the secant procedure:

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Figure 10.4 Schematic representation of the bisection method. Note that the curve shown here is given just for illustrative reasons and its shape doesn’t represent the real dependency given by the Blasius system of equations.

– – – –

Calculate F03 using Eq. (10.30). Run the ODE solver with the new F03 . Obtain G3N . Calculate |εG | for G3N . If |εG | is more than the chosen error threshold, repeat the secant procedure for every consecutive F0k+1 until convergence.

10.3.2.3

Bisection method for the iteration towards finding the accurate F |η=0

The bisection method is even simpler in concept and implementation than the secant method, however, it often leads to slower convergence. Its principle is based on the observation that a function changes signs on opposite sides of its roots. We consider here the error function εG given by Eq. (10.27). As pointed out above, GN can be considered as a function of F0 , and our task can be formulated as finding the root F0 of εG (F0 ) = G|η=10 − GN (F0 ). To start the iterations of the bisection method, like with the secant method considered above, two initial guesses for F0 have to be made, F01 and F02 . We also have to calculate G1N and G2N . After this is completed, a bracketing procedure has to be performed. To make sure that the root of the function εG is on the interval defined by F01 and F02 , εG has to change its sign on this interval. This can be checked using the condition εG (F01 )εG (F02 ) < 0.

(10.31)

If condition (10.31) is not fulfilled, it means that εG (F01 ) and εG (F02 ) have the same sign and the root of εG (F0 ) is not within the interval [F01 , F02 ]. In this case, new F01 and F02 have to be found. If the condition of Eq. (10.31) is fulfilled, the iterations of the bisection method can be started. It is assumed that the interval on which the root of εG can be found can be made consecutively smaller and smaller, by dividing it into two intervals on each iteration

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k + 1 and checking the sign of the function εG on each end of the smaller intervals. Schematically the process is shown in Fig. 10.4. F 1 +F 2

We determine F03 using F01 and F02 as F03 = 0 2 0 . With F03 the new G3N can be obtained using the ODE solver (Eqs. (10.26) or any other method of choice). After that εG (F03 ) can be calculated. If |εG | is not small enough to be within a user-specified error threshold, it means that the first bisection didn’t give the value of F0 that leads to GN → G|η=10 . The bisection procedure has to be repeated. First, the bracketing is performed again, and the signs of εG (F01 ) εG (F03 ) and         εG (F03 ) εG (F02 ) are compared. If εG F01 εG F03 > 0 and εG F03 εG F02 < 0, the new bisection will take place on the interval [F03 , F02 ]. Otherwise, the new bisection will be performed on the interval [F01 , F03 ]. Every new F0k+1 can be found using the old F0k as     F0l + F0k if εG F0l εG F0k < 0; 2 k + Fu     F 0 if εG F0k εG F0u < 0. F0k+1 = 0 2 F0k+1 =

(10.32)

Here F0u and F0l are the upper and the lower bounds of the interval on which F0k was found. When such an F0k+1 is found that |εG | is within a user-specified error threshold, the numerical solution of the systems of equations defined by Eq. (10.25) is complete, and the global solution that satisfies all the boundary conditions is found. In summary, the iterative procedure of the numerical solution for the Blasius equation via shooting method in combination with the bisection method is as follows: 1) Make two initial guesses for the absent boundary condition F |η=0 , F01 and F02 . 2) Run two initial iterations of the ODE solver using Eqs. (10.26) or equations of any other method of choice. Obtain G1N and G2N . 3) Calculate |εG | for both G1N and G2N using Eq. (10.27). If |εG | for G1N and G2N are more than the chosen error threshold start the bisection procedure: – Perform bracketing to make sure that εG (F01 ) εG (F02 ) < 0 and confirm that the     desired F0 can indeed be found on the interval [F01 , F02 ]. If εG F01 εG F02 > 0, repeat steps 1–3. – – – –

F 1 +F 2

Calculate F03 using as F03 = 0 2 0 . Run the ODE with the new F03 . Obtain G3N . Calculate |εG | for G3N . If |εG | is more than the chosen error threshold, repeat the bisection procedure for every consecutive F0k+1 using Eq. (10.32) until convergence.

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Figure 10.5 Initial input section of the program shooting_method.m.

10.4 Implementation of the shooting method for the solution of the Blasius equation 10.4.1 Program description The Matlab program shooting_method.m presents the implementation of the shooting method for the solution of the Blasius equation. Fig. 10.5 shows the initial input section of the program. First, the user has to select between the bisection and the secant method described in Sect. 10.3.2 by setting the value of the parameter method. The next input parameter err_tol sets the value of the error threshold for εG (see Eq. (10.27)). Then the number of the discretization points for the ODE solver is selected as well as the value of η∞ (parameters N and eta_end, respectively). In the next step the boundary conditions G|η=0 = 0 (f2(1)=0), H |η=0 = 0 (f3(1)=0), and G|η=∞ = 1 (EndBC=1) are set. The parameters f1_guess1 and f1_guess2 define F01 and F02 , respectively. These are the values of the first and the second guess for the absent boundary condition F |η=0 . After the initialization procedure, the program enters the iteration loop of the shooting method. The maximum allowed number of iterations k_max is set to k_max=20. In practice, with an appropriate choice of the initial guess values for F01 and F02 , both the secant and the bisection methods require less than 20 iterations for full convergence. The secant and the bisection method procedures in the program follow the steps described at the end of the Sects. 10.3.2.2 and 10.3.2.3. The ODE solver selected for this implementation is the fourth-order Runge–Kutta method (see Eqs. (10.18)). The values of εG and F0k on each iteration k are being printed on the screen (see Fig. 10.6). In case of the failed bracketing procedure for the initial guess values F01 and F02 in the bisection method, the program is terminated, and the error message is printed (see Fig. 10.7). In this case, the user should select different values for F01 and F02 in the input section.

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Figure 10.6 Typical output of shooting_method.m.

Figure 10.7 Error message for the failed initial bracketing in the bisection method.

At the end of its execution, the program presents the solution obtained for the equation system (10.25). The functions F = f  , G = f  , H = f are shown on the interval η ∈ [0 , η∞ (see Fig. 10.8). The solution presented in Fig. 10.8 was obtained with the input parameter values from Fig. 10.5. Furthermore, the program also demonstrates the convergence behavior of the selected numerical methods. Fig. 10.9 shows a comparison between the bisection and the secant method convergence. The bisection method requires 15 iterations to satisfy the convergence criterion |εG | < 10−4 set by the input parameter err_tol. The secant method can obtain a converged solution after six iterations. The initial guess values for F01 and F02 are the same in both simulations. Thus, here, the secant method demonstrates its superiority over bisection.

10.4.2 Results, interpretation, and sensitivity study Table 10.1 presents the comparison between the solution values for F |η=0 obtained with the present program and the “generally accepted” value F |η=0 = 0.4696 that can be found in various references, e.g., [1], [4]. The convergence criterion for |εG | was varied between |εG | < 10−4 and |εG | < 10−6 . Both the secant and the bisection method with the convergence criterion |εG | < 10−6 exactly recover the generally accepted solution. The secant method needed seven iterations to obtain this solution, and 21 iterations were needed with the bisection method. Here again, the secant method seems to be more suited for the numerical solution of the Blasius equation. As mentioned in the previous section, 6 and 15 iterations were needed to obtain a converged solution with the secant and bisection method, respectively, for |εG | < 10−4 . The differences between these solutions and the “generally accepted” one are still relatively small (see Table 10.1). Furthermore, the comparison between the various solutions obtained here and the solutions given by the finite differences and the finite element methods (last two columns in Table 10.1) also shows a good agreement.

Table 10.1 Comparison between different numerical solutions for the Blasius equation.

Solution value for F |η=0 Error relative to the generally accepted solution

Generally accepted solution [1], [4] 0.4696

Presented program secant   εG  < 10−4 0.469631

Presented program secant   εG  < 10−6 0.469600

Presented program bisection   εG  < 10−4 0.469568

Presented program bisection   εG  < 10−6 0.469600

Finite difference method [7]

Finite element method [8]

0.4695999886

0.46964178

–

−3.10E–05

0

3.20E–05

0

1.14E–08

−4.18E–05

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 10.8 Solution of the Blasius equation system (10.19) obtained with the bisection method and fourth-order Runge−−Kutta ODE solver.

Figure 10.9 Convergence behavior of the bisection and the secant methods. Absolute error vs. iteration number.

Lastly, we will examine the sensitivity of both methods to the choice of the initial guess values for F01 and F02 . The values used in all the calculations shown above are 0 and 1.3, respectively. With the expansion of the interval between F01 and F02

Numerical techniques for the solution of the laminar boundary layer equations

253

to 0 and 2.3, the secant method needs 8 iterations to obtain a converged solution for |εG | < 10−6 (1 more than for F02 = 1.3). The bisection method needed 20 iterations (1 less than for F02 = 1.3). With F02 = 3.3 the secant method needed eight iterations and the bisection method 23. It can be concluded that both methods are sensitive to the choice of F01 and F02 , but if the solution is within the initial [F01 , F02 ] interval, the number of iterations doesn’t vary significantly. Nonetheless, the secant method may essentially be regarded as a numerical technique that significantly accelerates convergence in iterative calculations. Thus, the presented program can find the solution of the Blasius equation with state-of-the-art accuracy. The physical interpretation of this solution follows from the formulation of the Blasius equation (see Sect. 10.1). The function F = f = f (η) is the dimensionless streamfunction, η is the dimensionless wall distance (η ∼ y/δ (x)), and G = f  = u/U ∞ is the dimensionless flow velocity. Thus, the G(η) plot demonstrated in Fig. 10.8 represents the self-similar boundary layer velocity profile on a flat plate.

10.5 Application of the finite-element method for one-dimensional unsteady heat equation After demonstrating a shooting method solution for the Blasius equation, this section describes the application of the finite element method for the unsteady heat equation. This simple, yet important example will illustrate the typical numerical solution procedures of the boundary layer Eqs. (10.1)–(10.2) in the general case when a self-similar solution of Blasius-type equations cannot be obtained. Recall the one-dimensional unsteady heat equation introduced in Sect. 10.1, namely ∂u ∂ 2u − ν 2 = 0. ∂t ∂x

(10.33)

Here we will consider the finite-element method numerical solution of this equation. For the general introduction into the finite-element method, the reader can consult Sect. 10.3.1 as well as numerous literature references, e.g., [3], [5], [4]. First, the spatial domain x ∈ [0, 1] is discretized into a uniform grid with spacing x. After selecting the basis functions according to Eq. (10.23), substituting the discretized numerical solution (10.24) into the original equation, and taking the inner product of this equation with the basis functions, the integrated form of the resulting semidiscrete heat equation is [4]:    1 du  2 du  1 du  ui+1 − 2ui + ui−1 + + =ν . (10.34)    6 dt i−1 3 dt i 6 dt i+1 x 2 A time integration scheme still needs to be selected for the left-hand side of the above equation. Using the Crank–Nicolson scheme [5], [3], [4] for the time integra-

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 10.10 Input section of the program diffusion_equation.m.

tion, a second-order implicit scheme, we can obtain the following algebraic equation:  2  1  1  n+1 − uni + − uni+1 ui−1 − uni−1 + un+1 un+1 i i+1 6 3 6   t  n+1 t  n n+1 n+1 n n =ν u . − 2u + u − 2u + u u + ν i i+1 i−1 i i+1 i−1 2 x 2 2 x 2 (10.35) Thus, to obtain a numerical solution of the boundary value problem formulated by Eqs. (10.5)–(10.7) the algebraic system of equations (10.35) has to be solved. That can be done using various matrix inversion algorithms. Since system (10.35) after appropriate rearrangement results in a tridiagonal matrix, Thomas algorithm (see [3]) was used for the program in the current work.

10.6

Practical implementation of the finite-element method for one-dimensional unsteady heat equation

The Matlab program diffusion_equation.m presents the implementation of the finiteelement method solution for the unsteady one-dimensional heat equation discussed in the previous section. Fig. 10.10 shows the initial input section of the program. First, the user has to define the number of spatial discretization points Nx and the number of time steps Nt. After that, the discretization of the spatial solution interval x ∈ [0, 1] is performed and the step size x is determined. It should be mentioned that Eq. (10.33) is nondimensionalized, i.e., both time and space coordinates in the solution are dimensionless and should not be associated with the real physical units. Parameter nu determines the value of the diffusion coefficient ν of Eq. (10.33). 2 The time step size is determined using the formula t = koeff_tν( x) . Here koeff_t is the user-defined timestep coefficient that connects the timestep size with the spatial

Numerical techniques for the solution of the laminar boundary layer equations

255

Figure 10.11 Typical solution of the unsteady heat equation (10.33) – comparison between the numerical solution and the exact solution. Diffusion coefficient ν = 0.001.

grid size and the diffusion coefficient. The value of this parameter cannot be too large to ensure an accurate numerical solution. Finally, the initial condition determined by Eq. (10.7) is applied. The time integration procedure begins after all the coefficients of the tridiagonal matrix determined by Eq. (10.35) are set. First, the right-hand side of the matrix equation is computed, then the Thomas algorithm matrix inversion is performed. Finally, the solution is plotted, as shown in Fig. 10.11. The user has the option to select the time interval of the graphical solution output. This is determined by the parameter deltaplot of the program. In the example shown in Fig. 10.11, the value of this parameter is deltaplot=20, i.e., the program plots the solutions on every 20th time step. Caption “plot dt=6.00” in Fig. 10.11 indicates the corresponding physical time interval. It can be seen in Fig. 10.11 that the numerical solution reproduces the exact solution of the unsteady heat equation at every time step. This solution illustrates the temperature diffusion over time, corresponding to the given value of the diffusion coefficient ν. Fig. 10.12 shows the impact of the value of the diffusion coefficient. The value was changed from ν = 0.001 to ν = 0.002. The timestep coefficient koeff_t was adjusted accordingly in order to keep the same size of the physical time step. It can be seen how the increased value of the diffusion coefficient results in the “flatter” temperature profile over time. Thus, the one-dimensional unsteady heat equation was solved here using a finiteelement method. The numerical algorithm chosen for implementation leads to a solution that matches the exact solution with a high degree of accuracy. The program presented here can be enhanced and used as a starting point of the finite-element method code for more complex equations.

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 10.12 Typical solution of the unsteady heat equation (10.33) – comparison between the numerical solution and the exact solution. Diffusion coefficient ν = 0.002.

10.7 Summary and outlook A comprehensive review of the numerical methods typically used for the solution of the Blasius or Falkner–Skan type equations was presented in this work. First, different solution methods for ordinary differential equation (ODE) initial and boundary value problems were discussed. Numerous related topics, such as method stability, convergence, and error analysis, were also elaborated. The direct application of the discussed methods for the numerical solution of the Blasius equation was outlined. Furthermore, a program for the numerical solution of the Blasius boundary value problem has been created and presented. This program represents the implementation of the shooting method, which is by far the most popular algorithm for the numerical solution of the Blasius or Falkner–Skan equations. Both the secant and the bisection algorithm for the shooting method were implemented along with a fourth-order Runge–Kutta ODE solver. The program can recover the solution of the Blasius equation with state-of-the-art accuracy. Finally, this work provides an example of a finite-element solution of the onedimensional unsteady heat equation. This example illustrates a solution algorithm of the boundary-layer PDEs in cases when they cannot be transformed into ODEs. It should be mentioned here that finite-element or finite-difference methods can also be applied for the solution of Blasius equation. Generally, despite the dominance of the shooting method for the solution of the Blasius boundary value problem, there are several examples of the successful employment of different numerical algorithms. Those examples represent the most active research topics in this area. For the most

Numerical techniques for the solution of the laminar boundary layer equations

257

part, they are related to the attempts to use the finite-differences or the finite-element methods. References [9], [7] are devoted to the solution of the Falkner–Skan and Blasius equations using a finite-difference method. As mentioned in Sect. 10.3.1, one of the challenges for solving the considered boundary value problems numerically is the semiinfinite solution interval η ∈ [0 , ∞). Instead of truncating it to the finite computational interval η ∈ [0 , η∞ as it is done in the shooting methods, references [9], [7] employ the coordinate transformation that maps the physical domain [0, ∞) directly onto the computational domain [0, 1]. However, the coordinate transformation in both cases led to the mathematical complications that had to be appropriately addressed. In [9] the coordinate transformation introduced for the Falkner–Skan equation was ξ = e−η with ξ = 0 corresponding to η = ∞ and ξ = 1 corresponding   to η = 0. Bedf  df  df  = −ξ dξ  , the boundary condition dη  = 1 could cause in this case dη  η→∞

ξ =0

η→∞

not be satisfied. An additional change of variables was needed in order to solve this problem. Thereafter, another change of variables and a number of mathematical operations was necessary to derive a convenient finite-difference stencil. In [7] the Crocco–Wang coordinate transformation was used for the solution of the Blasius equation. However, this transformation introduced the end-point singularity in the η = 0 boundary condition. This singularity impacted the accuracy of the finite-difference method adversely even when very fine computational meshes were used. The η = 0 boundary value had to be set to a small nonzero value, and additional corrections to the solution had to be introduced. As discussed in Sect. 10.3.1, application of the finite-element method to find a numerical solution of the Blasius or Falkner–Skan equations poses challenges that are similar or sometimes even greater than in the application of the finite-difference method. The examples found in the literature mostly are related to the development of finite-element codes not for the isolated self-similar cases like those that are defined by the Blasius equation but for more general sets of the boundary-layer equations [10], [11]. In these cases, the employment of a finite-element method is justified by the possibility to use the code for a variety of boundary layer applications. The Blasius or Falkner–Skan solutions follow naturally as one of the variety of solutions produced by the boundary layer code. The papers devoted to just a solution of the Blasius equation using the finite-element method, like [8] that was published in 1982, do not represent a subject of modern research. Additionally, [8] deals with an unusual implementation of the finite-element method that combines it with the shooting method but doesn’t avoid any of the finite-element method challenges like the application of the inner product or large matrix inversion. The iterative shooting method aspect increases the computational cost of the finite-element code. In light of the discussion on the current research, two other papers should be mentioned. Reference [12] presents a solution to the Blasius equation using the spectral method. The reader may also refer to [14,15] for other variants of the spectral method. Using global orthogonal polynomials as the basis functions allow for more natural treatment of the semiinfinite solution interval than in the previous examples such as the piecewise linear basis functions typically used in the finite element method. We

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will not go into further details here as an introduction into spectral method specifics goes beyond the scope of the current review and involves more complex mathematical topics. The reader can consult the appropriate references for more details. Reference [13] presents a more exotic approach as the Blasius equation is successfully solved there using a neural network algorithm. A change of variables is used to handle the infinite solution interval. The reader may consult [13] for more details. Thus, it can be stated that laminar boundary layer equations continue to attract researchers to investigate different numerical algorithms and remain one of the important test cases in numerical engineering mathematics.

References [1] F.M. White, Viscous Fluid Flow, McGraw-Hill, Inc., 1991. [2] H. Blasius, Grenzschichten in Flussigkeiten mit kleiner Reibung, Zeitschrift für Angewandte Mathematik und Physik 56 (1908) 1–37. [3] S.C. Chapra, R.P. Canale, Numerical Methods for Engineers, McGraw Hill Education, 2015. [4] P. Moin, Fundamentals of Engineering Numerical Analysis, Cambridge University Press, 2010. [5] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipies – the Art of Scientific Computing, Cambridge University Press, 2007. [6] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 2010. [7] A. Asaithambi, Numerical solution of the Blasius equation with crocco–wang transformation, Journal of Applied Fluid Mechanics 9 (5) (2016) 2595–2603. [8] A.R. Wadia, F.R. Payne, A finite element-shooting analysis of laminar boundary layers, International Journal of Computer Mathematics 12 (1982) 153–159. [9] A. Asaithambi, A second-order finite-difference method for the Falkner–Skan equation, Applied Mathematics and Computation 156 (2004) 779–786. [10] J.A. Schetz, E. Hytopoulos, M. Gunzburger, Numerical solution of the incompressible boundary-layer equations using the finite-element method, Transactions of ASME: Journal of Fluids Engineering 114 (1992). [11] P. Rana, R. Bhargava, O. Beg, Finite element simulation of unsteady magnetohydrodynamic transport phenomena on a stretching sheet in a rotating nanofluid, Proceedings of the Institution of Mechanical Engineers, Part N ,Journal of Nanoengineering and Nanosystems 227 (2) (2012). [12] J.P. Boyd, The Blasius function in the complex plane, Experimental Mathematics 8 (4) (1999). [13] I. Ahmad, M. Bilal, Numerical solution of Blasius equation through neural networks algorithm, American Journal of Computational Mathematics 4 (2014). [14] S.S. Motsa, G.T. Marewo, P. Sibanda, S. Shateyi, An improved spectral homotopy analysis method for solving boundary layer problems, Boundary Value Problems 2011 (3) (2011). [15] S. Shateyi, J. Prakash, A new numerical approach for MHD laminar boundary layer flow and heat transfer of nanofluids over a moving surface in the presence of thermal radiation, Boundary Value Problems 2014 (2) (2014).

On a selection of convective boundary layer transfer problems

11

Anselm Oyem Department of Mathematical Sciences, Federal University Lokoja, Lokoja, Kogi State, Nigeria

11.1 Introduction This work is intended to survey the concept of convective heat transfer boundary layer fluid flow and solve problems associated with convective heat and mass transfer, and Darcy, Dufour, Soret, Newtonian, and non-Newtonian boundary layer flows. These convective flows can be laminar or turbulent, free, forced or mixed convection in nature. The fluid flow determines the velocity field at every point in the region, which could either be steady or unsteady. In this context, the property of fluid (liquid or gas) does not have a preferred shape which makes it less fundamental in the study of fluid dynamics [1–3]. The main ideas of the boundary layer approximation for natural and forced convections are similar in character. The main difference is that the pressure outside a boundary layer is not determined by the main stream conditions and is hydrostatic, and the velocity beyond the boundary layer is zero. Also, it is assumed that any freeconvective flows, and mass and energy transfers by this flow, are concentrated in the main-stream in a thin layer near the boundary layer. Outside this layer, the fluid is assumed to be immobile, as gradients along the surface are assumed to be much smaller than those normal to the surface.

11.2 Viscous dissipation and magnetic field effects on convection flow over a vertical plate Consider a steady flow of an incompressible viscous free convection over a vertical plate with magnetic field and viscous dissipation effects in the presence of thermal conductivity. The physical coordinates (x, y) are chosen such that x-axis is taken along the vertical plate in the upward direction and the y-axis is normal to the plate. Using Boussinesq and boundary layer approximations, equations for mass, momentum and energy take the following form [4]: ∂u ∂v + = 0, ∂x ∂y Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00019-0 Copyright © 2020 Elsevier Ltd. All rights reserved.

(11.1)

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Applications of Heat, Mass and Fluid Boundary Layers

u

σ B02 u ∂u ∂u ∂ 2u +v = υ 2 + gβ (T − T∞ ) − , ∂x ∂y ρ ∂y

u

μ ∂T κ ∂ 2T ∂T + +v = 2 ∂x ∂y ρcp ∂y ρcp



∂u ∂y

(11.2)

2 (11.3)

.

These equations are subject to the constraints: u = 0, v = 0, T = Tw at y = 0, u → 0, T → T∞ as y → ∞.

(11.4)

In transforming equations (11.1)–(11.4) using streamfunction ψ(x, y), the following dimensionless variables are introduced: $ η=y

U0 , 2υx

θ (η) =

T − T∞ , T w − T∞

ψ=

/

2xυU0 f (η) ,

u = U0 f  (η) . (11.5)

Substituting Eq. (11.5) into Eqs. (11.1)–(11.4), we obtain ∂ψ ∂ ∂y ∂x





σ B02 ∂ψ ∂ψ ∂ ∂ψ ∂ 2 ∂ψ ∂ψ − =υ 2 + gβ (T − Tw ) − , (11.6) ∂y ∂x ∂y ∂y ∂y ρ ∂y ∂y

μ ∂ψ ∂T κ ∂ 2T ∂ψ ∂T + − = ∂y ∂x ∂x ∂y ρcp ∂y 2 ρcP



∂ ∂ψ ∂y ∂y

2 .

(11.7)

The boundary conditions associated with Eqs. (11.6)–(11.7) are: ∂ψ ∂ψ = 0, − = 0, T = Tw at y = 0, ∂y ∂x

(11.8)

∂ψ = 0, T → T∞ as y → ∞. ∂y

(11.9)

Using Eqs. (11.6) and (11.7), we arrive at the following nonlinear ordinary differential equations: f  + ff  + Grθ − Mf  = 0,

(11.10)

θ  + P rf θ  + P rEcf  2 = 0.

(11.11)

These equations are subject to the constraints: f = 0, f  = 0, θ = 1 at η = 0, f  → 0, θ → 0 as η → ∞.

(11.12)

On a selection of convective boundary layer transfer problems

261

  Table 11.1 Numerical values of skin friction Cf and Nusselt number (N u) at different values of the governing parameters. Pr 0.72 2 0.72 0.72 0.72

Gr 2.0 2.0 4.0 2.0 2.0

M 1.0 1.0 1.0 3.0 1.0

Ec 0.1 0.1 0.1 0.1 1.1

Cf 1.2924 1.1054 2.3404 0.9507 1.3976

Nu −0.3810 −0.5348 −0.4242 −0.3135 −0.0089

In practical applications, the physical quantities of interest are respectively the skinfriction, Cf , and Nusselt number, N u. These can be written in dimensionless form as: Cf =

τw = ρυ



η f  (0) , yρυ

qw η = −θ  (0) , y k(Tw − T∞ )     ∂T where τw = μ ∂u is the shear stress on the plate and q = −κ w ∂y ∂y Nu =

y=0

(11.13)

y=0

is the

rate of heat transfer over the plate. We now solve Eqs. (11.10) and (11.11) using the Runge–Kutta fourth-order scheme, along with the shooting method, implemented in MATLAB to obtain the numerical solution, by reducing the higher-order nonlinear differential equations to a system of first-order differential equations. To illustrate the method, let f = y (1), f  = y (2), f  = y (3), θ = y (4), and θ  = y(5) be such that 

dy(1)  dη y(1)=0

= y (2)



dy(2)  dη y(2)=0

= y (3)



dy(3)  dη y(3)=s



dy(4)  dη y(4)=0



dy(5)  dη y(5)=s

= −y (1) y (3) − Gry (4) + My (2) 1

= y (5) = −P ry (1) y (5) − P rEcy (3)2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(11.14)

2

where s1 and s2 are the guess values for the boundary conditions to be satisfied. Based on Eq. (11.14), the following can be observed: an increase in Ec and M leads to an increase in the rate of heat transfer while the rate decreases with increasing P r and Gr. Also the effects of the prescribed parameter on the skin friction coefficient Cf shows that an increase in P r and M leads to a decrease in the skin friction coefficient while it increases as Ec and Gr increase. (See Table 11.1, Figs. 11.1, 11.2, 11.3.)

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 11.1 Variation of f  (η) against η for different values of Ec.

Figure 11.2 Variation of velocity distribution against η for various values of M.

Figure 11.3 Temperature θ against η for various values of M.

On a selection of convective boundary layer transfer problems

263

11.3 Free convection mhd flow past a semiinfinite flat plate Consider a two-dimensional, incompressible free convection flow past a continuously moving semiinfinite flat plate with magnetic field. The physical coordinates (x, y) are chosen such that x-axis is taken along the plate and the y-axis is normal to the plate. Under the Boussinesq and boundary layer approximation, the flow is governed by the continuity, momentum, energy and concentration equations as [5,6]: ∂u ∂v + = 0, ∂x ∂y

u

σ B02 u ∂u ∂ 2u ∂u +v = υ 2 + gβ (T − T∞ ) − , ∂x ∂y ρ ∂y

∂T μ ∂T κ ∂ 2T u + +v = 2 ∂x ∂y ρcp ∂y ρcp u

(11.15)



∂u ∂y

(11.16)

2 ,

∂C ∂ 2C ∂C +v =D 2 . ∂x ∂y ∂y

(11.17)

(11.18)

These equations must satisfy the following boundary conditions: u = U0 , v = 0, T = Tw , C = Cw at y = 0, u = 0, T → T∞ , C → C∞ as y → ∞.

(11.19)

The set of partial differential equations is then converted into ordinary differential equations by using the streamfunction ψ(x, y) and the following similarity variables: $ / U0 C∞ T∞ , ψ = xυU0 f (η) , φ = C − , (11.20) ,θ = T − η=y υx Tw − T∞ C w − C∞ where θ is the dimensionless stream temperature and f is the dimensionless velocity parameter. The streamfunction satisfies the continuity equation (11.15), and the final transformed equations (11.16)–(11.18) with boundary conditions (11.19) using the similarity variable (11.20) are: 1 f  + ff  + Grθ − Mf  = 0, 2

(11.21)

 2 1 θ  + P rf θ  + P rEc f  = 0, 2

(11.22)

1 φ  + Scf φ  = 0. 2

(11.23)

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Applications of Heat, Mass and Fluid Boundary Layers

Table 11.2 Numerical values of governing parameters effects on f  (0), θ  (0), and φ  (0). Gr 1 2 2.5

Pr 0.71

M 0.1

Sc 0.3

Ec 1

7 10 1 2 0.6 0.8 1.5 2

f  (0) 0.4619 1.5891 2.5916 0.1109 0.0788 −0.3500 −0.8706 0.4619 0.4619 0.5413 0.6417

−θ  (0) 0.3152 −0.1645 −1.1565 0.8110 0.9282 0.2543 0.0954 0.3152 0.3152 0.2216 0.0810

−ϕ  (0) 0.2766 0.3384 0.3836 0.2324 0.2298 0.2217 0.1936 0.4232 0.5005 0.2858 0.2965

Figure 11.4 Velocity profile for different values of Ec.

Also, the initial conditions are: f = 0, f  = 1, θ = 1, φ = 1 at η = 0, f  = 0, θ = 0, φ = 0 as η → ∞.

(11.24)

The set of Eqs. (11.21)–(11.23), together with boundary conditions (11.24), can be numerically solved by the Runge–Kutta fourth-order scheme, along with the shooting method, implemented using MATLAB. A step size η = 0.001 is used to obtain the required accuracy for the numerical solution with a 10−7 criterion of convergence. (See Table 11.2, Figs. 11.4, 11.5, 11.6.) The effects of viscous dissipation in the presence of a magnetic field on free convection flow of an incompressible viscous laminar fluid past a continuously moving semiinfinite flat plate have been resolved and it is observed that an increase in the viscous dissipation parameter Ec results in an increase in velocity and temperature but a decrease in concentration distribution. Also, an increase in the heat transfer coefficient

On a selection of convective boundary layer transfer problems

265

Figure 11.5 Temperature profile for different vales of Eckert number Ec.

Figure 11.6 Concentration profile for different values of Ec.

results in an increase in Gr but Gr decreases with an increase in P r, M, Sc, and Ec. Finally, Sherwood number decreases with an increase in Gr, Sc, Ec while it increases with an increase in P r and M.

11.4

Convection of non-darcy flow, Dufour and Soret effects past a porous medium

Darcy’s empirical flow model represents a simple linear relationship between flow rate and pressure drop in a porous media as any deviation from the Darcy flow scenario is termed a non-Darcy flow. Similarly, the importance of the Soret (thermal-diffusion) and Dufour (diffusion-thermo) effects on fluids with very light and medium molecular weight can never be overemphasized. In order to illustrate this concept, the following example is considered. A steady two-dimensional laminar free convection flow of viscous incompressible variable viscosity and variable thermal conductivity fluid along a porous vertical sur-

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Applications of Heat, Mass and Fluid Boundary Layers

face in the presence of suction and heat generation and absorption is described by the following equations [22]: ∂u ∂v + = 0, ∂x ∂y

u

(11.25)



∂u ∂u 1 ∂ ∂u +v = μ(T ) + gβ (T − T∞ ) + gβ ∗ (C − C∞ ) ∂x ∂y ρ ∂y ∂y b∗ μ (T ) 1 u − u2 , − ρ K K

(11.26)



∂T 1 ∂ ∂T 1 ∂qr q ∂T +v = κ(T ) − + u (T − T∞ ) ∂x ∂y ρCP ∂y ∂y ρCP ∂y ρCP +

u

DKt ∂ 2 C , CP CS ∂y 2

∂C ∂ 2 C DKt ∂ 2 T ∂C , +v =D 2 + ∂x ∂y Tm ∂y 2 ∂y

(11.27)

(11.28)

subject to the boundary conditions: u = Bx m , v = −Vw , T = CW , C = CW at y = 0,

(11.29)

u → 0, T → T∞ , C → C∞ as y → ∞.

(11.30)

In obtaining the boundary layer solution for the flow past the porous medium, a similarity solution is used to emphasize the assumptions undertaken by introducing stream functions and dimensionless variables. Using the Rosseland approximation for radiation, we have qr =

−4σ ∂T 4 . 3k ∗ ∂y

(11.31)

The energy equation then reduces to u

∂ 2T 1 ∂T ∂κ(T ) ∂T ∂T ∂T 1 κ (T ) 2 + +v = ∂x ∂y ρCP ρCP ∂y ∂T ∂y ∂y 16σ T 3 ∂ 2 T q DKt ∂ 2 C + ∗ ∞ + . − T + (T ) ∞ 3k ρCP ∂y 2 ρCP CP CS ∂y 2

(11.32)

Firstly, we introduce the streamfunctions, which satisfy the continuity equation (11.25): u=

∂ψ ∂ψ ,v = − . ∂y ∂x

On a selection of convective boundary layer transfer problems

267

Then, in terms of the streamfunction, Eqs. (11.26), (11.28)–(11.30), and (11.32) become ∂ψ ∂ ∂ψ ∂ψ ∂ ∂ψ − ∂y ∂x ∂y ∂x ∂y ∂y ∂ 3ψ ∂θ ∂ 2 ψ + υ ∗ [1 + (1 − θ ) ξ ] 3 + gβθ (Tw − T∞ ) = −υ ∗ ξ 2 ∂y ∂y ∂y

∗ b∗ ∂ψ 2 υ [1 + (1 − θ) ξ ] ∂ψ ∗ , + gβ φ (Cw − C∞ ) − − K ∂y K ∂y

(11.33)

∂ψ ∂θ ∂ψ ∂θ − ∂y ∂x ∂x ∂y 2 2 ∂ θ ∂θ κ∗ q 16σ T 3 ∂ 2 θ κ∗ ε + + θ + ∗ ∞ = [1 + θ ε] 2 2 ρCP ∂y ρCP 3k ρCP ∂y ρCP ∂y +

DKt (Cw − C∞ ) ∂ 2 φ , CP CS (Tw − T∞ ) ∂y 2

∂ψ ∂φ ∂ψ ∂φ ∂ 2 φ DKt (Tw − T∞ ) ∂ 2 θ . − =D 2 + ∂y ∂x ∂x ∂y Tm (Cw − C∞ ) ∂y 2 ∂y

(11.34)

(11.35)

These equations are subject to the constraints: ∂ψ ∂ψ = Bx m , = Vw , θ = 1, φ = 1 at y = 0, ∂y ∂x

(11.36)

∂ψ → 0, θ → 0, φ → 0 as y → ∞. ∂y

(11.37)

Introducing the following dimensionless variables: $ √ B T − T∞ C − C∞ η=y ,φ = , , ψ = υ ∗ xBf (η) , θ = ∗ υ x Tw − T∞ C w − C∞

(11.38)

Eqs. (11.33)–(11.37) reduce to a system of coupled nonlinear ordinary differential equations 3 1 d 2f dθ d 2 f d f + + JGr ξ θ + JGT ξ φ f − ξ [1 + ξ − θ ξ ] dη dη2 2 dη2 dη3 Fs df 2 [1 + ξ − θ ξ ] df = 0, (11.39) − − D a Re dη Da dη 1 + θε +

4 3N

2 d 2φ d 2θ 1 dθ dθ + + $P rθ + Pr Df 2 = 0, (11.40) P rf + ε 2 2 dη dη dη dη

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 11.7 Velocity profiles for several values of local Darcy Da .



d 2φ dη2

dφ 1 + Sc Sr + Sc f 2 dη



d 2θ dη2

= 0.

(11.41)

Whereby the equations are subject to the following constraints: df = 1, f = fW , θ = 1, φ = 1 at y = 0, dη

(11.42)

df → 0, θ → 0, φ → 0 as y → ∞. dη

(11.43)

The mere fact that the original partial differential equations have been reduced to a pair of ordinary differential equations confirms the assumption that similarity solutions do exist. The set of coupled nonlinear ordinary differential equations are numerically solved using Runge–Kutta fourth-order technique, along with the shooting method. The computer program MATLAB was used to implement this procedure with a step-size η = 0.05 to compute skin-friction coefficient and Nusselt and Sherwood numbers which are respectively proportional to f  (0), θ  (0), and φ  (0) as shown in Table 11.3. (See Figs. 11.7, 11.8, 11.9.)

11.5

Unsteady mhd convective flow with thermophoresis of particles past a vertical surface

An unsteady one-dimensional hydrodynamic convective flow of a viscous incompressible, electrically conducting and chemically reacting fluid flow past a porous vertical

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269

Figure 11.8 Temperature profiles for several values of local Darcy Da .

Figure 11.9 Concentration profiles for several values of local Darcy Da .

heated surface moving through a binary mixture is presented as: ∂v = 0, ∂y

(11.44)



σ ∗ B02 ∂u ∂u ∂u +v = μ + gβ (T − T∞ ) + gβ ∗ (C − C∞ ) − u, ∂t ∂y ∂y ρ

(11.45)

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Applications of Heat, Mass and Fluid Boundary Layers

Table 11.3 Effects of variation of D a . Da f  (0) θ  (0) φ  (0) 0.25 −0.6948909153462 −0.0847901517352 −0.2322796609776 0.50 0.0569874691501 −0.1248722156789 −0.2557853344647 0.75 0.4441734747311 −0.1443460599340 −0.2689197433020 1.00 0.6961089859273 −0.1563124614536 −0.2775192130596 1.25 0.8772472106404 −0.1645105919899 −0.2836324649220 ξ = ε = 1.2; Fs = 1; Re = 200; J GR = 1; J GT = 1; P r = 0.72; N = 0.7; Df = 0.03; $ = 0.2; Sc = 0.22; Sr = 0.4; fW = 0.3



∂T 4σ α 2 4 ∂T 1 ∂ ∂T − H RA − T +v = κ + ∂t ∂y ρcp ∂y ∂y ρcp ρcp   Q0 (T − T∞ ) −g √y 2 ϑt , + e ρcp ∂ 2C ∂C ∂ ∂C +v + [Vτ (C − C∞ )] = Dm 2 − RA . ∂t ∂y ∂y ∂y

(11.46)

(11.47)

Based on Eqs. (11.44)–(11.46), some assumptions are made: the flow is assumed to be in the x-direction, which is taken along the semiinfinite plate and y-axis is normal to it. Since the surface is of infinite length in the x-direction, all the physical quantities (i.e., velocity, temperature, and species concentration) are assumed to be independent of x. A uniform magnetic field is applied in the direction perpendicular to the surface. The fluid is assumed to be slightly conducting; the induced magnetic field is negligible in comparison with the applied magnetic field. It is further assumed that there is no applied voltage, so that electric field is absent. Eqs. (11.44)–(11.46) must be solved subject to the following boundary conditions: ⎫ t ≤ 0 : u (y, 0) = 0, T (y, 0) = Tw , C (y, 0) = Cw ⎬ t > 0 : u (0, t) = U0 , T (0, t) = Tw , C (0, t) = Cw . (11.48) ⎭ t > 0 : u (∞, t) → 0, T (∞, t) → T∞ , C (∞, t) → C∞ Since, this formulation is based on the notion that just as temperature gradient constitutes the driving potential for heat transfer, species concentration gradient in a mixture provides the driving potential for mass transfer [7], the effect of thermophoresis is usually prescribed by means of the average velocity, which a particle will acquire when exposed to a temperature gradient [8,9]. From Eq. (11.44), it is worth noticing that velocity component (v) is either constant or a function of time hence, following [10], velocity component along y-axis is considered as √ v = −c ϑt, (11.49) where c > 0 and c < 0 are known as suction and injection, respectively. From Eq. (11.46), the relationship between the activation energy and the rate at which a

On a selection of convective boundary layer transfer problems

271

reaction proceeds is accounted for by using

EA Q = (− H ) RA , RA = Kr exp − Cn. RG T

(11.50)

It is valid to consider the mathematical model of a temperature-dependent viscosity model which was developed using experimental data, together with the mathematical model of temperature-dependent thermal conductivity [11–14]: μ (T ) = μ∗ [a + b (Tw − T )] and

κ (T ) = κ ∗ [a + δ (T − T∞ )] ,

(11.51)

where μ∗ and κ ∗ are the constant values of the coefficient of viscosity and thermal conductivity at the free stream, respectively. A case where a = 1 is considered with (b, δ) > 0. The thermophoretic velocity parameter in Eq. (11.4) is given as [15,16]: Vτ = −

k T h ∂T Tref ∂y

and τ = −

k T h Tw , ϑTref

(11.52)

where k T h is the thermophoretic coefficient ranging from 0.2 to1.2 and defined by the theory [15,17] as:   fluid 2Cs κκdiff·P + Ct Kn Cu ,  (11.53) kT h = fluid + 2Ct Kn (1 + 3Cm Kn ) 1 + κ2κdiff·P where Cs , Cm , Ct are constants, κfluid , κdiff·P , Cu , and Kn are the fluid thermal conductivity, thermal conductivity of the diffused particle, Cunningham correction factor, and Knudsen number, respectively. Substituting Eqs. (11.49)–(11.53) into Eqs. (11.44)–(11.47), we have  2  ξ ∂θ ∂u ∂ u ∂u θ ∂u ∗ ξ − ϑ∗ +v =ϑ 1+ξ − ∂t ∂y θw θw ∂y ∂y ∂y 2 + gβ (T − T∞ ) + gβ ∗ (C − C∞ ) − ρcp

∂T ∂T +v ∂t ∂y

σ ∗ B02 u, ρ

(11.54)

 ∂ 2θ 1 λ λ ∂θ 2 κ ∗ T∞ λθ − =κ 1+ + T∞ 2 θw θw 4ϑt ∂η 4ϑt θw ∂η

EA 4 − 4σ α 2 θ 4 T∞ + (− H ) Kr exp − RG T ∞

EA 1 n × exp φ n C∞ 1− T ∞ RG θ    y + Q0 T∞ (θ − 1) exp −g , (11.55) 1 2 (ϑt) 2 ∗



272

Applications of Heat, Mass and Fluid Boundary Layers

∂C ∂C C∞ ∂θ ∂φ 1 τ ∂ 2θ 1 +v + C∞ (φ − 1) + ∂t ∂y θw ∂η2 4t θw ∂η ∂η 4t

2 ∂ C EA EA 1 n . (11.56) φ n C∞ exp 1− = Dm 2 − Kr exp − RG T ∞ T ∞ RG θ ∂y Introducing the following dimensionless variables: f (η) =

Gr =

h=

u y Tw C Cw tQ0 T ,η = √ ,θ = , θw = ,φ = , φw = ,χ = , U0 T∞ T∞ C∞ C∞ ρcp 2 ϑt

4tσ ∗ B02 4tgβ ∗ κ∗ 4tgβ , λ = δTw , ξ = bTw , , Gc = ,M = ,γ = U0 b U0 b ρ ρcp 4 µcp ϑ 4σ α 2 T∞ EA (− H ) C∞ , P r = = ∗ , Ra = 4t ,ω = , ρcp T∞ γ κ ρcp T∞ T ∞ RG

Da = 4tKr e

−R

EA G T∞

n−1 C∞

into Eqs. (11.54)–(11.56), the following dimensionless nonlinear ordinary differential equations are obtained:   ξ dθ df df θ ξ d 2f − + 2 (η + c) 1+ξ − 2 θw dη θw dη dη dη ξ ξ + Gr (11.57) (θ − 1) + Gc (φ − 1) − Mf = 0 θw θw 

 λ d 2θ λ dθ 2 dθ λθ − + + 2 Pr (η + c) 2 θw θw ∂η θw dη dη

1 φ n − P rθ 4 Ra + 4χ (θ − 1) exp (−gη) = 0, + P rhDa exp ω 1 − θ (11.58)

1+



d 2φ dφ τ d 2θ 1 n + 2Sc + c) − Sc − 1) − ScDa exp ω 1 − φ (η (φ dη θ θw dη2 ∂η2 τ dθ dφ − Sc (11.59) θw dη dη subject to the constraints: f (0) = 1, θ (0) = θw (> 1) , φ (0) = φw (> 1) ,

(11.60)

f (∞) → 0, θ (∞) → 1, φ (∞) → 1,

(11.61)

On a selection of convective boundary layer transfer problems

273

Table 11.4 Comparison values of Cf with Sastry and Murti [19] using ξ = λ = τ = χ = 0,     Gr θξ = Gc φξ = Da = ω = Ra = θw = φw = 0.1, P r = 0.71, Sc = 0.22, h = n = 1. w

w

f  (0) when c = 0.1 in Sastry and Murti [19] M =5 M = 10 M =5 M = 10

−2.635097137 −3.475594470 f  (0) when c = 0.1 −2.418368678 −3.265944783

Shooting method along with quadratic interpolation present at f  (0) when c = 0.1 −2.635097136618 −3.4755944595667 f  (0) when c = −0.1 −2.418368675767 −3.265944783052

Shooting method with linear interpolation present at f  (0) when c = 0.1 −2.6314758242887 −3.4738132412511 f  (0) when c = −0.1 −2.418388801774 −3.265943456084

Figure 11.10 Effect of variable thermo-physical properties: (A) velocity profile and (B) temperature profile when Gr = Gc = −1.

where Da, Ra, ω, Gr, Gc, χ, Kr , P r, Sc, M, and τ are the Damköhler number, radiation parameter, activation energy parameter, modified thermal Grashof number, modified solutal Grashof number, heat source parameter, chemical reaction rate, Prandtl number, Schmidt number, magnetic field parameter, and thermophoretic parameter, respectively. The quantities of physical interest is this problem are local skin friction Cf , Nusselt number N u, and Sherwood number Sh defined as [18]  √ √    2ϑ t ∂u  df  2 ϑt ∂T  dθ  = , Nu = − = − , Cf = √  dη η=0 T∞ ∂y  dη η=0 U0 ϑ ∂y y=0 y=0  √ 2 ϑt ∂C  Sh = −  C∞ ∂y 

=− y=0

 dφ  . dη η=0

(11.62)

Furthermore, we employ the Runge–Kutta method with shooting technique coded in MATLAB to obtain the numerical results [19,20]. These are displayed in the table below with graphical representations. (See Table 11.4, Figs. 11.10, 11.11, 11.12.)

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 11.11 Variation of skin friction coefficient for different values of variable thermo-physical properties (viscosity and thermal conductivity) together with Gr and Gc .

Figure 11.12 (A) Effects of magnetic field parameter on velocity, (B) variation of skin friction coefficient for different values of magnetic field parameter against buoyancy parameters (Gr and Gc ).

When variable thermo-physical properties are accounted for in the transport phenomena of fluid flow along a vertical surface moving through a binary mixture in the presence of exponential heat source, thermal and solutal Grashof numbers are an important yardstick to control skin friction drag which arises from the friction of the fluid against the skin of the vertical surface that is moving through the binary mixture. The convergence rate of the new approach (Quadratic Interpolation) of finding roots while shooting is faster than the rate of the secant method (Linear Interpolation) [21].

11.6 Thermal conductivity effects on compressible boundary layer flow over a vertical plate Thermal conductivities of materials vary dramatically both in magnitude and temperature from one material to another due to differences in sample sizes. In this problem,

On a selection of convective boundary layer transfer problems

275

we will develop insights about the effects of varying different thermal conductivity and heat transfer through a laminar boundary layer flow of a viscous fluid over a body of arbitrary shape and arbitrary specified surface temperature. The difference in the temperature will initiate the physical contact between the particles, creating kinetic energy and momentum. The equations describing the steady flow of compressible, laminar two-dimensional boundary layer flow, under the assumption that the viscosity μ is proportional to the absolute temperature T and the Prandtl number P r is unity, is given as [23]: ∂ ∂ (ρu) + (ρv) = 0, ∂x ∂y

(11.63)



∂u ∂u 1 ∂p 1 ∂ ∂u u +v =− + μ , ∂x ∂y ρ ∂x ρ ∂y ∂y

(11.64)



2 ∂T ∂p ∂ ∂T ∂u ∂T ρcp u , +v −u = κ +μ ∂x ∂y ∂x ∂y ∂y ∂y

(11.65)

p = ρRT ,

(11.66)

where

μ = μ0

T , T0

(11.67)

subject to the boundary conditions: u = v = 0, T = Tw at y = 0, u = U1 , T = T1 as y → ∞,

(11.68)

where U1 is the main-stream velocity taken as the velocity in the irrotational motion of an incompressible fluid. Thus, if a is the radius of the cylinder, then x  U1 (x) = U∞ sin . (11.69) a In obtaining a solution describing the flow and solving the heat transfer equations (11.63)–(11.65), we reduce the problem to almost an incompressible form by applying the Stewartson’s transformations [24]: a1 ∂ψ Y= , (11.70) a0 ∂y √ ∂ψ . ρu = ρ0 v0 ∂y

(11.71)

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Applications of Heat, Mass and Fluid Boundary Layers

In  applying   the transformation, we take into consideration that ρ = constant, T μ0 T0 ≡ μ0 pp0 , and by the Eulerian equation of motion of an inviscid flow (assuming no force with steady flow), Eqs. (11.63) and (11.64) become u

∂u ∂u 1 ∂p +v =− , ∂x ∂y ρ ∂x

(11.72)





a12 ∂ ∂T ∂ 2T ∂κ ∂T κ = 2 +κ 2 . ∂y ∂y ∂Y a0 v0 ∂Y ∂Y

(11.73)

If there is a constant flow along the x-direction, and the pressure gradient term is assumed to be known from Bernoulli’s equations and is applied to the outer inviscid flow, we have

∂u dU1 ∂u 1 ∂ ∂u u +v = U1 + μ . (11.74) ∂x ∂y dx ρ ∂y ∂y Taking u = U1 and T = T1 based on Eq. (11.68), applying streamfunction to Eq. (11.70), and by the power law of an isentropic process, Eq. (11.73) becomes

a1 a0

3γ −2 γ −1

∂ψ ∂ 2 ψ ∂ψ ∂ 2 ψ − ∂y ∂y∂x ∂x ∂y 2

5γ −3 dU1 a1 γ −1 T ∂ 3ψ + U1 . (11.75) = 3 a0 T0 dx ∂Y

Multiplying Eq. (11.64) by u, adding the result to Eq. (11.65), taking cp = c and the function S as relating to the absolute temperature T with Mach number relation

 

γ −1 2 γ −1 2 γ −1 2 v T u2 T 1+ − = 1+ M1 S = M1 1 − 2 − 1, M1 , M1 = , 2 T1 2 T 2 a U1 1 (11.76)

Eq. (11.72) becomes

a1 a0



∂ψ ∂S ∂ψ ∂S − ∂Y ∂x ∂x ∂Y

=

a12 a02



∂ 2S ∂κ ∂S +κ 2 . ∂Y ∂Y ∂Y

(11.77)

Now considering a flow in which the Mach number is 1, we replace a0 /a1 by unity and obtain the equations describing the flow and heat transfer as ∂ψ ∂ 2 ψ dU1 ∂ 3ψ ∂ψ ∂ 2 ψ = + U1 − (1 + S) , 2 3 ∂Y ∂Y ∂x ∂x ∂Y dx ∂Y

(11.78)

∂ψ ∂S ∂ψ ∂S ∂κ ∂S ∂ 2S − = +κ 2, ∂Y ∂x ∂x ∂Y ∂Y ∂Y ∂Y

(11.79)

Table 11.5 Values of surface temperature parameter Sw at α = 0.3. ξ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

S(0.5) 5.0000000E-01 4.4444689E-01 3.8838514E-01 3.3255827E-01 2.7811006E-01 2.2642120E-01 1.7889744E-01 1.3674414E-01 1.0077685E-01 7.1312747E-02 4.8166178E-02

S(1.0) 1.0000000E + 00 8.9238086E-01 7.8260478E-01 6.7208666E-01 5.6314838E-01 4.5873482E-01 3.6198873E-01 2.7574259E-01 2.0203875E-01 1.4180915E-01 9.4806832E-02

S(1.5) 1.5000000E+00 1.3429429E+00 1.1814220E+00 1.0173874E+00 8.5425412E-01 6.9655567E-01 5.4934589E-01 4.1739678E-01 3.0435345E-01 2.1208689E-01 1.4046393E-01

S(2.0) 2.0000000E+00 1.7955634E+00 1.5839714E+00 1.3675704E+00 1.1507399E+00 9.3953727E-01 7.4098518E-01 5.6200573E-01 4.0817178E-01 2.8262670E-01 1.8557053E-01

S(2.5) 2.5000000E+00 2.2498478E+00 1.9896258E+00 1.7219556E+00 1.4520443E+00 1.1873624E+00 9.3687876E-01 7.0978643E-01 5.1385276E-01 3.5382077E-01 2.3047875E-01

Table 11.6 Values of surface temperature parameter Sw at α = 0.075. ξ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

S(0.5) 5.0000000E-01 4.4124862E-01 3.8260933E-01 3.2493129E-01 2.6944047E-01 2.1752479E-01 1.7049698E-01 1.2937907E-01 9.4753962E-02 6.6710296E-02 4.4882868E-02

S(1.0) 1.0000000E+00 8.8085809E-01 7.6153821E-01 6.4395795E-01 5.3088268E-01 4.2541588E-01 3.3043867E-01 2.4811385E-01 1.7956629E-01 1.2479788E-01 8.2827638E-02

S(1.5) 1.5000000E+00 1.3193565E+00 1.1378411E+00 9.5861856E-01 7.8626629E-01 6.2589984E-01 4.8220615E-01 3.5860142E-01 2.5671571E-01 1.7630552E-01 1.1555046E-01

S(2.0) 2.0000000E+00 1.7571043E+00 1.5122498E+00 1.2699910E+00 1.0369548E+00 8.2053891E-01 6.2746069E-01 4.6250009E-01 3.2776381E-01 2.2261742E-01 1.4420080E-01

S(2.5) 2.5000000E+00 2.1943569E+00 1.8852922E+00 1.5788623E+00 1.2839490E+00 1.0104798E+00 7.6741602E-01 5.6101325E-01 3.9383333E-01 2.6471959E-01 1.6958809E-01

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Applications of Heat, Mass and Fluid Boundary Layers

subject to the conditions ∂ψ Tw ∂Y = 0, S = T1 − 1 = Sw at Y = 0, ∂S ∂Y = U1 (x) , S → 0 as Y → ∞.

ψ=

(11.80)

In order to solve Eqs. (11.78)–(11.80), we introduce Merkin’s dimensionless variables [25] given as: √ v0 1 1 x √ Re 2 , ψ = v0 Re 2 ξf (ξ, η) , S (x, Y ) = S (ξ, η) . (11.81) ξ = ,η = Y a R Applying the dimensionless variables in Eq. (11.81) at the stagnation point ξ = 0, Eqs. (11.16)–(11.18) reduce to coupled nonlinear ordinary differential equations f  + ff  − f  2 + S + 1 = 0,

(11.82)

κS  + κ  S  + f S  = 0,

(11.83)

with conditions f (0) = f  (0) = 0, S (0) = Sw , f  (∞) = 1, S (∞) = 0.

(11.84)

11.7 Conclusion Convective heat transfer problems have been surveyed and presented in this work. Numerical results have also been presented. The reader may refer to refs. [26–29] for further insights.

References [1] K.S. Roger, Introductory Lectures on Fluid Dynamics, 2008, June 13 version. [2] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, USA, 1990. [3] O.G. Martnenko, P.P. Khramtso, Free-Convective Heat Transfer, Springer-Verlag, Berlin, Heidelberg, Netherlands, 2005. [4] O.A. Oyem, Effects of Thermo-Physical Properties on Free Convective Heat and Mass Transfer Flow Over a Vertical Plate, PhD Thesis, Department of Mathematical Sciences, Federal University of Technology Akure, 2016. [5] P. Geetha, M.B.K. Moorthy, Viscous dissipation effect on steady free convection and mass transfer flow past a semi-infinite flat plate, Journal of Computer Sciences 7 (2011) 1113–1118. [6] O.A. Oyem, Viscous dissipation effect on free convection flow past a semi-infinite flat plate in the presence of magnetic field, Theoretical Mathematics and Applications (ISSN 1792-9687) 6 (3) (2016) 101–118.

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[7] F. Incropera, Fundamentals of Heat and Mass Transfer, John Wiley and Sons, New York, 1985. [8] I.L. Animasaun, Effects of thermophoresis, variable viscosity and thermal conductivity on free convective heat and mass transfer of non-darcian MHD dissipative Casson fluid flow with suction and nth order of chemical reaction, Journal of the Nigerian Mathematical Society 34 (2015) 11–31, https://doi.org/10.1016/j.jnnms.2014.10.008. [9] A.J. Chamkha, A.F. Al-Mudhaf, I. Pop, Effect of heat generation or absorption on thermophoretic free convection boundary layer from a vertical flat plate embedded in a porous medium, International Communications in Heat and Mass Transfer 33 (2006) 1096–1102, https://doi.org/10.1016/j.icheatmasstransfer.2006.04.009. [10] O.D. Makinde, Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate, International Communications in Heat and Mass Transfer 32 (2005) 1411–1419, https://doi.org/10.1016/j.icheatmasstransfer.2005.07.005. [11] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, London, 1987. [12] K.K. Sivagnana-Prabhu, R. Kandasamy, R. Saravanan, Lie group analysis for the effect of viscosity and thermophoresis particle deposition on free convective heat and mass transfer in the presence of suction/injection, Theoretical and Applied Mechanics 36 (2009) 275–298, https://doi.org/10.2298/TAM0904275S. [13] I.L. Animasaun, A.O. Oyem, Effect of variable viscosity, Dufour, Soret and thermal conductivity on free convective heat and mass transfer of non-darcian flow past porous flat surface, American Journal of Computational Mathematics 4 (2014) 357–365, https:// doi.org/10.4236/ajcm.2014.44030. [14] J. Charraudeau, Influence de gradients de properties physiques en convection force application au cas du tube, International Journal of Heat and Mass Transfer 18 (1975) 87–95, https://doi.org/10.1016/0017-9310(75)90011-3. [15] L. Talbot, R.K. Cheng, R.W. Scheffer, D.P. Wills, Thermophoresis of particles in a heated boundary layer, Journal of Fluid Mechanics 101 (1980) 737–758, https://doi.org/10.1017/ S0022112080001905. [16] R. Tsai, A simple approach for evaluating the effect of wall suction and thermophoresis on aerosol particle deposition from a laminar flow over a flat plate, International Communications in Heat and Mass Transfer 26 (1999) 249–257, https://doi.org/10.1016/S07351933(99)00011-1. [17] G.K. Batchelor, C. Shen, Thermophoretic deposition of particles in gas flowing over cold surfaces, Journal of Colloid and Interface Science 107 (1985) 21–37, https://doi.org/10. 1016/0021-9797(85)90145-6. [18] O.D. Makinde, P.O. Olanrewaju, W.M. Charles, Unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture, Afrika Mathematica 21 (2011) 1–17. [19] D.R.V.S.R.K. Sastry, A.S.N. Murti, A double diffusive unsteady MHD convective flow past a flat porous plate moving through a binary mixture with suction or injection, Journal of Fluids (2013) 935156. [20] S. Gill, A process for the step-by-step integration of differential equations in an automatic digital computing machine, Mathematical Proceedings of the Cambridge Philosophical Society 47 (1951) 96–108, https://doi.org/10.1017/S0305004100026414. [21] I.L. Animasaun, Dynamics of unsteady MHD convective flow with thermophoresis of particles and variable thermo-physical properties past a vertical surface moving through the binary mixture, Open Journal of Fluid Dynamics 5 (2015) 106–120.

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[22] I.L. Animasaun, O.A. Oyem, Effect of variable viscosity, Dufour, Soret and thermal conductivity on free convective heat and mass transfer of non-Darcy flow past porous flat surface, American Journal of Computational Mathematics 4 (2014) 357–365, https:// doi.org/10.4236/ajcm.2014.44030. [23] M.A. Hossian, I. Pop, T.Y. Na, Effect of heat transfer on compressible boundary layer flow over a circular cylinder, Acta Mechanica 131 (1998) 267–272. [24] K. Stewartson, Correlation incompressible and compressible boundary layers, Proceedings of the Royal Society of London. Series A 200 (1949) 84–100. [25] J.M. Merkin, Mixed convection from a horizontal circular cylinder, International Journal of Heat and Mass Transfer 20 (1977) 73–77. [26] J.D. Hoffman, Numerical Methods for Engineers and Scientist, 2nd ed., Marcel Dekker, Inc., New York, 2001. [27] O.A. Oyem, A.J. Omowaye, O.K. Koriko, Combined effects of viscous dissipation and magnetic field on MHD free convection flow with thermal conductivity over a vertical plate, Daffodil International University Journal of Science and Technology 10 (1–2) (2015) 21–26. [28] O.A. Oyem, Effects of Heat Transfer on Compressible Boundary Layer Flow Over a Circular Cylinder With Variable Thermal Conductivity, M. Tech Thesis, Department of Mathematical Sciences, Federal University of Technology, Akure, 2011. [29] O.O. Anselm, K.O. Koriko, Thermal conductivity and its effects on compressible boundary layer flow over a circular cylinder, International Journal of Research and Reviews in Applied Sciences 15 (2013) 89–96.

Advanced fluids – a review of nanofluid transport and its applications

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Leye M. Amooa , R. Layi Fagbenleb a Stevens Institute of Technology, Hoboken, NJ, United States, b Mechanical Engineering Department and Center for Petroleum, Energy Economics Law, CPEEL (A John D. and Catherine T. McArthur Center of Excellence), University of Ibadan, Ibadan, Oyo State, Nigeria

12.1 Introduction This chapter is motivated by the rapid and progressive developments in nanofluids. Since Masuda and Choi properly characterized nanofluids, research has grown significantly as reflected in the particularly large number of papers concerned with nanofluids that have been published especially in the fields of thermal-fluid-mass transfer, material sciences, chemistry, and nanotechnology. Due to their potentially superior thermophysical properties and heat transfer enhancements, technological implications may be significant. In this review, we emphasize selected relevant and important topical areas and articles while addressing current and future potential applications of nanofluids. Nonetheless, every review is subjective in terms of material covered and references cited, and this review is not different in this regard. A review of all the publications about nanofluids and every area of nanofluids would be near-impossible and well beyond space limitations. The disposition of this article is as follows. Some historical perspectives, together with some fundamentals of nanofluids, are discussed in Sect. 12.1. Section 12.2 briefly discusses the current understanding of nanofluids. Classification of nanofluids is discussed in Sect. 12.3. The thermophysical properties and relations of nanofluids are discussed in Sect. 12.4.1. Section 12.5 discusses convective heat transfer of nanofluids, which includes considerations of porous media nanofluid flow, mass transfer in nanofluids, nanofluid flow in magnetic fields, and turbulent nanofluid flow. Section 12.6 addresses current and future applications of nanofluids, while Sect. 12.7 addresses some research gaps and open questions in nanofluids. A conclusion section sums up the review with key points. Amongst the many reviews in the literature, this review provides a perspective that is less focused on dense theoretical or mathematical aspects but rather on real-world practical implications, one that should find relevance with engineers in general and especially thermal-fluid engineers. Heat transfer may be defined as the microscopic transfer of random motions (molecules) due to temperature differences, not due to macroscopic forces (i.e., Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00020-7 Copyright © 2020 Elsevier Ltd. All rights reserved.

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work) or mass transfer, and accompanied by entropy transfer (ds ≥ dq T ). Macroscopically, there is a net moment of fluid particles, and this is regarded as mass transfer. Nonetheless, heat transfer permeates nearly every sector of the global economy such as utilities (energy conversion and efficiency), air transportation, air quality, automotive and aerospace transportation, petroleum and coal products, and computer and electronic products. Energy and environmental considerations are key drivers in the performance improvement of thermal designs and systems. High heat flow processes create a growing demand for new techniques and technologies that enhance heat transfer. Thermal engineering problems of applied boundary layer theory, for example, have recently intensified significantly due to the promising nature of this advanced fluid type known as nanofluid. The continuous need to enhance heat transfer through various means is an intense area of research that can be expected to endure in the foreseeable future. Submicron-scale thermo-fluid-mass transfer, especially on nanofluids is a subject of active research that continues to grow rapidly, as evidenced by the now voluminous work on the subject in literature. Nanofluids have excited and continue to excite the heat transfer literature. Though that excitement has cooled somewhat, numerous theoretical and experimental research continues due to various gaps on the subject that may have significant technological and practical implications. A nanofluid connotes a heat transfer liquid with stably suspended or dispersed nanoparticles. It can be defined as the mixture formed when nanoparticles (e.g., Al, Cu, etc.) are stably suspended within common fluids such as water, oil, ethylene glycol, ethylene glycol-based liquids (antifreeze), refrigerants, heat transfer liquids, polymer solutions, bio-fluids, and others. They may be considered as advanced and next generation working fluids owing to the realm of possibilities to enhance thermal performance as compared to common working fluids such as water. They are a smart type of fluid. The word “smart” implies adaptability and the capacity to tune or tailor nanofluids to numerous applications. Nanoscale materials (i.e., nanoparticles, nanofibers, nanotubes, nanowires, and nanorods) suspended in a base fluid (e.g., water, ethylene glycol, vegetable oil, PAO oil, transformer oil, naphthenic mineral oil, diathermic oil, paraffin oil, SAE oil, etc.) yield nanofluids, a state-of-the-art fluid technology that is gradually becoming a significant addition to established thermal-fluid engineering. Particle-based nanofluids have shown promise for use in a wide variety of areas. The use of conventional fluids with low heat conductivity (e.g., water, oil, ethylene glycol) as base fluids requires either high velocities of moving fluids or complex geometrical designs in the surfaces used to transfer heat to these fluids, thus leading to higher manufacturing and operating costs. However, water and air remain the most important fluids in engineering applications. A way to overcome this problem of higher costs may be through nanofluids. The application of nanofluids is a novel technique that increases the heat transfer rates of conventional fluids with the addition of nanoscale ( N u0 . The question here is whether we have increased the heat transfer coefficient. Certainly not because h is the same. But enhancements in heat transfer are constantly being estimated by certain researchers using this argument. The fact that N u1 is greater than N u 0 does not imply an increase in h. Suffices to say that the discovery of nanofluids has, in many ways, reexamined our understanding of heat transfer. Nonetheless, nanofluids have continued to be examined intensively as evidenced in the periodical literature.

12.2 Current understanding of nanofluids Research on the physics and understanding of nanofluids continues as a multidisciplinary undertaking. Nanofluids may very well become the next generation heat transfer medium. The growing repository of knowledge on this advanced and smart fluid makes it perhaps the most fascinating subject for well-established and emerging scientists and engineers. For some perspectives, a nanometer is a billionth of a meter, or approximately 1/80,000th of the diameter of a human hair, and ten times the diameter of a hydrogen atom. A nanofluid is considered a two-component (or solid–liquid) mixture that is incompressible, without no chemical reaction, viscously dissipative, negligible heat transfer and, with thermal equilibrium between solid nanoparticles and the base fluid without slip occurring between them [26,27]. In nanofluids, the velocity of the particles is approximately equal to the velocity of the base fluid. Nanofluids

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are steady and unchanging suspensions of nanomaterials (nanoparticles, nanorods, nanotubes, nanowires, nanofibers, nanosheets, other forms of nanocomposites, nanodroplets, or nanobubbles) in regular fluids. Furthermore, nanofluids are obtained from various materials such as metals (Cu, Ni, Al, Ag, and Au), oxide ceramics (Al2 O3 , CuO, TiO2 , CuO, SiO2 , Fe2 O3 , Fe3 O4 , and BaTiO3 ), nitride ceramics (AIN, and SiN), carbide ceramics (SiC and TiC), carbon nanotubes, CaCO3 , graphene and composite materials, such as alloyed nanoparticles (Al70 Cu30 ) and nanoparticle core-polymer shell composites [28,29]. These materials are sometimes referred to as nanoparticles or quantum dots. Nanofluids are uniformly engineered, and stably suspended colloids, or nanoscale particles (typically 2–100 nm) in a base fluid (e.g., water) [26]. They are a passive means of enhancing heat transfer. Colloidal dispersions constitute a dispersed phase distributed in a dispersing medium. Nanofluids are thus nanocolloids with dilute suspensions of nanoscale particles. These particles that comprise a nanofluid are also smaller than the wavelength of light and are thus optically transparent when no aggregation is present. The optical properties of nanofluids may be advantageous for applications such as solar energy by significantly increasing efficiency. If particles are small enough (> kf and ϕ 300 W/cm2 [264], necessitating the consideration of liquid cooling. Conventional liquid coolants are being enhanced with nanoparticles to meet the cooling requirements of high power electronic systems. Thus, nanofluids represent an enhanced dimension to cooling techniques for electronics. Furthermore, chips have become much denser due to their increased number of functions. Therefore, recent electronic designs are more tightly packed, leading to reduced cooling ability. Several new electronics have heat management problems due to their high rate of heat production and at the relatively inadequate surface area for cooling. Nanofluids have higher heat transfer capabilities due to their higher convective heat transfer coefficients compared to conventional industrial fluids. Recent studies have demonstrated that nanofluids can raise the heat transfer coefficient of coolant by increasing the coolant’s thermal conductivity. This was proved using the cooler developed by [138] which was fitted with a microchannel heat sink together with nanofluids. For everyday electronics product used, such as the personal computer, the importance of cooling is more pronounced due to the higher heat production of the central processing unit (CPU). One recommended solution to the problem of cooling is the application of heat pipes. When nanofluids were used as the working substances for normal heat pipes, it was discovered that there was a higher cooling efficiency compared to when water was used. Specifically, with an equal volume of each substance, the heat resistance of the heat pipe reduced when nanofluids with suspended gold nanoparticles were used instead of water. These suspended nanoparticles appear to have an affinity to the vapor bubble when it is formed. Thus, it can be deduced that the size of the produced vapor bubble is much smaller for nanofluids than for normal liquids. This could be the main reason for the lower thermal resistance of the heat pipe [213]. The superthermal conductivity of the liquid–metal fluid can be further improved if certain conductive nanomaterials are included [213]. Liquid metals with low boiling points are considered a perfect base fluid for the production of ultra-efficient conductive liquids that can be used as coolants in many heat transfer application areas. Researchers at Intel Corporation have experimented with possible uses of nanofluids for cooling high-heat flux supercomputers among other potential

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electronic cooling applications [206]. Other applications of nanofluids high heat flux cooling systems are discussed in [212].

12.7.4 Applications of nanofluids in medicine In medicine and medical applications, researchers at MIT have developed nanoparticles that could treat brain cancer [269]. Nanofibers are also being developed towards bioengineering of the heart ventricle [292]. Mass transfer applications of nanofluids in medicine also include molecular drug delivery using carbon nanotubes [270]. Magnetic nanofluids discussed earlier also find use in medical imaging, cancer therapy, and cell sorting. There are also developments for nanoparticles with antibacterial properties. Magnetic nanoparticles having the same size have now been developed for breast cancer clinical trials. Being able to produce particles of the same size is important, and they function by sticking to breast cancer cells allowing them to detect and remove the smallest of metastases [293,294]. Also, small molecules called microRNA have been linked to the onset of blood-borne cancers, whereby it has been challenging to detect them in a patient’s blood. In this regard, researchers in Australia developed gold-plated nanoparticles that function as dispersible electrodes, which were modified with DNA, furnishing them with the electrochemical signal that aligns with the microRNA they intend to detect in blood [295]. Nanoparticle drug delivery, especially nano-barcoding system, is being developed to barcode nanoparticles to optimize drug-delivery [296]. Researchers recently developed mRNA nanoparticles or a nanoparticle-mediated delivery approach, which serve as super-small-scale delivery vehicles that can send messages of genetic information into cells. The researchers observed that there was significant suppression of tumor growth and progression of prostate cancer. This includes prostate cancer in bones, which is the most common site to which prostate cancer can spread [297]. People who have arthritis, especially rheumatoid arthritis, a disease that causes inflammation of joints and severe pain know too well that the medications available only treat the symptoms and doesn’t attack the disease at the molecular level. Recently, a team of bioengineering researchers at the University of California developed a nanosponge embedded draped in white blood cells that work to absorb the inflammatory proteins to treat the disease better and help patients manage the disease better [298]. Other biomedical applications of nanofluids are examined in [447–450].

12.7.5 Other notable areas of application of nanofluids Nanofluids that have been described as thermally smart fluids, used for applications that include mitigating hotspots in microelectromechanical systems are discussed in [211]. Researchers at Texas A&M University have examined the opportunities and applications of nanofluids for cooling, thermal energy storage, and sensing in microelectromechanical systems (MEMS) [182]. MEMS and fluid flows have been reviewed in [265]. MEMS find use in various industries such as aerospace, automotive, chemical, and medical. Nanoscale MEMS, otherwise known as NEMS, is another promising

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future direction for nanoparticles in several industries. As fuel delivery systems in the automotive industry, for example, nanoparticles can enhance the flow and performance of such systems. The transformers that wheel electrical energy would experience enhancements through better cooling [205]. Nanoparticles for three-dimensional printing of parts by metal inkjet was demonstrated at the Formnext show in Germany in November 2016 [184]. Nanoscale structures for water purification, otherwise known as Turing structures, named after famed mathematician Alan Turing, have also been examined in [266], showing enhanced water transport through polyamide membranes. Furthermore, several industrial activities, especially for industrial cooling, the use of nanofluids would produce large budget savings and a reduction in harmful chemicals released to the environment. Optical nanosensors or nanostructures, which are 50 times thinner than human hair have vast applications such as detecting and managing plant and animal diseases, the ripeness of fruits, and in noninvasive medical diagnosis [267]. The physics of nanofluids with applications to microchannels cooling and lubrication is considered in [431]. The way we cool and heat our buildings are also expected to be influenced significantly through the use of nanofluids [439]. Industrial manufacturing and production activities such as machining, welding, and drilling may also be influenced by nanofluids [440–442]. From the discussion earlier on graphite nanofluids, the various manufacturing processes involved in production of graphene fibers such as stretching, especially, hot stretching to approximately 3000 K (or 2726.85◦ C) together with the boundary condition of Newtonian heating or heating from below a surface constitutes an active area of study in fluid-thermal-mass boundary layer research [98]. Further, the evident boundary layer or edge effects in composite materials is integral to the production of high strength-to-weight components. In the lamination theory of composite materials, interlaminar stresses and their analysis is important to the structural integrity of several parts and components. These interlaminar stresses are understood to exhibit a boundary layer phenomenon restricted to the free edge and extending inward, a distance equal to the laminate thickness. Innerlaminar shear stresses are very high at this free edge. An important outcome of this boundary layer phenomenon of edge effects is in the production processes of the laminate stacking sequence, which invariably influences the magnitude and character of the interlaminar stresses [268]. Further, areas of application include ultrafast water transport to desalinate seawater, and nanofiltration, in heating, ventilation and air conditioning systems (HVAC), absorption chillers [437,438], nanorefrigerants [271] in refrigeration systems [435, 436] and vegetable oil-based nanorefrigerants, for boiler flue gas temperature reduction, pharmaceutical processes, organic Rankine cycle (ORC) units, and nuclear power plants. A fairly comprehensive assessment of areas in which this novel fluid can be used to optimize heat transfer has also been examined as well [223]. High-density heat flux systems such as microelectromechanical systems or MEMS and nanoelectromechanical systems or NEMS that require increased heat removal efficiency for stability and better performance may also be achieved using nanofluids. Before there can be a more penetrating practical application of nanofluids, some key issues have to be addressed. Towards industrial applications, and for each nanofluid type, well-defined correlations (experimental or theoretical) for thermal

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conductivity, viscosity, density, specific heat, and Nusselt number are necessary. The long-term sustainability and stability of nanofluids also need to be demonstrated.

12.8 Research gaps and outlook While research, in general, is a demanding undertaking, the field of nanotechnology and nanofluids is especially a demanding research area considering the numerous factors above. Ultimately, this review highlights that more research remains to be done on nanofluids. Nanofluids are a new type of heat transfer material that requires further research and may potentially replace conventional fluids in many applications due to their superior thermal performance. There continues to be a strong need for theoretical and simulation studies in nanofluids, overcoming or enhancing current understanding of nanofluids, and that may provide an improved basis for fully understanding the scatter observed in experimental results. In the ideal sense, scientists and researchers with the characteristic nature of the human mind being more amenable to the linear than the nonlinear may wish to establish a linear relationship that explains the underlying reasons for enhancements due to nanofluids. However, there is little that is simple about nanofluids, considering the different nanoparticle materials used to make them and a host of other intricacies. Nanofluids exhibit novel thermal transport phenomena that are not simple to characterize or understand [215]. There are certainly more transport mechanisms in nanofluids to discover that may cause their enhancements. Quantitative information from which to infer the dominant mechanisms for heat transfer enhancement under convective conditions remains elusive and constitutes a research gap. Nanofluids researchers understand that several fundamental questions remain unanswered. These include the most efficient nanoparticles and liquid combination and cost of production, the most acceptable and optimum knf values, optimal nanoparticle size for each nanoparticle type to yield highest thermal conductivity enhancement, and the simplest and most sophisticated models that thoroughly describe the particle properties and heat transfer variables of nanofluids. It is evident from the review of relevant literature that there is a need and perhaps even a race to establish a new robust theory of the efficient enhancement of the thermal conductivity and viscosity of nanofluids, which is a necessary undertaking. A robust explanation that may settle the discrepancies between experimental findings in nanofluids and classical theories for estimating the effective thermal conductivity of suspensions remains elusive and constitutes a significant research gap in our comprehensive understanding of nanofluids. While the knowledge and understanding of the physics of nanofluids continue to evolve, it may, however, be challenging to develop a singular versatile model that can predict all trends observed from the significant scatter observed in experimental works. In another regard, it is evident that standards must be set in terms of experimentation. What appears to be lacking in experimental nanofluid literature is perhaps standardization efforts in nanofluid preparation methods through to experimental techniques, and this may be the reason for the different results. Standardization and the

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teachings of the Toyota Way [217], for example, suggest that standardization not only aids in achieving consistent and even results every time, but it also aids in identifying the factors that are limiting the achievement of consistent results every time. From the first authors’ experiences, standardization would also drive innovation, reduce cost, and accelerate the uptake of technologies such as nanofluids. This equates to standardization in preparation, production, and indeed experimentation on nanofluids. This point has also been addressed in [218]. Various nanofluid types produced differently or sourced from different suppliers will naturally yield varying experimental results, and this is evident in the literature [220]. Such variations may lead to the development of novel and more efficient experimental methods and also to fully ascertain the accuracy of experimental results. A primary objective remains to align theoretical calculations with experimental data better to yield better than merely qualitative agreement. Many controversies remain, leading some researchers to query whether mathematics contributes to the rather high enhancements observed [37,124]. It is for this reason that notable researchers tend to call for a more comprehensive theory to fully explain the behavior of nanoparticle–fluid mixtures [220]. In another regard, the authors in [221] have also called for new experimental methods for characterizing and understanding nanofluids. Likewise, the authors of [39] concluded that more work must be done on the nature of heat transfer in nanofluids, citing the lack of a hybrid model that effectively characterizes the several influencing parameters of the thermal conductivity of nanofluids, and the lack of a reliable database and experimental data on properties of nanoparticles. With regards to the viscosity of nanofluids, a review of theoretical, empirical and numerical models was examined in [85], the results of which indicate that there is a lack of investigation into the stability of different nanofluids when viscosity is the objective. For example, assessment of nanofluid stability and sustainability in engineered devices, e.g., cooling systems remains to be fully demonstrated. Also, it has been expressed that systems using nanofluids will most likely require more maintenance than those using traditional coolants. While this point is debatable, it needs to be further demonstrated if the benefits of heat transfer enhancement outweigh the maintenance penalty and necessitate a robust study to elucidate the problem better. Environmental assessments of nanofluid disposal are also a necessary undertaking. In summary, many gaps exist in the nanofluid literature that present opportunities for both experimental and theoretical multi-faceted and multi-disciplinary research.

12.9 Conclusion We have presented above salient aspects of nanofluids and reviewed the current status of developments and potential applications of advanced fluids known as nanofluids. It has not been the intent of this paper to give a thorough treatment of nanofluids. Rather, the aim has been to introduce the subject, presenting research studies and challenges, and various practical applications of nanofluids. The reader is referred to the open literature for more penetrating discussions on the subject.

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Nanoscale sciences may be causing significant changes to technology, as stated earlier. Future research may already be moving beyond nanotechnology. Nanofluids are simply nanoparticles suspended in fluids that display many improved properties at low nanoparticle concentrations compared to conventional fluids. There can be little doubt in the potential applications of nanofluids in the industry towards energy conservation and sustainability. Nanofluids are being developed to drive product and process solutions that also use less material and energy and with enhanced thermalfluid-mass performance. It is observed from the various nanofluid types available, that they may all be application-specific. Better guidelines for selection or a selection criterion of particular colloids or nanoparticles for specific applications may be necessary. In this article, we established a common repertoire of knowledge to advance the science and physics of nanofluids. The attempt to answer the many unanswered questions on nanofluids continues to be an interesting task of modern scientists and engineers for the foreseeable future. Much of the research on nanofluids is aimed at developing a complete understanding of the physics such that they can be applied where heat transfer improvement would be most useful. Numerous areas remain to be conquered in nanofluids through experiment, theory, and computation. There has been a significant amount of research and innovation on the topic of nanofluids due to its promising heat transfer features. Several studies have demonstrated how efficient nanofluids are in heat transfer machines such as heat exchangers. There continues to be particular interest in nanoscale and nanofluid applied to energy systems. The promise that nanofluid flow in porous media which manifests in improving the performance of solar systems, for example, should be rigorously studied. By any measure, the uptake and utilization of nanofluids will lead to significant energy conservation and emissions reduction. Miniaturization and laminarization of devices, together with energy efficiency, constitute focal domains for the development of advanced heat transfer materials. Very controlled physical experiments (standardization), together with rigorous theoretical analysis, have an important role to play in characterizing nanofluids effectively. There continues to be a need for better scientific arguments in the debate about nanofluids. The most outstanding feature of the future could be a possible change in the general outline of several thermofluid systems for improvements in efficiency and reductions in emissions. A current positive development is that several topical areas of nanofluids continue to be intensely investigated by both academic and industry researchers (e.g., GM, and Intel). Research in nanofluids must naturally center around how to ensure the in-depth understanding that may lead to their uptake. Some emerging themes from this review are – The potential applications of nanofluids are vast, and their impact on cooling and efficiency may be significant. – Nanofluids may be application specific necessitating a classification based on their areas of applications. – Guidance on the optimal nanoparticle volume fraction may be necessary, according to the system of the application. – In one regard, more disciplined (standardized) and multidisciplinary experimental nanofluid research is needed. In another regard, well developed theoretical con-

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structs of thermophysical properties of nanofluids (e.g., thermal conductivity) may provide guidelines for better experiments and to gain more understanding. Establishing a more robust experimental research database on nanofluid properties may lead to the development of better understanding and theoretical constructs of nanofluids. Heat generation rates at the micro- and macro-levels – from electronic circuitry to automotive engines are increasing. Better cooling of this systems promotes efficient energy utilization and reduction of harmful emissions to the environment. Thermal conductivity and viscosity of nanofluids are two key parameters driving efficiencies towards several applications. Long-term sustainability studies of nanofluids is a necessary undertaking. Significant experimental and theoretical heat transfer enhancements have been observed for various nanofluid types.

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[459] K.A. Hamid, W.H. Azmi, M.F. Nabil, R. Mamat, K.V. Sharma, Experimental investigation of thermal conductivity and dynamic viscosity on nanoparticle mixture ratios of TiO2 –SiO2 nanofluids, Int. J. Heat Mass Transf. 116 (2018) 1143–1152. [460] M.F. Nabil, W.H. Azmi, K. Abdul Hamid, R. Mamat, F.Y. Hagos, An experimental study on the thermal conductivity and dynamic viscosity of TiO2 –SiO2 nanofluids in water: ethylene glycol mixture, Int. Commun. Heat Mass Transf. 86 (2017) 181–189. [461] M. Afrand, Experimental study on thermal conductivity of ethylene glycol containing hybrid nano-additives and development of a new correlation, Appl. Therm. Eng. 110 (2017) 1111–1119. [462] M. Hemmat Esfe, S. Saedodin, W.M. Yan, M. Afrand, N. Sina, Erratum to: study on thermal conductivity of water based nanofluids with hybrid suspensions of CNTs/Al2 O3 nanoparticles, J. Therm. Anal. Calorim. 125 (2016) 565. [463] N.N. Esfahani, D. Toghraie, M. Afrand, A new correlation for predicting the thermal conductivity of ZnO–Ag (50%–50%)/water hybrid nanofluid: an experimental study, Powder Technol. 323 (2018) 367–373. [464] S. Sarbolookzadeh Harandi, A. Karimipour, M. Afrand, M. Akbari, A. D’Orazio, An experimental study on thermal conductivity of f-MWCNTs–Fe3 O4 /EG hybrid nanofluid: effects of temperature and concentration, Int. Commun. Heat Mass Transf. 76 (2016) 171–177. [465] S.K. Mechiri, V. Vasu, A. Venu Gopal, Investigation of thermal conductivity and rheological properties of vegetable oil based hybrid nanofluids containing cu–zn hybrid nanoparticles, Exp. Heat Transf. 30 (2017) 205–217. [466] M. Hemmat Esfe, A.A. Abbasian Arani, M. Rezaie, W.M. Yan, A. Karimipour, Experimental determination of thermal conductivity and dynamic viscosity of Ag–MgO/water hybrid nanofluid, Int. Commun. Heat Mass Transf. 66 (2015) 189–195. [467] S. Akilu, A.T. Baheta, K.V. Sharma, Experimental measurements of thermal conductivity and viscosity of ethylene glycol-based hybrid nanofluid with TiO2 –CuO/C inclusions, J. Mol. Liq. 246 (2017) 396–405. [468] M.H. Esfe, M.H. Hajmohammad, Thermal conductivity and viscosity optimization of nanodiamond-CO3 O4 /EG (40:60) aqueous nanofluid using NSGA-II coupled with RSM, J. Mol. Liq. 238 (2017) 545–552. [469] M.H. Esfe, W.M. Yan, M. Akbari, A. Karimipour, M. Hassani, Experimental study on thermal conductivity of DWCNT–ZnO/water–EG nanofluids, Int. Commun. Heat Mass Transf. 68 (2015) 248–251. [470] M.H. Esfe, S. Esfandeh, M. Rejvani, Modeling of thermal conductivity of MWCNT-SiO2 (30:70%)/EG hybrid nanofluid, sensitivity analyzing and cost performance for industrial applications, J. Therm. Anal. Calorim. 2 (2017). [471] M.H. Esfe, M. Rejvani, R. Karimpour, A.A. Abbasian Arani, Estimation of thermal conductivity of ethylene glycol-based nanofluid with hybrid suspensions of SWCNT–Al2 O3 nanoparticles by correlation and ANN methods using experimental data, J. Therm. Anal. Calorim. 128 (2017) 1359–1371. [472] M.H. Esfe, P.M. Behbahani, A.A.A. Arani, M.R. Sarlak, Thermal conductivity enhancement of SiO2 –MWCNT (85:15%)–EG hybrid nanofluids: ANN designing, experimental investigation, cost performance and sensitivity analysis, J. Therm. Anal. Calorim. 128 (2017) 249–258. [473] M.H. Esfe, S. Esfandeh, S. Saedodin, H. Rostamian, Experimental evaluation, sensitivity analyzation and ANN modeling of thermal conductivity of ZnO–MWCNT/EG–water hybrid nanofluid for engineering applications, Appl. Therm. Eng. 125 (2017) 673–685.

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On a selection of the applications of thermodynamics

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L.M. Amoo University of California, Los Angeles, Los Angeles, CA, United States

13.1 Introduction Thermodynamics is the characterization of three key quantities – energy, work, and heat, and in the transformations that occur between these quantities. The subject serves as a bedrock for studies in fluids and heat transfer. The applications of thermodynamics are vast. In what follows are selected discussions of areas of application of thermodynamics and its principles. The discussions are simple, yet significant to our daily lives and critical to our understanding of the subject and ways to improving our environment through efficient energy utilization.

13.2 Internal combustion engines – Otto and Diesel cycles Sometime during the late 19th century, innovations in internal combustion engine (ICE) technologies were providing new opportunities for applying this new source of power in ways that could help move men and materials. By the fin de siècle, the internal combustion engine had become the most promising new technology for providing easier transportation as well as transportation modes such as air flight that would not be possible otherwise. The ICE continues to be a mainstay driving the modern economy. The ICE is still largely powered by fossil fuels. However, alternative energy forms to power the ICE have been examined in [1]. Such alternative forms of energy include electric vehicles or hybrid electric vehicles (evident in places such as US, Europe, etc.), and natural gas powered ICE vehicles (evident in places such as India, UAE, etc.). This also demonstrates that the dynamics of a nation or economy will dictate the appropriate alternative energy forms to transition towards, to power the ICE from the conventional use of gasoline that still largely dominates, that is, alternative forms that are economical, yet sustainable. In what follows, we discuss the ICE thermodynamically. The ideal air-standard Otto and Diesel cycles, respectively, represent the processes that occur in spark ignition and compression ignition engines. The most common spark-ignition engine fuel is gasoline, and the most common compression ignition engine fuel is diesel fuel. The air-standard ideal cycles include simplifications of the actual processes to make them tractable for a basic thermodynamic analysis. The main Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00021-9 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Figure 13.1 Ideal air-standard Otto cycle.

simplifications are: only air as the working fluid (no fuel vapor mixed with the air prior to combustion, and no products of combustion after), air is treated as an ideal gas, no combustion or exhaust processes (the entire cycles are analyzed as closed systems, and heat addition and heat rejection replace combustion and exhaust), ideal compression and expansion which means that the processes are assumed to be adiabatic and reversible, instantaneous heat addition for the Otto cycle, constant pressure heat addition for the Diesel cycle, and instantaneous heat rejection for both Otto and Diesel cycles. The most significant approximation of those listed is the assumption of ideal compression and expansion. This simplification can be handled more accurately with the use of isentropic compression and expansion efficiencies, which assumes that the processes are still adiabatic but no longer assumes that they are reversible. While each of these simplifications introduces some inaccuracy into the model compared to the processes of real internal combustion engines, the simplified, idealized cycles still accurately reflect general trends in the behavior of actual engines and provide a solid foundation for developing more accurate models. These advantages, along with the accessibility, transparency, and ease of use, make the ideal air-standard cycles very useful for engine modeling. The ideal air-standard Otto cycle consists of four processes: (1 to 2) isentropic compression, (2 to 3) constant volume heat addition, (3 to 4) isentropic expansion, (4 to 1) constant volume heat rejection. Fig. 13.1 shows the Otto cycle in P-v and T-s coordinates. These four processes represent, respectively, the compression stroke, combustion of the air–fuel mixture, the expansion stroke, and exhaust and intake processes. In the ideal air-standard Diesel cycle (Fig. 13.2), the instantaneous (constant volume) heat addition of the Otto cycle is replaced with constant pressure heat addition, while the other three processes are identical to those of the Otto cycle. The rationale for using constant volume heat addition in the Otto cycle is that the spark-ignited combustion process proceeds fast enough to be well-approximated by instantaneous heat addition. For the Diesel cycle, the injection of fuel is controlled by fuel injectors and occurs continuously as the piston descends from top dead center. A constant pressure heat addition process approximates the simultaneous expansion and combustion. The bottom dead center (BDC) and top dead center (TDC) in Figs. 13.1 and 13.2 imply the bottommost and topmost positions inside a cylinder whereby the motion from either TDC to BDC or BDC to TDC represents the completion of one stroke. For a closed system, the first law of thermodynamics can be written as a Qb

−a Wb = m ∗ (ub − ua ),

(13.1)

On a selection of the applications of thermodynamics

385

Figure 13.2 Ideal air-standard Diesel cycle.

where Q is the heat transfer, W is the work interaction, m is the mass, and u is the specific internal energy. Applying this to the first process of the Otto and Diesel cycles (isentropic compression) results in this expression for the specific work of compression   k−1 , (13.2) 1 w2 = cv ∗ T1 ∗ 1 − rv where cv is the constant volume specific heat, T1 is the temperature, k is the specific heat ratio, and rv is the compression ratio defined as rv = V1 /V2 where V1 and V2 are the cylinder volumes at bottom dead center and top dead center, respectively. The compression ratio is usually set by the geometric design and construction of the engine and is unchangeable for a particular engine. The first law applied to the constant volume heat addition step of the Otto cycle results in this expression for the specific heat transfer 2 q3

= cv ∗ (T3 − T2 ) .

(13.3)

Similarly, for the remaining two steps of the Otto cycle   −(k−1) , and 3 w4 = cv ∗ T3 ∗ 1 − rv 4 q1

= cv ∗ (T1 − T4 ) .

(13.4)

A cycle efficiency is calculated in terms of the heat input and net work η≡

wnet 1 w2 + _3w4 = . qin 2 q3

(13.5)

Using the expressions for heat and work given above, this can be rearranged to be η = 1 − rv−(k−1) .

(13.6)

This remarkable result shows that for the ideal air-standard Otto cycle, the cycle efficiency depends only on the compression ratio. Furthermore, the cycle efficiency rises monotonically with increasing compression ratio. While the values calculated for efficiency are higher for the ideal air-standard Otto cycle than can be achieved in real engines, the dependence of efficiency on compression ratio is identical. Thus, engine manufacturers try to use the highest compression ratio possible to maximize

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efficiency, but since a mixture of air and fuel vapor is compressed in a spark ignition engine, a compression ratio that is too high will result in spontaneous ignition of the fuel before the piston reaches the top of the stroke. This is called pre-ignition, or knock, and is very hard on the engine and must be avoided. This limits the compression ratio for spark ignition engines to around 10–12. For the Diesel cycle, the analysis of the first process (isentropic compression) is identical to that of the Otto cycle. For the second process (constant pressure heat addition), an additional parameter is defined. The cutoff ratio is defined as (13.7)

rc= V3 , V2

where V3 is the volume of the cylinder when fuel addition is “cut off.” With this parameter and the first law, the heat addition during the second process can be expressed as 2 q3

= cp ∗ T2 ∗ (rc − 1) .

(13.8)

The expansion work during this process is 2 w3

= R ∗ T2 ∗ (rc − 1) ,

(13.9)

where R is the gas constant for air. The heat rejection step is identical to that of the Otto cycle. The Diesel cycle efficiency, defined the same as that of the Otto cycle, can be written   rck − 1 η = 1 − rv−(k−1) . (13.10) (rc − 1) ∗ k Written in this form, the expression for Diesel cycle efficiency resembles the expression for Otto cycle efficiency, but with an extra term, dependent only on the cutoff ratio. The term in square brackets is always greater than unity, so for the same compression ratio, the Diesel cycle efficiency will always be less than the Otto cycle efficiency. Of course, no one would build a compression ignition engine with the same compression ratio as a spark-ignition engine. While the spark-ignition engine designer must be careful to avoid overcompression leading to knock, the compression–ignition engine relies on the high air temperature at the end of compression to ignite the fuel that is sprayed in by the fuel injectors. Compression ratios for compression ignition engines can be in the range of 18–24. Entropy generation, or the closely related concept of exergy destruction, occurs in all real processes according to the second law of thermodynamics. For the closed system analysis appropriate to the Otto and Diesel cycles, the second law can be expressed σgen = s2 − s1 −

q , Tbound

(13.11)

On a selection of the applications of thermodynamics

387

where s is the property entropy, q is the specific heat transfer, and Tbound is the temperature at which heat transfer occurs. According to the second law, the entropy generation, denoted σgen , is positive in all real processes, zero in ideal processes, and can never be negative. This equation can be applied to each process of the Otto and Diesel cycles as follows. For all of the isentropic processes (1–2 and 3–4 in both cycles), the entropy generation is zero. This is in keeping with the assumed ideal nature of the process. Combining the first and second laws for the constant volume heat transfer processes, the minimum entropy generation can be expressed as

T3 1 −1+ σgen = cv ∗ ln T2 T3 /T2

(13.12)

for the heat addition process and

T1 1 −1+ σgen = cv ∗ ln T4 T1 /T4

(13.13)

for the heat rejection process. In both cases, the entropy generation would be a minimum (zero) when the temperature change is zero (T3 = T2 or T1 = T4 ). Of course, that is just the degenerate case of no heat transfer at all. For all real cases with finite heat transfer, the entropy generation is positive for both heat addition and heat rejection, as we would expect. For the constant pressure heat addition step of the Diesel cycle, the entropy generation can be expressed as

1 . σgen = cp ∗ ln (rc ) − 1 + rc

(13.14)

We further emphasize that the entropy generation is positive for all realistic values of the cutoff ratio. The ICE is a time-tested machine. Though notoriously resistant to further improvements in efficiency, recent and significant technological improvements have been achieved by researchers showing power and fuel efficiency improvements greater than 10% [2,3]. In summary, the Otto cycle contains two adiabatic and two isochoric processes while the Diesel cycle involves two adiabatic, one isobaric, and one isometric process. Taken together, the ideal air-standard Otto and Diesel cycles provide a foundation for the analysis of internal combustion spark-ignition and compression–ignition engines. They provide a succinct and manageable thermodynamic representation of the real processes involved in these engines and accurately reflect general behavioral trends of the underlying physics. Simple formulas for efficiency, performance, and entropy generation can be derived from a simple thermodynamic analysis.

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13.3

Applications of Heat, Mass and Fluid Boundary Layers

Electrical power generation – ideal basic Rankine cycle

Some variation on the basic Rankine cycle is used for power generation in virtually all electricity generating steam power plants. This includes utility-scale coal-fired and nuclear power plants, as well as some natural gas-fired, biofuel, and concentrated solar energy plants. In the other predominant use of natural gas for electricity generation – gas turbine generators, – the gas turbine is often paired with a Rankine bottoming cycle which uses the exhaust heat from the gas turbine as the heat source for the Rankine cycle. The same cycle is sometimes implemented using a working fluid with a lower boiling point than water for utilization of lower temperature heat sources such as solar or geothermal energy. These implementations are commonly termed organic Rankine cycle. Organic Rankine cycle which is simply a Rankine cycle that uses a low-temperature boiling liquid as the working fluid rather than water. Many refrigerants, propane, butane, ammonia, and others can also be used. The organic Rankine cycle is the most common and commercially available cycle, especially suited to low-grade heat sources for producing work (electricity generation, shaft power, etc.). Others were based loosely on the Stirling engine cycle, especially the Ringbom Stirling for low-temperature differentials that have been proposed and demonstrated [4, 5]. The organic Rankine cycle finds use in maritime applications [6], solar thermal energy [7], for the conversion of low to medium grade heat sources into useful power [8,9], and the use of advanced working fluids, such as nanofluids [10]. The market penetration and potential of the organic Rankine cycle are examined in [11]. The ideal basic Rankine cycle consists of four processes. Low-temperature liquid water is raised to a high pressure with a pump. It is directed into a boiler which heats the water to the boiling temperature, boils it to vapor, and superheats it. The high pressure, high-temperature steam goes through an expansion – usually in a steam turbine – and drops in pressure and temperature while producing work. The low-pressure steam is then condensed back to a liquid, and the cycle begins again. The thermodynamic processes of the ideal basic Rankine cycle are shown in Fig. 13.3. The pump process (1–2) is assumed to start with saturated liquid water at the condenser pressure and proceed adiabatically and reversibly (isentropically) to the entrance of the boiler. The boiler (2–3) heats the water to the boiling point at the boiler pressure and continues heating it into superheated steam. This high-pressure steam is used to produce work (3–4), which is the whole purpose of the cycle. The low-pressure exhaust from the expansion is condensed (4–1) back to a saturated liquid. Each of these processes can be analyzed as an open system using the first law of thermodynamics ˙b aQ

−a W˙ b = m ˙ ∗ (hb − ha ) ,

(13.15)

˙ is the rate of heat transfer, W˙ is the power, m where Q ˙ is the mass flow rate, and h is the specific enthalpy. For the adiabatic pumping process, the liquid water is assumed to be incompressible, and the specific work can be expressed as 1 w2

= v ∗ (p1 − p2 ) ,

(13.16)

On a selection of the applications of thermodynamics

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Figure 13.3 T-s diagram for the ideal Rankine cycle.

where v is the specific volume, and p is the pressure. Using this with the first law fixes the state of the liquid entering the boiler as h2 = h1 + v ∗ (p2 − p1 ) .

(13.17)

The boiler process is assumed to take place at constant pressure, and the specific heat transfer can be expressed as 2 q3

= h3 − h2 .

(13.18)

For the ideal cycle, the expansion process is assumed to proceed adiabatically and reversibly (isentropically) and the specific work is 3 w4

= h3 − h4 .

(13.19)

The condensation of the water back to liquid is also assumed to take place at constant pressure, and the specific heat transfer for this process is 4 q1

= h1 − h4 .

(13.20)

The Rankine cycle efficiency is η=

wnet 1 w2 +3 w4 = . qin 2 q3

(13.21)

In practice, the pump work required for step (1–2) is usually a tiny fraction of the total cycle work and is sometimes neglected in calculating cycle efficiency. In the boiler (2–3) heat is added to the cycle from whatever primary heat source is being used. For coal-fired plants, the boiler consists of a large space—sometimes several stories tall— lined with water-carrying pipes. The coal is ground into a fine powder and injected with combustion air into a large tornado of flame at the center of the boiler space. For nuclear and high-temperature solar plants some intermediate heat transfer fluid is usually used to bring heat from the source (nuclear decay heat or concentrating mirrors) to boil the Rankine cycle process water in a heat exchanger. Heat transfer fluids commonly used in nuclear power plants include a separate high-pressure water stream or sometimes a liquid metal. Molten salts are often used as the heat transfer fluid for high-temperature solar installations. The maximum cycle temperature (state 3) is limited by material strength considerations when the heat source is nuclear decay or a

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Applications of Heat, Mass and Fluid Boundary Layers

combustion process. The steam expansion device (3–4) is usually a steam turbine in modern utility-scale power plants. The turbine is normally connected directly to an electric generator which in turn is connected to the power grid. The turbine speed is precisely controlled by plant operators to determine the amount of energy which feeds into the electrical grid at any given time. Liquid water droplets can be extremely damaging to the blades of a high-speed turbine, so the quality of the steam (state 4) must be maintained close to the saturated vapor line. Condensers usually use ambient air or water from the ocean, a lake, or a river to remove the heat from the process water. Air cooled condensers tend to have high capital costs because of the large surface area needed. Water cooled condensers can be more compact but require careful consideration of the environmental effects of returning warm water to a natural water body. Sometimes evaporative cooling is used to enhance the effectiveness (reduce the size) of an air-cooled condenser if a limited source of water is available. Even with a readily available source of water, there are two main reasons to condense and recycle the process water rather than simply discarding the steam at the outlet of the turbine and taking fresh water into the pump. First, the water used in a steam power plant must be exceptionally pure. The evaporation rate in the boiler is so high that even trace amounts of minerals or metals will cause unacceptably fast scaling of the boiler pipes. There is a large investment in purifying the process water from leach contaminants and purifying any makeup water needed. Thus, the process water is far too valuable to discard. Second, the temperature maintained in the condenser is in the neighborhood of ambient air or water temperatures for which the saturation pressure is a fraction of a psi. If the steam were discarded to the atmosphere, it could only be expanded to about 14 psi. The additional pressure decrease provided by maintaining relatively large vacuum pressure in the condenser translates directly into additional work per unit mass of circulating steam. In the ideal Rankine cycle, the pump (1–2) and the turbine (3–4) processes are isentropic, so there is no entropy generation associated with them. The entropy generation is a qb σgen = sb − sa − , (13.22) Tb where s is the property entropy, q is the heat transfer, and Tb is the boundary temperature; this occurs exclusively in the two heat transfer steps in the ideal cycle. In both cases, most of the heat transfer occurs at the saturation temperature of the respective boiler and condenser pressures. (This neglects the subcooled and superheated sections in the boiler and neglects potential slight superheating at the beginning of the condenser.) So, a reasonable approximation to the entropy generation of an ideal Rankine cycle is given by σgen = σgen,boiler + σgen,condenser     hfg −hfg = sfg − + −sfg − , Tb boilerpressure Tb cond.pressure

(13.23)

where Tb is the boundary temperature corresponding to the respective component heat transfer, and the subscript f g represents the difference between saturated vapor and

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saturated liquid for the respective property at the appropriate component pressure. Again, this approximation neglects the entropy generation due to superheating in the boiler, which may be significant. The ideal Rankine cycle provides a basic foundation to understand some processes in a steam power plant. Real power plants virtually always employ reheat and regeneration and are also subject to many real considerations like pressure drops through the heat exchanger pipes, water purification processes, water loss from leaks, noncondensable gas infiltration, nonisentropic expansion, and others. The reheating process consists of an expansion to some intermediate pressure, after which the water is reheated to the maximum temperature, then expanded again to lower pressure. Regeneration utilizes some high energy steam from the expansion side of the cycle to heat the subcooled liquid on the other side, saving the high-temperature heat for the boiling and superheat parts of the cycle. The ideal basic Rankine cycle provides a foundation for a more complete and detailed power plant analysis and incorporates most of the elements necessary for such an analysis. Even in the basic form, it provides insights into efficiency trends and controlling and limiting mechanisms for power plant operation and analysis.

13.4 Refrigeration systems – ideal vapor compression refrigeration cycle In terms of the total number of installed machines, by far the most common mechanical cooling system is the vapor compression refrigeration cycle. It is commonly used in residential, commercial, and automobile air conditioning systems, refrigerators and freezers, ice making, industrial process cooling, and many other applications. Most of the common refrigeration systems are based on the vapor compression cycle. Some advances in refrigeration cycle technologies have been examined in [12]. The ideal vapor compression cycle, shown in Fig. 13.4 on T-s coordinates, consists of four processes: (1–2) is an isentropic compression of refrigerant vapor, (2–3) is an isobaric cooling and condensation of the hot, high-pressure vapor to a saturated liquid, (3–4) is an isenthalpic expansion to a low-temperature liquid–vapor mixture, while (4–1) is an isobaric boiling of the mixed liquid and vapor back to saturated vapor. It is the last process, the boiling of the low-temperature refrigerant that is the purpose of the whole cycle—the heat input into the refrigerant corresponds to heat being taken away from the cooled space. A mechanical compressor accomplishes the compression step (1–2). Reciprocating, screw, and centrifugal compressors are all commonly used. A single heat exchanger usually accomplishes the cooling step (2–3) termed a condenser. It is common to reject the heat from the hot refrigerant to the ambient air, so many condensers have air-side fins to expedite the heat transfer. The expansion step (3–4) usually occurs through an expansion valve, capillary tube, orifice, or another throttling device. This adiabatic expansion results in a significant drop in temperature of the refrigerant. The cold refrigerant consisting of a mixture of liquid and vapor is passed through another

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Applications of Heat, Mass and Fluid Boundary Layers

heat exchanger, termed an evaporator, for the final step of the cycle (4–1). The evaporator removes heat from the cold conditioned space by boiling the cold refrigerant liquid into vapor. The saturated vapor leaving the evaporator is returned to the compressor to complete the cycle. Each step of the ideal vapor compression refrigeration cycle can be analyzed as an open system using the first law of thermodynamics: ˙b aQ

−a W˙ b = m ˙ ∗ (hb − ha ) ,

˙ is the rate of heat transfer, W˙ is the power, m where Q ˙ is the mass flow rate, and h is the specific enthalpy. Applying this to the first step (1–2), the specific input work for compression can be expressed as = h1 − h2 .

1 w2

(13.24)

For the ideal vapor compression cycle analysis, state 1 is taken as the saturated vapor at the evaporator pressure, and state 2 is calculated assuming an isentropic compression from state 1 to the condenser pressure. The specific heat transfer from the condenser can be expressed as: 2 q3

= h3 − h2 .

State 3 is assumed to be a saturated liquid at the condenser pressure. The expansion process is assumed to occur adiabatically while producing no work, which results in an isenthalpic process, h3 = h4 . Finally, the cooling effect of the cycle can be expressed as 4 q1

= h1 − h4 ,

or, for the total cooling capacity, ˙1 4Q

=m ˙ ∗ (h1 − h4 ) .

(13.25)

By far the largest use of electricity or other external work in a real refrigeration system occurs in the compressor. While many systems utilize fans to enhance the heat transfer processes at the condenser and evaporator, the energy used by such fans is normally negligible relative to the energy use of the compressor. While the ideal vapor compression refrigeration cycle assumes that the compression process is adiabatic and reversible (that is, isentropic) in real compressors, there are significant deviations from this assumption. First, no real process is reversible. If the process is still assumed to be adiabatic, it will proceed to the point 2a in Fig. 13.5. This compression process can be handled with an isentropic compressor efficiency defined as η=

ws h1 − h2s = , wa h1 − h2a

(13.26)

where ws is the work that would have been needed for an isentropic compression, and wa is the work required to arrive at point 2a. If the efficiency is known, this equation

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can be used to determine point 2a, or, if the compressor outlet state (2a) is known, the efficiency can be calculated. A second deviation from the ideal cycle is that most real compressors are cooled in some way, so the adiabatic assumption would not be accurate. Small compressors often have fins to reject heat to the atmosphere, while some very large compressors are built with a liquid cooling jacket. Cooling tends to move the compressor outlet toward the point 2b shown in Fig. 13.5. Of course, irreversibilities still tend to move the outlet state toward points 2s or 2a. The cooling effect has two benefits. Less compressor work is required to compress the refrigerant to the condenser pressure, and less cooling is needed (that would translate to a smaller condenser size) to condense the refrigerant to state 3. Other less significant deviations from the ideal cycle include pressure drops in the condenser and evaporator. No refrigerant would flow in a truly constant pressure process, and the greater the mass flow rate, the higher the pressure drop. These pressure drops are normally small relative to the pressure difference across the cycle and can be neglected without a large loss of accuracy. Real refrigeration machinery also usually includes mechanical provisions for making sure that no vapor enters the expansion device and that no liquid enters the compressor. In terms of the mass flow rate, a relatively small amount of vapor passing through the small flow area of the expansion device would essentially halt the flow momentarily which could cause transient pressure fluctuations and undue load on the compressor. For the compressor, any significant presence of essentially incompressible liquid phase could cause damage. Simple gravity-driven vertical separators are often employed for these purposes. This leads to another deviation from the ideal cycle in which it was assumed that saturated liquid enters the expansion device, and saturated vapor enters the compressor. With the mechanical separators ensuring that only liquid enters the expansion device and only vapor enters the compressor, it is common to have a few degrees of subcooling of the liquid entering the expansion device, and a few degrees of superheat for the vapor entering the compressor. Entropy generation is present in three of the four steps of the ideal vapor compression cycle. The isentropic compression process occurs without any entropy generation. For the steady-state, open system analysis for each of the remaining three steps, the specific entropy generation can be written as σgen = sb − sa −

a qb

Tb

,

(13.27)

where s is the specific entropy, q is the specific heat transfer, and Tb is the boundary temperature at which the heat transfer occurs. For the condenser, the sign on q is negative and the quantity −q/Tb is larger in magnitude than the quantity (s3 –s2 ) which is negative, resulting in a positive entropy generation in the condenser. For the evaporator, the sign on q is positive, but the quantity (s1 –s4 ) has a larger positive magnitude than the −q/Tb term, so again, the entropy generation is positive. For the adiabatic process through the expansion device, the specific entropy generation is (s4 –s3 ), which is always positive as can be seen clearly in Fig. 13.4. The thermodynamic analysis of the ideal vapor compression refrigeration cycle can be used directly for the initial design of refrigeration systems. Thermodynamic

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 13.4 T-s diagram for the ideal vapor compression refrigeration cycle.

Figure 13.5 Compression process.

˙ 1, optimization of refrigerators has been examined in [13]. If a refrigeration load, 4 Q and evaporator and condenser temperatures are specified, then the first law analysis of the cycle as presented earlier yields a required refrigerant mass flow rate. Refrigerant property tables provide evaporator and condenser pressures from the specified tem˙ 3 , and peratures. These can be used to calculate the condenser rejection heat rate, 2 Q the power required for the compressor, 1 W˙ 4 . The required power and mass flow rate enables the selection of a compressor. In summary, it is the specified refrigeration load and the calculated condenser heat transfer rate that enables the sizing and selection of heat exchangers.

13.5

Gas turbine systems – ideal air-standard Brayton cycle

Modern gas turbine engines were developed extensively through the first half of the 20th century and today are used in a wide variety of applications. Since they were first applied in the 1930s, they have revolutionized the electrical power generation industry [14]. In the area of propulsion, they are used for most military and commercial aircraft, and are used widely in marine propulsion and some large ground vehicle applications. They are also used extensively for electricity generation in utility-scale and smaller power plants. The recent availability of low-cost natural gas for fuel and attributes of comparatively fast and easy start-up and shut-down (relative to a large Rankine cycle facility) have boosted the number of gas turbine machines used for meeting both base load and demand spikes in the electrical generation industry. A common configuration uses a gas turbine engine connected to a generator with the exhaust heat from the tur-

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bine used as the primary heat source for a Rankine cycle with a second generator. The overall efficiency of these combined cycle plants can be very high. In fact, combined cycle power systems are the most efficient heat engines and are expected to displace their single cycle counterparts over the coming years. This is due to the type of thermal efficiencies that have been observed. For example, the Mitsubishi Hitachi power system, M501JAC, a 57 MW combined cycle unit, demonstrated a thermal efficiency of 64%. Similarly, the GE-9HA.02, an 826 MW combined cycle engine, also achieved a thermal efficiency of over 64%. Siemens, another engine manufacturer, forecasts that their H-L class gas turbine will have efficiencies of almost 65%. Furthermore, based on data collated from Forecast International (FI), a market firm based in Newtown, Connecticut, USA, on the global gas turbine market including both aviation (commercial and military), and nonaviation (electrical power, mechanical and marine), production in 2017 was valued at 84.3 billion USD, up from 77.1 billion USD in 2016. It is predicted that by 2032, this value would reach 100 billion USD, a 19% growth over 15 years. This undoubtedly continues to demonstrate the significance of gas turbine systems [14]. A gas turbine engine operates by compressing air to high pressure, combusting fuel in the air stream which increases pressure and temperature, then expanding the highpressure exhaust gases through a turbine to produce work. Some or all of the work produced by the turbine is used to power the compressor. These steps are common to all gas turbine engines, but the rest of the configuration depends on the engine’s design and purpose. In turboprop engines, helicopter engines, and engines used for marine and ground transportation and electricity generation, the turbine extracts all of the work that is practical to get from the hot gas stream. Some of that work is used to power the compressor, and the rest is directed through a gearbox which in turn is connected to the end use: a propeller for aircraft and marine propulsion, a rotor for helicopters, a generator for electricity generation. Turbofan engines which are used for propulsion in most large commercial aircraft operate similarly, but the work beyond that required for the compressor is used to power a ducted fan [15]. In turbojet engines, which are widely used in high-performance military aircraft, the engine is designed so that the turbine only extracts enough work from the gas stream to power the compressor. The high enthalpy gas stream exiting the turbine is directed to a nozzle and the resultant thrust from the high-speed gas stream is used to propel the aircraft directly. The ideal air-standard Brayton cycle is used as a basic model representing the processes in a gas turbine engine. The cycle is shown on T-s coordinates in Fig. 13.6. The ideal air-standard Brayton cycle consists of four steps. It is assumed that only air is used for the working fluid throughout the cycle. The first step represents the compressor process (1–2) by an adiabatic reversible (isentropic) compression. The second step replaces the addition of fuel and combustion with a constant pressure heat addition (2–3). The expansion through the turbine is represented by an adiabatic and reversible expansion (3–4). In a real engine, the exhaust gases at the exit of the turbine are directed to the exhaust or nozzle outlet. In the Brayton cycle, the exhaust and intake processes are replaced with a constant pressure heat rejection process (4–1).

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Applications of Heat, Mass and Fluid Boundary Layers

Each step of the Brayton cycle can be analyzed as an open system using the first law of thermodynamics ˙b aQ

−a W˙ b = m ˙ ∗ (hb − ha ) ,

˙ is the rate of heat transfer, W˙ is the power, m where Q ˙ is the mass flow rate, and h is the specific enthalpy and subscripts a and b refer to initial and final states, respectively. For the compression process (1–2), using the assumption of constant specific heat, the specific work can be expressed as

  k−1 k , (13.28) 1 w2 = cp ∗ T1 ∗ 1 − rp where cp is the constant pressure specific heat, T is the temperature, k is the specific heat ratio, and rp is the pressure ratio defined as p2 /p1 . The specific heat addition during the constant pressure heat addition process (2–3) is 2 q3

= cp (T3 − T2 ) .

(13.29)

The expansion work (3–4) is

 − k−1 k , 3 w4 = cp ∗ T3 ∗ 1 − rp

(13.30)

and the heat rejection in the final step of the cycle (4–1) is 4 q1

= cp (T1 − T4 ) .

(13.31)

The overall cycle efficiency can be expressed as η=

 − k−1 wnet k . = 1 − rp qin

(13.32)

Similar to the case of the ideal air-standard Otto cycle discussed in Sect. 13.2, the cycle efficiency of the ideal air-standard Brayton cycle depends only on the pressure ratio for a given gas and increases monotonically with increasing pressure ratio. From a practical standpoint, the maximum temperature that an engine can reach is limited by material properties in the first row of turbine blades (essentially at point 3 in Fig. 13.6). Since the maximum temperature is limited, raising the pressure ratio too high results in a small net work, even though the cycle efficiency is high. Conversely, a very lowpressure ratio also has a small net work in addition to low efficiency. This is illustrated in Fig. 13.2. On T-s coordinates, the area enclosed by a cycle is proportional to the net work, and it is apparent in Fig. 13.7 that both high and low-pressure ratios yield less specific net work than an intermediate value. For given inlet (T1 ) and maximum (T3 ) temperatures, an expression can be derived for the pressure ratio which results in maximum net specific work   rp max network =



T3 T1

k 2∗(k−1)

.

(13.33)

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While efficiency is always important, for aircraft propulsion applications operating near the pressure ratio for maximum net specific work, it is also very important to maximize the power to weight ratio for the engine. For applications where the power to weight ratio is not as large a concern, as opposed to aircraft propulsion where it is the largest concern, real gas turbine engines regularly employ regeneration, and intercooling and reheat. In regeneration, a heat exchanger, termed a regenerator, is used to capture heat from the exhaust stream and add it to the air stream after compression and before the heat addition step. This reduces the amount of supply heat (fuel) required and improves the efficiency at the cost (including capital cost, maintenance, added weight, and pressure drop) of the additional heat exchanger. Intercooling and reheat are employed in multistage machines to generate more work with the same peak temperatures and pressures. In a two-stage device, the process air would be compressed part of the way to the maximum pressure, then cooled at constant pressure, and then compressed the rest of the way. This is termed intercooling. For reheat, starting at the peak temperature, the air would be expanded part-way from the to the low pressure, then heated again (more fuel) at constant pressure, and finally expanded the rest of the way. These processes are illustrated in Fig. 13.8. The cost lies with additional equipment: compressors, turbines, and heat exchangers. Intercooling and reheat alone would tend to lower overall cycle efficiency, so they are never used unless regeneration is also used. For the ideal air-standard Brayton cycle, the compression and expansion steps are assumed to be adiabatic and reversible (isentropic) and so they have no associated entropy generation. The second law of thermodynamics can be expressed as σgen = sb − sa −

a qb

Tb

,

(13.34)

where s is the property entropy, q is the specific heat transfer, and Tb is the boundary temperature at which heat transfer occurs. For the constant pressure heat addition (2–3) and heat rejection (4–1), the second law of thermodynamics can be applied to yield these expressions for the specific entropy generation   T3 T 3 − T2 σgen = cp ∗ ln − (13.35) T2 Tb for heat addition, and   T1 T 1 − T4 − σgen = cp ∗ ln T4 Tb

(13.36)

for heat rejection. In one regard, entropy generation analysis functions as a design tool in optimizing for efficiency and work in engineering systems [16], especially in gas turbine systems. However, in another regard, it has been suggested that thermodynamic optimization is to be based on maximum thermal efficiency or maximum work output rather than an entropy-based design approach [17]. Invariably, there are special circumstances or cases where minimum entropy production leads to maximum work production.

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Figure 13.6 Ideal air-standard Brayton cycle.

Figure 13.7 Maximum net work.

Figure 13.8 Intercooling and reheat.

The ideal air-standard Brayton cycle provides a simple and succinct representation for the main processes in a gas turbine engine. It faithfully reflects the same trends for efficiency, properties, and output found in actual engines and provides a good starting point for more detailed analyses. The simple thermodynamic analysis presented here allows for an understanding of the overall cycle and can be used for the initial design. Further refinement can be added as needed, including nonideal compression and expansion, variable specific heats, actual combustion processes rather than simple heat addition, heat losses, flow friction, and more.

13.6

Desiccant and subcooling dehumidification

Controlling the humidity in a conditioned space can be important for a wide variety of reasons. Traditional concerns include moisture damage during marine transport, hu-

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399

Figure 13.9 Dehumidification processes.

midity damage for moisture-sensitive artifacts, and process improvement for certain manufacturing concerns such as confectioneries, electronics, and pharmaceuticals, product protection from degradation for some foods, grains, and seeds, and protection from corrosion. Human comfort is important in potential high humidity venues such as refrigerated warehouses, ice rinks, hotels, supermarkets, hospitals, and retail establishments. There is also interest in maintaining moisture control in lower humidity environs such as offices and residential buildings. In traditional cooling systems, dehumidification is achieved by simply cooling the moist air stream below its dewpoint so that liquid water condenses out of the air. This process is familiar to anyone who has seen moisture condense on a glass of ice water on a humid day. The approximate process is illustrated on a psychrometric chart in Fig. 13.9 as processes 1–2b–3. Initially, the dry bulb temperature of the moist air decreases, while the moisture content remains constant. The dry bulb temperature continues to decrease as moisture begins to condense out of the air onto the cooling coil, resulting in simultaneous cooling and condensation and a decrease in the moisture level of the air. To deliver air at the desired condition (state 3), some form of reheating must be used. Desiccant dehumidification systems remove moisture from the air by collecting the water vapor directly onto a desiccant material. In a desiccant system, the dry desiccant material collects water vapor directly from the air, until it becomes saturated. The desiccant is moved to another air stream, and water vapor is expelled by raising the temperature (this step is called “regeneration”), after which, it is ready to be brought back to absorb more water vapor. The entire process involves only water vapor – no liquid is ever condensed. In the review on liquid desiccant air conditioning systems, Liu et al. [18] suggest that the liquid desiccant, instead of the air, must be cooled (or heated) for better dehumidification performance. In another review by Sultan et al. [19], the researchers indicate that dehumidification air conditioning systems can be operated by solar thermal energy for moderate humid climates. Specific to solar thermal systems, the work of Guo et al. [20] examined dehumidification techniques that could be applied to improve solar thermal cooling. Also, heat and mass transfer processes for dehumidification and cooling with useful mathematical models developed have been discussed in [21]. The effect of key parameters was also examined, show-

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Applications of Heat, Mass and Fluid Boundary Layers

ing improved efficiencies. Perhaps the most definitive source for dehumidification and heating, ventilation, and air conditioning systems is the four-volume ASHRAE Handbook [22]. Another useful text in this regard, which has been used by the author in undergraduate and postgraduate studies is in [23]. The approximate path of the process air through a desiccant dehumidification device is shown in Fig. 13.9 for the same inlet and final conditions as were shown for the subcooling system. This time, the process follows 1–2a–3. Note that the desiccant dehumidification (1–2a) process significantly increases the dry bulb temperature of the process air. The path from point 1 to point 2a is close to a line of constant enthalpy. After the dehumidification process, the process air must undergo a sensible cooling process to reach the final desired condition. Each step of these dehumidification processes can be analyzed as an open system using the first law of thermodynamics   ˙ b −a W˙ b = ˙ ∗ h) − ˙ ∗ h) , (13.37) (m (m aQ outlets

inlets

˙ is the rate of heat transfer, W˙ is the power, m where Q ˙ is the mass flow rate, and h is the specific enthalpy. The law of conservation of mass is also used   m ˙= m. ˙ (13.38) inlets

outlets

For the sensible heating and cooling processes shown in Fig. 13.9, the first law reduces to ˙b aQ

=m ˙ ∗ (hb − ha ) ,

(13.39)

˙ is the heat removal or addition rate, m where Q ˙ is the mass flow rate of dry air, and h is the specific enthalpy per unit mass of dry air. For the desiccant dehumidification process, the first law becomes: hin = hout where h is the specific enthalpy per unit mass of dry air. For the cooling and condensation process, the first law in conjunction with the conservation of mass yields ˙b aQ

=m ˙ ∗ [hb − ha + (ωa − ωb ) ∗ hw ] ,

(13.40)

where ω is the humidity ratio, and hw is the specific enthalpy of the liquid condensate produced. The entropy generation associated with each process can be derived from the second law of thermodynamics which can be expressed as σgen = sb − sa −

a qb Tbound

(13.41)

where s is the property entropy, q is the specific heat transfer, and T bound is the boundary temperature at which heat transfer occurs. For the sensible heating and cooling processes, the specific entropy generation can be expressed as Tb a qb σgen = cp ∗ ln (13.42) − Ta Tbound

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where cp is the constant pressure specific heat of the moist air expressed on a per-unitmass of dry air basis, and the entropy generation is also on a per-unit-mass of dry air basis. For the isenthalpic desiccant dehumidification process, this expression reduces to Tb , (13.43) σgen = cp ∗ ln Ta and for the cooling and condensation process, the entropy generation can be approximated as Tb a qb − σgen = cp ∗ ln + (ωa − ωb ) ∗ sw , (13.44) Ta Tbound where sw is the specific entropy of the liquid condensate. Most commercial desiccant dehumidification systems use as their working material either a solid adsorbent or a liquid absorbent. Briefly, absorption is a process in which the nature of the absorbent is changed, either physically, chemically, or both. The change may include the formation of a hydrate or phase change. An adsorbent, on the other hand, does not change physically or chemically during the sorption process. Silica gels and zeolites are often used in commercial desiccant equipment. Other solid desiccant materials include activated alumina’s and activated bauxites. Solid desiccant materials are arranged in a variety of ways in desiccant dehumidification systems. A large desiccant surface area in contact with the air stream is desirable, and a way to bring regeneration air to the desiccant material is necessary. The most common configuration for commercial space conditioning is the desiccant wheel. The desiccant wheel rotates continuously between the process and regeneration air streams. The wheel is constructed by placing a thin layer of desiccant material on a plastic or metal support structure. The support structure, or core, is formed so that the wheel consists of many small parallel channels coated with desiccant. Both “corrugated” and hexagonal channel shapes are currently in use. The channels are small enough to ensure laminar flow through the wheel. A sliding seal must be used on the face of the wheel to separate the two streams. Typical rotation speeds are between 6 and 20 rotations per hour. Typical wheel diameters vary from one foot to over 12 feet. Air filters are an important component of solid desiccant systems. Dust or other contaminants can interfere with the adsorption of water vapor and quickly degrade the system performance. All commercial systems include filters and maintenance directions for keeping the filters functioning properly. Some materials that function as liquid absorbents are ethylene glycols, sulfuric acid, and solutions of the halogen group such as lithium chloride, calcium chloride, and lithium bromide. A generic configuration for a liquid desiccant system is: The process air is exposed to a concentrated desiccant solution in an absorber, usually by spraying the solution through the air stream. As the solution absorbs water from the air stream, the concentration drops, and the weak solution is taken to a regenerator where heat is used to drive off the water (which is carried away by a regeneration air stream), and the concentrated solution is returned to the absorber.

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Applications of Heat, Mass and Fluid Boundary Layers

For solid or liquid systems, regeneration energy might be drawn from a variety of sources. Certainly, waste plant or process heat is the most economical if it can be used. Desiccant systems employing solar heat for regeneration have been proposed and tested. Hybrid systems have been designed which use the condenser heat from vapor compression systems or the engine jacket heat from gas engine driven systems as the source of regeneration energy. Of course, the long-term economics, including capital costs, maintenance costs, and operating costs or savings, must be considered when selecting a source of regeneration energy. Desiccant dehumidification supports the overall effort toward energy conservation by introducing flexibility in energy sources and potentially improving efficiency in some applications. Care must, however, be taken in their application recognizing that it does not always reduce total energy use— sometimes it could, but other times it might use more energy but from more convenient or appropriate sources. Both traditional subcooling and desiccant dehumidification are widely used in the HVAC industry and have different (sometimes complementary) strengths and weaknesses. Understanding the two processes enables appropriate choices for design, application, and analysis.

13.7

Evaporative cooling

In a traditional vapor compression refrigeration system, the bulk of the required external work is used by the compressor. In the right conditions, an evaporative cooling system can achieve a significant temperature depression with no compressor. The main strength of evaporative cooling systems is their very low operating costs relative to refrigerated air systems. Also, they don’t require fin and tube heat exchangers or special refrigerants, so they tend to have low capital costs compared to vapor compression refrigeration systems. The main weaknesses of evaporative cooling systems are that they only work well in very dry environments, and they can add significant humidity to the conditioned space. At first blush, one may think evaporative cooling is similar to natural convective cooling. However, not quite, although there are situations where both are present. Natural convection refers to situations where a buoyancy difference resulting from a temperature difference in a fluid causes movement of the fluid. Evaporative cooling refers to the cooling effect associated with the evaporation of a liquid. There are certain situations where the evaporative cooling effect causes a temperature (and associated buoyancy) difference. This combination of effects is used in some natural draft cooling towers. However, they are not necessarily related. For example, most evaporative coolers associated with space conditioning use forced movement of the air (by a fan or blower) and do not rely on any natural convection driven movement at all. Evaporative coolers work by exposing the process air to evaporating liquid water. This is usually done by blowing the air over a wetted media or through a water spray or mist to maximize the evaporation area. The water evaporation process drops the temperature of the combined (air and evaporated water vapor) moist air stream and

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the cooler, moister air is delivered to the conditioned space or used for other purposes. The process can be analyzed by using the first law of thermodynamics for an open system   ˙ 2 −1 W˙ 2 = ˙ ∗ h) − ˙ ∗ h) , (m (m 1Q outlets

inlets

˙ is the rate of heat transfer, W˙ is the power, m where Q ˙ is the mass flow rate, and h is the specific enthalpy. The law of conservation of mass is also used   m ˙= m. ˙ inlets

outlets

Combining the first law with the conservation of mass and applying them to the evaporative cooling process yields h2 − h1 = (ω2 − ω1 ) ∗ hw ,

(13.45)

where ω is the humidity ratio, and hw is the specific enthalpy of the liquid water that is used. This process is illustrated on a psychrometric chart in Fig. 13.10. Starting at point 1, the process proceeds nearly along a line of constant wet bulb temperature to a final condition which is usually around 85–95% relative humidity in a well-designed system. The sensible temperature depression achieved by this process is also illustrated in Fig. 13.10. Note that the moisture content of the air at point 2 is considerably higher than point 1 as seen in both the relative humidity and the humidity ratio. One description of the process is that “dryness” is being traded for “coolness.” In the desert Southwest and mountain West of the United States, and some other parts of the world, dry, hot ambient conditions often provide conditions favorable to the use of evaporative cooling for space conditioning. With the ambient air already excessively dry, the addition of some moisture can be welcome along with the cooling effect. In coastal areas and much of the Midwest, Eastern and Southern United States, for example, higher humidity levels make evaporative cooling much less attractive. Adding more moisture to the already damp air is usually unwelcome, and the cooling effect is significantly reduced. This is illustrated in Fig. 13.10 by process 1a–2a. Point 1a is at the same temperature as point 1, but with a higher humidity ratio. Proceeding again along a line of the constant wet bulb to around 85–95% relative humidity results in a much smaller temperature depression. Equipment configurations for evaporative coolers vary widely, but often a small pump is used to circulate water from a reservoir through sprayers, misters, or dripping onto a wetted media. The damp conditions and standing water in the reservoir sometimes lead to a distinctive “damp” smell in the process air giving rise to the colloquial name of “swamp coolers.” Because of their predominant use in hot and dry regions for space conditioning, they are also sometimes termed “desert coolers.” In addition to space conditioning in appropriate climates, evaporative cooling is employed for industrial applications in cooling towers and cooling equipment or processes. They can also be used to create hybrid systems to improve the efficiency of

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Applications of Heat, Mass and Fluid Boundary Layers

vapor compression systems by either providing cooler air to the condenser or precooling the process air through a nonmixing heat exchanger. While still subject to the limitations on the magnitude of the available temperature depression, such applications are not limited by occupant comfort considerations associated with very humid air. The entropy generation associated with the evaporative cooling process can be derived from the second law of thermodynamics which can be expressed as σgen = s2 − s1 −

1 q2

Tbound

,

(13.46)

where s is the property entropy, q is the specific heat transfer, and T bound is the boundary temperature at which heat transfer occurs. For the evaporative cooling process, this becomes T2 σgen = cp ∗ ln (13.47) + (ω2 − ω1 ) ∗ sw , T1 where sw is the specific entropy of the liquid water, ω is the humidity ratio, and cp is the constant pressure heat capacity of the moist air. Fig. 13.11 demonstrates the maximum possible (i.e., the theoretical limit if the air could be brought to 100% relative humidity) temperature depression as a function of the initial temperature and initial relative humidity. At the right side of the figure (high initial relative humidity), there is very little temperature depression available, regardless of the initial temperature. Conversely, on the left side of the figure (low initial relative humidity), there is substantial potential for temperature depression with even the relatively modest initial temperature of 70°F, 10% relative humidity yielding 23°F temperature depression. That potential increases rapidly with increasing initial temperature reaching 37°F for air that is initially at 100°F, 10% relative humidity. Of course, it isn’t realistic to assume that the air could be brought to complete saturation (100% relative humidity) in practical operating equipment. The following equation: T85% = −7.353◦ F + 0.4476 ∗ T + 0.1404 ∗ ϕ − 0.006256 ∗ T ∗ ϕ

(13.48)

gives the temperature depression assuming a conservative 85% relative humidity of the exit air. The temperature depression at the exit is given as a function of inlet temperature (T , in °F) and inlet relative humidity (ϕ, in %). This simple empirical equation is applicable over approximately the same range of inlet conditions shown in Fig. 13.11. In addition to the significant advantages of severely reduced operating costs and reduced capital costs mentioned previously, evaporative cooling systems have several other advantages over vapor compression refrigeration systems. Normally, a welldesigned evaporative cooling system for space conditioning would have a much higher flow-through of ventilation air than a corresponding refrigerated air system. This tends to provide a more pleasant atmosphere in the conditioned space. Evaporative cooling systems use only water, which is much cheaper, more readily available, and less

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Figure 13.10 Evaporative cooling.

Figure 13.11 Maximum possible temperature depression.

toxic than most refrigerants. Finally, routine maintenance of evaporative systems is well within the skillset of most building maintenance workers and many residential homeowners, whereas maintenance of vapor compression systems normally requires a trained and certified technician. The disadvantages of evaporative cooling systems include the performance issues and climate limitations discussed previously. The higher conditioned space humidity levels can increase corrosion rates and encourage the growth of mold and bacteria. High moisture levels in both the conditioned space and inside the evaporative cooling equipment can, in poorly maintained equipment, lead to unpleasant odors or even health hazards. Finally, evaporative coolers need a continuous external water supply which can be an issue in the dry climates for which they are best suited. Evaporative cooling systems can be an effective alternative or enhancement to vapor compression refrigeration systems in some circumstances. They can be used for cost-effective stand-alone air conditioning in hot, dry climates and can be combined with vapor compression systems for improved efficiency in many more applications. They can also be used for commercial and industrial processes where the high moisture content of the air is not a concern.

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Applications of Heat, Mass and Fluid Boundary Layers

Entropy generation in boundary layer flow and heat transfer

The usefulness of the application of the second law of thermodynamics in engineering and other disciplines is now well recognized. The laws of thermodynamics allow us to define the limits of performance of a system. Real world systems are limited by the total cost of the end product, whereby cost is a function of many parameters, with energy being one of them [40]. Mathematically, thermodynamics offers more simplicity than Newtonian mechanics, being mostly algebraic and linear. Entropy generation analysis is a well-established undertaking in thermodynamics, not only of academic importance but also of enormous importance in real-world engineering systems. Also, the second law continues to be crucial, not only for efficiency studies but also to cost accounting and economic analyses. A second law approach to any system represents the true efficiency of a system. Entropy is not a conserved quantity, but one that may be produced. It is a quantity that is positive in irreversible processes and zero in reversible processes. For example, sustainability objectives and initiatives across the world towards a more sustainable environment demand an optimal use of resources whereby entropy generation assessments play a significant role. Entropy generation has always been an essential part of thermodynamic studies and to the thermodynamicist. The physical basis of entropy generation is lost work. Heat transfer analyst and fluids engineers have often ignored entropy generation by simply pretending it does not exist. Only recently have they both embraced the consideration of entropy generation in their analyses discovering that there are advantages in accounting for it and disadvantages to ignoring it. Energy conservation or the first law of thermodynamics suggests that there is no energy loss in a system, due to its consideration of that system being an isolated system. However, we know that real systems are not isolated from the ambient and do experience energy loss in transport processes. Energy system optimization was once based on energy (first law) analysis; today, it is based on exergy and entropy analysis. Exergy is available work and the useful part of energy. Entropy generation is understood to be a loss of exergy according to the Guoy–Stodola theorem [24]. Entropy generation and exergy destruction are two concepts that are firmly rooted in the fundamentals of the second law of thermodynamics, which is universally applicable. They are well-established means for analyzing thermal systems of all kinds. Entropy generation allows for or requires a global systems approach in characterizing all the components of a system. Further, the entropy generation of the thermal-fluid-mass transport is assessed to determine the depreciation of the energy that is transferred in the form of heat. While energy is a state property, heat transfer is a process quantity. The second law approach provides for a superior understanding compared to the first law approach. It is one of many nonequilibrium thermodynamic approaches (other types are endoreversible thermodynamics, quantum thermodynamics, finite time thermodynamics, rational thermodynamics, etc.) to assess real-world systems [25]. The second law of thermodynamics is a concise statement of the natural phenomenon of irreversibility. Characterizing the entropy generation of a system is significantly essential to the evolution and sustainability of various technologies; those numerous

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technologies today being possible due to the correct application of the laws of thermodynamics. In the preceding sections, we discussed thermodynamic cycles that relate to different practical applications such as power plants. Similar to how the second law and entropy generation analysis is shown to be available for the evaluation of cycles and processes, it is also available at a much more fundamental level in evaluating the entropy generation in convective heat transfer processes, for which boundary layer theory is applied. Entropy generation in boundary layer flow is the focus of much of the current research. The boundary layer is an open thermodynamic system. There are losses to the environment in the real world or practical systems. The losses are due to irreversibility. Irreversibility is otherwise known as the destruction of useful energy (exergy) or entropy generation. Entropy generation due to irreversibility is the most useful measure of inefficiencies and loss in engineering systems. Losses in nonideal or real systems manifest as thermal loss due to heat transfer across a finite temperature difference, shock losses, viscous dissipation (boundary layers and shear layers), etc. More broadly, irreversible processes include drag (friction), heat addition, diffusion of species, chemical reactions, Joule heating (current through a resistor), friction between solid surfaces, fluid viscosity in a flow, heat diffusion, unconstrained expansion (or compression) and many more. An increase in entropy production is directly related to efficiency. This energy loss reduces the efficiency of the several cycles above, and as such, the purpose of assessing entropy generation in convective boundary layer flows is to find avenues for optimization. Entropy generation of laminar boundary layer research may provide, for example, answers to the question of what shapes a body must take to optimize thermal-fluid-mass transfer and delay a transition to separation. Also, in an ideal sense, we want heat transfer maximization with entropy generation minimization. The advancement of anything or any technology is towards higher efficiency. This section briefly examines the characterization and assessment of entropy generation in convective boundary layer flows. The governing equations for local entropy generation due to flow and heat transfer can be expressed in different coordinate systems can be expressed as follows according to Bejan [26] Sgen = Sgen (friction) + Sgen (thermal) + Sgen (other effects) .

(13.49)

The third term, and perhaps subsequent terms regarded as entropy generation due to other effects, in Eq. (13.49) considers flow situations that may have other effects on the problem such as viscous dissipation, fluid damping (Darcy and non-Darcy), electromagnetic effects, chemical effects, and so on. It is also observed from Eq. (13.49) that entropy generation is additive. Expanded, for a normal fluid without any effects and in the Cartesian coordinate system (x, y, z), this is represented as 

2

2

2  k ∂T ∂T ∂T  + + S˙gen = 2 ∂x ∂y ∂z T 8 

2

2

 ∂νy μ ∂νx 2 ∂νx + + + 2 T ∂x ∂y ∂z

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∂νy ∂νx + + ∂y ∂x

2



∂νx ∂νz + + ∂z ∂x

2



∂νx ∂νz + + ∂z ∂y

2 9 .

(13.50)

To obtain the total entropy generation (S˙gen ), one integrates the local entropy generation distribution over the volume as [44]   ˙ (13.51) Sgen = Sgen dV . The preceding equation demonstrates the importance of both thermal conductivity and viscosity on entropy generation. In what follows, we discuss some selected research works on entropy generation in boundary layer flow, heat, and mass transfer. In general terms, the authors of [45] discussed second law analysis of convective heat and mass transfer problems. In a review that examined both natural and mixed convection flows in porous media towards applications in energy systems, the researchers concluded that an analysis of entropy generation is essential to reduce energy consumption. The application of magnetic force, or in systems that employ a magnetic effect, magnetism, functions to also minimize entropy generation [27]. The consequent effect of entropy generation of Blasius flow under thermal radiation and viscous dissipation was examined in [28], showing that an increase of the thermal radiation parameter reduces entropy production while viscous dissipation increases entropy production. Entropy analysis of MHD boundary layer flows with convective surface boundary conditions has also been considered in [48] revealing that entropy generation decreases with an increase in Prandtl number. Furthermore, towards the optimization of flows in porous media a succinct review is presented in [39] discussing the perspectives of the traditional local thermal equilibrium (LTE) and local thermal non-equilibrium (LTNE) modeling techniques, and notable challenges and progress towards modeling of entropy generation in porous media. Regarding the use of advanced fluids, such as nanofluids, a basic, yet the major objective is to achieve a higher exergetic efficiency, i.e., optimal efficiency with minimal entropy increase. It is for this reason that studies not only study nanofluid flow and heat transfer in boundary layers but also evaluate entropy generation. Research efforts on the entropy generation of nanofluids started circa 2010. It has been expressed that to improve system performance further even when nanofluids are used as working fluids, researchers have to emphasize entropy generation [37]. There are now several conferences and special journal issues dedicated to the subject of entropy generation with nanofluids as the working fluid. In an analytical study of entropy generation of nanofluid flow over a flat plate, researchers indicate that entropy generation is a function of volume fraction or concentration, Prandtl (P r) number, Eckert (Ec) number, and Reynolds (Re) number, whereby entropy generation increases with an increase in volume concentration. It was also found that due to the higher density of Cu nanoparticles, as compared to other nanoparticles, it generated more entropy in a base fluid of water [29]. Unsteady nanofluid MHD flow considering thermal radiation, viscous dissipation, and the chemical reaction was considered in [46] showing

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the buoyancy ratio and mixed convection parameter exhibiting an opposite behavior to the Bejan number. In a study examining mixed convection nanofluid flow over vertical cylinders, it is shown that the role of suction is to increase entropy generation while injection reduces entropy generation. Also, it was shown that thermal stratification also intensifies the entropy generation [30]. The performance of hybrid nanofluids from an entropy generation perspective has also been investigated in [31]. The researchers showed approximately 26.5% reduction in entropy generation using hybrid nanofluids (MWNCT+Fe3 O4 in water), as compared to the base fluid with a 0.3% volume concentration. A further increase in the volume concentration of the hybrid nanofluid was also shown to decrease the total entropy generation. However, in another study for simple nanofluids (e.g., Al2 O3 and TiO2 ) [32], it is shown that entropy generation parameter increases as a function of an increase in nanoparticle volume concentration, the magnetic parameter, the mass suction/injection parameter, the Brinkman and Hartmann dimensionless numbers. In [33], it is also demonstrated that entropy generation is a function of Re, P r, Lewis (Le), Brinkman (Br) numbers, and thermophoresis parameter. Entropy generation analysis of magnetite nanofluids in a base fluid of engine oil and ethylene glycol was presented in [42] with the analysis revealing that Fe3 O4 ethylene glycol based nanofluid is more suitable than Fe3 O4 engine oil based nanofluid. The influence of heat generation or absorption and a partial velocity slip on the entropy generation of flows over the stretching surface, which finds applications in the several engineered systems was examined in [43]. The authors concluded that heat generation, among other effects, reduces entropy generation. The reader may also refer to [49] for other considerations of entropy analysis of nanofluid flows. The essence of these studies is to establish those parameters that result in an overall reduction of entropy. Entropy generation due to non-Newtonian flow around a spherical particle at low Reynolds number was considered in [34] showing high entropy generation rate close to stagnation point around the spherical particle. A second grade MHD nanofluid flow is considered in [47] with various effects and showing that the entropy generation is enhanced by large values of the magnetic parameter and an increasing Reynolds number. The authors in [35] considered both a Casson fluid and Casson ferrofluid on a convectively heated upper surface of a paraboloid boundary layer. Their study revealed better heat transfer performance for Casson ferrofluid as compared to Casson fluid. Furthermore, using the homotopy analysis method (HAM), the authors in [36] considered a Casson nanofluid considering Navier slip and a convective boundary condition showing that an increase in N b, N t, Bi, Re, and Br all lead to an increase in entropy generation number. Entropy generation of nanofluid flow over a flat plate using the homotopy perturbation method (HPM) together with the variational iteration method (VIM) has been presented in [38] considering Cu, Al2 O3 , and TiO2 nanoparticles. It was shown that due to the high density of Cu, this particular nanoparticle generates more entropy as compared to Al2 O3 and TiO2 nanoparticles. A thermodynamic approach to time-dependent and time-independent non-Newtonian fluids is presented in [41].

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Conclusions

From this brief review, several applications of the subject of thermodynamics have been illustrated considering automotive, refrigeration, power generation, and so on. The review demonstrates that thermodynamics is evident in every facet of life and is indeed a manifestation of life as we know it. Regarding thermodynamic entropy generation in boundary layers, it is evident that almost all studies of entropy generation in boundary layer flows are limited to similarity flows perhaps because they offer the least mathematical complexity. The more rigorous and meaningful approach may be to investigate and characterize entropy generation in fluid flows with nonsimilarity. More so, many studies neglect effects such as fluid damping (Darcy and non-Darcy), perhaps treating them as non-essential and this may not be correct. Nevertheless, it is envisaged that the illustrative applications examined in this review will serve to crystallize the principles of thermodynamics further and will motivate the reader to apply it for their practical purposes, while also advancing the state-of-the-art.

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[35] J.V.R. Reddy, V. Sugunamma, N. Sandeep, Enhanced heat transfer in the flow of dissipative non-Newtonian Casson fluid flow over a convectively heated upper surface of a paraboloid of revolution, J. Mol. Liq. 229 (2017) 380–388. [36] M.H. Abolbashari, N. Freidoonimehr, F. Nazari, M.M. Rashidi, Analytical modeling of entropy generation for Casson nanofluid flow induced by a stretching surface, Adv. Powder Technol. 26 (2015) 542–552. [37] O. Mahian, A. Kianifar, C. Kleinstreuer, M.A. Al-Nimr, I. Pop, A.Z. Sahin, S. Wongwises, A review of entropy generation in nanofluid flow, IJHMT 65 (2013) 514–532. [38] A. Malvandi, D.D. Ganji, F. Hedayati, E.Y. Rad, An analytical study on entropy generation of nanofluids over a flat plate, Alex. Eng. J. 52 (2013) 595–604. [39] M. Torabi, N. Karimi, G.P. Peterson, S. Yee, Challenges and progress on modelling of entropy generation in porous media: a review, IJHMT 114 (2017) 31–46. [40] K.H. Mistry, J.H. Lienhard, An economics-based second law efficiency, Entropy 15 (2013) 2736–2765. [41] C. Huang, A thermodynamic approach to generalized rheological equations of state for time-dependent and time-independent non-Newtonian fluids, Chem. Eng. J. 3 (1972). [42] Z. Iqbal, E. Azhar, Z. Mehmood, E.N. Maraj, A. Kamran, Computational analysis of engine-oil based magnetite nanofluidic problem inspired with entropy generation, J. Mol. Liq. 230 (2017) 295–304. [43] A. Noghrehabadi, M.R. Saffarian, R. Pourrajab, M. Ghalambaz, Entropy analysis for nanofluid flow over a stretching sheet in the presence of heat generation/absorption and partial slip, J. Mech. Sci. Technol. 27 (3) (2013) 927–937. [44] H. Khorasanizadeh, M. Nikfar, J. Amani, Entropy generation of Cu–water nanofluid mixed convection in a cavity, Eur. J. Mech. B, Fluids 37 (2013) 143–152. [45] M. Mourad, A. Hassen, H. Nejib, B.B. Ammar, Second law analysis in convective heat and mass transfer, Entropy 8 (1) (2006) 1–17. [46] Y.S. Daniel, Z.A. Aziz, Z. Ismail, F. Salah, Entropy analysis in electrical magnetohydrodynamic flow of nanofluid with effects of thermal radiation, viscous dissipation, and chemical reaction, Theor. Appl. Mech. Lett. 7 (2017) 235–242. [47] H. Sithole, H. Mondal, P. Sibanda, Entropy generation in second grade magnetohydrodynamic nanofluid flow over a convectively heated stretching sheet with nonlinear thermal radiation and viscous dissipation, Results Phys. 9 (2018) 1077–1085. [48] O.D. Makinde, Entropy analysis for MHD boundary layer flow and heat transfer over a flat plate with a convective surface boundary condition, Int. J. Exergy 10 (2) (2012) 142–154. [49] G.S. Seth, A. Bhattacharyya, R.Kumar, A.J. Chamkha, Entropy generation in hydromagnetic nanofluid flow over a nonlinear stretching sheet with Navier’s slip and convective heat transfer, Phys. Fluids 30 (2018) 122003.

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Leye M. Amooa , R. Layi Fagbenleb a Stevens Institute of Technology, Hoboken, NJ, United States, b Mechanical Engineering Department and Center for Petroleum, Energy Economics Law, CPEEL (A John D. and Catherine T. McArthur Center of Excellence), University of Ibadan, Ibadan, Oyo State, Nigeria

14.1 Introduction Non-Newtonian fluids are fluids with a stress that can have a nonlinear and/or temporal dependence on the rate of deformation, unlike Newtonian fluids, which demonstrate a linear dependence. The literature reveals that interest in non-Newtonian fluids has grown since the 1940s and 1950s. Since the majority of raw materials and finished products from the processing industry (food, polymers, emulsions, slurries, etc.) are non-Newtonian fluids, it is becoming increasingly important to understand physical characteristics of these fluids [1]. Since most of the differences among the different categories of non-Newtonian fluids are related to their viscosity, which is a dominant physical property within the boundary layer region, a thorough understanding of the flow in the boundary layer is of considerable importance in a range of chemical and processing applications. The nature of boundary layer flow influences not only the drag at a surface or on an immersed object, but also the rates of heat and mass transfer when temperature or concentration gradients exist. The literature shows that there is a significant amount of research with the goal of understanding non-Newtonian flows through pipes and channels due to its relevance to the applications mentioned previously [2,3]. A limited body of research on external flows of non-Newtonian fluids also exists [4–6]. Section 14.2 of this chapter presents a review of selected research performed in relation to the behavior of non-Newtonian boundary layer flows and laminar heat transfer characteristics in non-Newtonian fluids. A summary of current research efforts is provided in Sect. 14.3, followed by a brief overview of future research prospects in this area in Sect. 14.4.

14.2

Background

The authors in [7] discerned three stages in the development of fluid mechanics. During the first stage, studies were focused on ideal fluids, that is, fluids without viscosity, Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00022-0 Copyright © 2020 Elsevier Ltd. All rights reserved.

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compressibility, or elasticity, with all remaining material properties kept constant. Such fluids are hypothetical and were used mainly for analysis. Despite the seemingly crude approximations, theories on inviscid and incompressible fluids led to groundbreaking results in many areas of science and engineering (e.g., accurate prediction of lift force, which paradoxically is a viscous effect). The next stage introduced viscous effects. Viscous effects and the equations used to model them were introduced in fluid mechanics studies in the first half the 19th century. These were buttressed by Ludwig Prandtl in 1904, who assumed that viscosity becomes important only in the boundary layer, formed in the direct vicinity of a solid surface. Prandtl recognized that in low-viscosity fluids, viscous effects could be localized to thin layers near boundaries, and subsequently, his student Blasius showed how the mathematics of this approximation could be developed for a flat plate. Hence, the flow domain was decomposed into a region of ideal fluid (far from the surface) and a viscous fluid (close to the surface). This approach is the basis for classical fluid dynamics. Lastly, the third stage, which is still an active area of research, addresses the departure from Newton’s linear law of viscosity. Its importance was appreciated at the beginning of the last century, as many industrial materials could not be accurately described with this simple relation. Two sources of non-Newtonian behavior can be distinguished. On a microscopic level, it is the molecular structure of fluid particles. Spherical and roughly spherical particles produce a Newtonian behavior, whilst the addition of long chains of particles may cause Newton’s approximation to become invalid. On a macroscopic level, mixtures such as emulsions and slurries may become non-Newtonian even though their components are Newtonian. Fluids can exhibit non-Newtonian behavior in several ways. They may be purely viscous, in that the stress depends on the rate of deformation in a nonlinear fashion, but there is no dependence on the history of the deformation. Viscous non-Newtonian fluids may be further classified as dilatant (shear-thickening, e.g., corn starch in water, or Oobleck) and pseudo-plastic (shear-thinning, e.g., nail polish, ketchup). They may be viscoelastic, in that the stress depends in a well-defined way on the history of the deformation; viscoelastic liquids are also called memory fluids and include fluids like lubricants, Silly Putty, and so on. Fluids may be thixotropic, in that the material properties are time-dependent at constant stress or deformation rate. This category includes fluids such as yogurt and a variety of gels. Other major types of non-Newtonian fluids include yield stress fluids (also called Bingham plastic fluids) that do not flow at all until a critical stress level is reached and liquid crystals that are anisotropic at rest [1]. These types of non-Newtonian fluids discussed above are presented in Fig. 14.1. As a consequence of having so many categories of fluids classified as nonNewtonian, one of the primary difficulties in any theoretical analysis of the motion of such fluids has been the lack of any generally acceptable equation of state between the stress tensor and state of flow of the system applicable to all these categories [8]. However, a few attempts have been made to arrive at a generalized formulation to describe the rheological properties of these fluids [9]. The following sections briefly describe the general characteristics of laminar boundary layers in the various categories of non-Newtonian fluids mentioned previously.

Overview of non-Newtonian boundary layer flows and heat transfer

415

Figure 14.1 The different types of non-Newtonian fluids.

14.2.1 Time-independent non-Newtonian fluid behavior Fluids in which the strain experienced at any point is purely dependent on the current value of the shear stress at that point are considered to be time-independent nonNewtonian fluids. They may be further categorized as the following: • shear-thinning, or pseudoplastic, fluids; • shear-thickening, or dilatant, fluids; and • viscoplastic fluids.

14.2.1.1

Shear-thinning, or pseudo-plastic, fluids

Shear-thinning is the most common type of time-independent non-Newtonian fluid behavior that can be observed. It is also called pseudoplasticity and is characterized by an apparent viscosity, which decreases with increasing shear rate, as shown in Fig. 14.2. The power-law model, or the Ostwald–de Waele model, is one of the more popular models used to describe the behavior of shear-thinning liquids. The apparent viscosity, μ, for the power-law fluid is given by the following relation: μ = mγ˙ ∗n−1 ,

(14.1)

where γ˙ is the shear rate. In this equation, m and n are empirical curve-fitting parameters known as the fluid consistency coefficient and the flow behavior index, respectively. For shear-thinning fluids, the value of n varies from 0 to 1. For values of n greater than 1, the model is applicable to shear-thickening fluids, while n = 1 represents Newtonian fluids. Thus, the smaller the value of n, the greater the degree of shear-thinning. Significant deviations from this model are observed at very low and very high shear rates, which are captured well by other models, such as, the Carreau and Ellis models. The application of boundary layer theory to an Ostwald–de Waele, or power-law, fluid was first described by Schowalter [10] and Acrivos et al. [4], where it was

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Figure 14.2 Shear-thinning behavior.

shown that self-similar solutions exist for the boundary layer flow of a power-law fluid when the flow is of the Falkner–Skan type. Specifically, Schowalter [10] examined the boundary-layer equations of a power-law fluid in the absence of body forces and showed the existence of self-similar solutions when the external velocity is of the form x m , where x is the distance along the surface of the body. Acrivos et al. [4] considered the boundary layer flow of a power-law fluid for the case m = 0, which corresponds to a flow along a flat plate. The continuity and momentum transport equations of a power-law fluid assuming a Cartesian co-ordinate system in dimensional form can be written as follows: ∂u∗ ∂v ∗ ∂w ∗ + + ∗ = 0, ∂x ∗ ∂y ∗ ∂z   ∗ ∂τy∗∗ x ∗ ∂τz∗∗ x ∗ ∂u ∂u∗ ∂u∗ ∂p ∗ ∂τ ∗∗ ∗ ρ u∗ ∗ + v ∗ ∗ + w ∗ ∗ = ∗ + x ∗x + + ∂x ∂y ∂z ∂x ∂x ∂y ∗ ∂z∗ (14.2)   ∗ ∗ ∗ ∗ ∗ ∂τy ∗ y ∗ ∂τz∗∗ y ∗ ∂p ∗ ∂τx ∗ y ∗ ∗ ∂v ∗ ∂v ∗ ∂v ρ u +v +w + + , = ∗+ ∂x ∗ ∂y ∗ ∂z∗ ∂y ∂x ∗ ∂y ∗ ∂z∗   ∂τy∗∗ z∗ ∂τz∗∗ z∗ ∂v ∗ ∂w ∗ ∂w ∗ ∂p ∗ ∂τ ∗∗ ∗ ρ w ∗ ∗ + v ∗ ∗ + w ∗ ∗ = ∗ + x ∗z + + , ∂x ∂y ∂z ∂z ∂x ∂y ∗ ∂z∗ variables (u∗ , v ∗ , w ∗ ) and p ∗ represent the components of fluid velocity and pressure inside the flow, respectively. Variables (x ∗ , y ∗ , z∗ ) are dimensional coordinates of a given location and ρ is the density. The components of the stress tensor are given by τij∗ . In dimensionless form, where L is taken as a typical length, U as typical speed, and ε as a scaling factor that is a function of the Reynolds number, the dimensionless

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variables can be expressed as x∗ , L u∗ u= , U

x=

y∗ , L v∗ v= , U y=

z∗ , L w∗ w= , U

z=

p=

p∗ . ρU 2

(14.3)

Thus, the continuity and momentum transport equations in dimensionless form can be written as ∂u ∂v ∂w + + = 0, ∂x ∂y ∂z   ∂τyx ∂τxx ∂u ∂u ∂p 1 ∂τzx ∂u +v +w =− + + + , u ∂x ∂y ∂z ∂x ρU 2 ∂x ∂y ∂z     ∂τxy ∂τyy ∂τzy 1 ∂p ∂v ∂v ∂v 1 =− ,  u +v +w + + + ∂x ∂y ∂z  ∂y ρU 2 ∂x ∂y ∂z     ∂τxz ∂τyz ∂τzz ∂w ∂w 1 ∂p 1 ∂w +v +w =− + + + .  u ∂x ∂y ∂z  ∂z ρU 2 ∂x ∂y ∂z

(14.4)

At sufficiently large Reynolds numbers, upon further simplification, the equations that describe boundary layer flow in three-dimensional form of a power-law fluid can be written as ∂u ∂v ∂w + + = 0, ∂x ∂y ∂z    n−3 ∂u ∂ 2 u ∂u ∂ 2 u ∂u δu ∂u ∂p ∂u 2 (n − 1) +v +w =− +κ + u ∂x ∂y ∂z δx ∂y ∂y 2 ∂z ∂y∂z ∂y  2 

 2 2 ∂u ∂ u ∂u ∂ u ∂ 2u ∂u ∂ u + +κ +(n − 1) + 2) , ∂y ∂z∂y ∂z ∂z2 ∂z ∂y 2 ∂z ∂p 0=− , ∂y ∂p 0=− . ∂z (14.5) The following equation defines k: 2 2 ∂u ∂u κ= + . ∂y ∂z

(14.6)

Assuming that the Carreau model of viscosity applies to the fluid, the boundary layer equations would differ slightly from the Ostwald–de Waele form. The dynamic viscosity of a Carreau fluid can be written as  n−1

2 μ∗ = μ∞ + (μ0 − μ∞ ) 1 + (K1 γ˙ ∗ )2 ,

(14.7)

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where μ0 is the zero-shear-rate viscosity, μ∞ is the infinite-shear-rate viscosity, γ˙ ∗ is the shear rate, and K1 is the characteristic time constant, sometimes referred to as the relaxation time. Adopting a similar approach to that used in power-law fluids, the boundary layer equations for fluids whose viscosity is governed by the Carreau model can be written as ∂u ∂v + = 0, ∂x ∂y

⎡ ⎤ 8

98

9 n−3 δu ∂p ⎣ ∂u 2 ∂u 2 2 ⎦ ∂ 2 u ∂u , +v =− + 1 + C0 1 + n λ 1+ λ u ∂x ∂y δx ∂y ∂y ∂y 2 0=−

∂p . ∂y (14.8)

∞ , and λ is the dimensionless The variable C0 is the viscosity ratio given by μ0μ−μ ∞ equivalent of K1 . Thompson and Snyder [11] examined the boundary-layer flow of a power-law fluid in the presence of fluid injection at the surface with the aim of determining the drag reduction potential of such non-Newtonian fluids. Solutions were found for a range of mass injection rates, the results indicating that the skin-friction coefficient decreases monotonically as the fluid index increases. For a fixed rate of fluid injection, it was shown that the percentage reduction in the skin-friction coefficient was greater for smaller values of the fluid index. Andersson and Toften [12] discussed aspects of obtaining a numerical solution to the laminar boundary layer equations for a power-law fluid. Their work provides a concise review of various techniques for finding solutions for laminar boundary-layer flow of power-law fluids and certain notable shortcomings of these techniques.

14.2.1.2

Shear-thickening, or dilatant, fluids

Dilatant fluids are similar to pseudoplastic systems in that they show no yield stress, but their apparent viscosity increases with increasing shear rate; thus, these fluids are also called shear-thickening fluids. This type of fluid behavior was originally observed in concentrated suspensions. At low shear rates, the liquid lubricates the motion of each particle past other particles and the resulting stresses are consequently small. At higher shear rates, on the other hand, the material expands or dilates slightly (as also observed in the transport of sand dunes) so that there is no longer a sufficient amount of liquid to fill the increased void and prevent direct solid–solid contacts that result in increased friction and higher shear stresses. This mechanism causes the apparent viscosity to rise rapidly with increasing rate of shear. Eqs. (14.2)–(14.8) can also be applied to shear-thickening fluids to understand the behavior of boundary layers in such instances. The term “dilatant” has also been used for all other fluids that exhibit increasing apparent viscosity with increasing rate of shear. Many of these, such as starch pastes,

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are not true suspensions and show no dilation on shearing. The above explanation therefore is not applicable, nevertheless such materials are still commonly referred to as dilatant fluids. Of the time-independent fluids, this subclass has received very little attention; consequently, little reliable data are available.

14.2.1.3

Viscoplastic fluids

This type of fluid behavior is characterized by the existence of a yield stress that must be exceeded before the fluid will deform or flow. Conversely, such a material will deform elastically (or flow en masse like a rigid body) when the externally applied stress is smaller than the yield stress. Once the magnitude of the external stress has exceeded the value of the yield stress, the flow curve may be linear or nonlinear but will not pass through origin, as seen in Fig. 14.1. The flow of muddy rivers is a typical example. Among many viscoplastic fluids, there is a special class called Bingham plastics. For Bingham plastic fluid, the shear stress beyond the yield stress is linearly proportional to the shear rate. If the yield stress approaches zero, the Bingham plastic fluid can be approximately treated as a Newtonian fluid. Mathematically, this model can be represented as: τ = τ0 + μγ˙ when τ ≥ τ0 ; and γ˙ = 0 when τ < τ0 ,

(14.9)

where τ is the shear stress and τ0 is the yield stress, below which the fluid behaves essentially like a rigid body. Regarding the flow of the Bingham fluid, the stress varies in space and time. There can be regions in the fluid where the yield stress is exceeded, and other regions where it is not. The boundaries between the two regions are the yield surfaces. Tracking the yield surfaces as the flow evolves is one of the most complicated problems associated with the Bingham model. The stability of the viscoplastic flows depends on what happens to these nonmaterial surfaces when a sudden perturbation is introduced.

14.2.2 Time-dependent non-Newtonian fluid behavior In practice, apparent viscosities may depend not only on the rate of shear but also on the time for which the fluid has been subjected to shearing, as seen in crude oils, emulsions, and so on. They are further classified based on whether the apparent viscosity decreases or increases with the amount of time they are subjected to shearing, as thixotropic and rheopectic fluids, respectively.

14.2.2.1

Thixotropic and rheopectic fluids

In thixotropic fluids, the shear stress (or apparent viscosity) decreases with increasing time of application of a constant shear rate. The macroscopic rheological properties of many complex fluids depend on their microscopic structure. If the flow curve is measured in a single experiment in which the shear rate is steadily increased at a constant rate from zero to some maximum value and then decreased at the same rate to zero again, a hysteresis loop of the form shown in Fig. 14.3 is obtained; the height, shape,

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and enclosed area of the hysteresis loop depend on the duration of shearing, the rate of increase and decrease of the shear rate, and the past kinematic history of the sample. Thixotropic behavior occurs if, when exposed to high shear rates, the microstructure gradually breaks down and the fluid becomes less viscous, and if, when exposed to low shear rates, the microstructure is gradually rebuilt and the fluid becomes more viscous. This is common for colloidal dispersions. The dynamics of the microstructure may occur on different timescales from the macroscopic flow. Coussot et al. [13] proposed a simple model to describe the rheological properties of a thixotropic fluid as   μ = μo 1 + λn ,

∂V ∂λ 1 = −α λ, ∂t θ ∂y

(14.10)

where V is the velocity, y is the distance normal to the flow direction, μ0 , n, θ and α are constant parameters for a given fluid, and μ is the apparent viscosity of the thixotropic fluid defined as μ=

τ ∂V ∂y

,

(14.11)

where τ is the shear stress. The degree of jamming of the thixotropic fluid can be represented by a single parameter λ which describes the instantaneous state of fluid structure. This parameter, λ, is also called the degree of jamming of the fluid. Barnes [14] provides an excellent review of thixotropic fluids, considering examples of coal suspensions, oils and lubricants, paints, detergents, clay suspensions, creams and pharmaceutical products, blood, and so on. Barnes also provides a theoretical and mathematical basis for the observed behavior from the point of view of microstructural changes, concluding that more work is needed to develop a comprehensive theory to describe such fluid systems. In rheopectic fluids, an opposite effect is observed, wherein the shear stress (or the apparent viscosity) decreases with increasing time of application of a constant shear rate. As seen in Fig. 14.3, the direction of the hysteresis loop is reversed compared to the thixotropic fluid.

14.2.3 Non-Newtonian laminar boundary layer flows Laminar boundary layers in non-Newtonian flows have been studied over the past half century. Bizzell and Slattery [15] provided a succinct explanation of boundary layer flows of non-Newtonian fluids and proposed an approximate solution of the boundary layer equations. The following subsections briefly describe the popular fundamental studies that have been undertaken to improve our understanding of the behavior of non-Newtonian flows over or through simple geometries such as flat plates, pipes, cylindrical annuli, porous surfaces, and so on.

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Figure 14.3 Behavior of thixotropic and rheopectic fluids, showing the hysteresis loops.

14.2.3.1

Laminar boundary layer flows through pipes

For all fluids entering a small pipe from either a much larger one or from a reservoir, the initial velocity profile is approximately flat, and it undergoes a progressive change until fully developed flow is established, as shown schematically in Fig. 14.4. The thickness of the boundary layer is theoretically zero at the entrance and increases progressively along the tube. The retardation of the fluid in the wall region must be accompanied by a concomitant acceleration in the central region in order to maintain continuity. When the velocity profile has reached its final shape, the flow is fully developed and the boundary layers may be considered to have converged at the centerline. It is customary to define an entry length, Le , as the distance from the inlet at which the centerline velocity is 99% of the velocity of the fully-developed flow. The pressure gradient in this entry region is different from that for fully developed flow and is a function of the initial velocity profile. There are two factors influencing the pressure gradient in the entry region: first, some pressure energy is converted into kinetic energy as the fluid in the central core accelerates, and second, the higher-velocity gradients in the wall region result in greater frictional losses. It is important to estimate both the pressure drop occurring in the region before flow has been fully developed and the extent of this entrance length. This situation is amenable to analysis by repeated use of the mechanical energy balance equation. Dodge and Metzner (1959) indicated that both the entrance length and the extra pressure loss for inelastic fluids were similar to those for Newtonian fluids. In the case of non-Newtonian flow, it is necessary to use an appropriate apparent viscosity. Although the apparent viscosity, μa , is defined in the same way as for a Newtonian fluid, it no longer has the same fundamental significance, and other, equally valid definitions of apparent viscosities may be made. In a flow through a pipe, where the shear stress varies with radial location, the value of μa also varies. The conditions near the pipe wall are most important. A fluid with a yield stress (e.g., Bingham plastic) will flow only if the applied stress (proportional to pressure gradient) exceeds the yield stress. There will be a solid

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Applications of Heat, Mass and Fluid Boundary Layers

Figure 14.4 Development of the boundary layer and velocity profile for laminar flow of a power-law fluid in the entrance region of a pipe.

Figure 14.5 Development of the boundary layer and velocity profile for laminar flow of a Bingham plastic fluid, demonstrating “plug-flow” at the core.

plug-like core flowing in the middle of the pipe where the applied shear stress is less than the yield stress, as shown in Fig. 14.5. Rudman and Blackburn [16] have implemented the non-Newtonian viscosity in their direct numerical simulation (DNS) code and validated for power-law fluids against laminar pipe flow and axisymmetric Taylor–Couette flow, both of which have analytical solutions. For the Herschel–Bulkley model, their code was validated against laminar pipe flow, and apart from ensuring correct viscosity estimates for known shear fields, no additional tasks were performed for the Carreau–Yasuda model. In all laminar simulation cases, the authors reported that numerical and theoretical velocity profiles agreed to within 0.01%, and the code is believed to accurately predict the laminar flow of non-Newtonian fluids with generalized Newtonian rheologies.

14.2.3.2

Laminar boundary layer flow over a flat plate

Yao and Molla [17,18] provided a realistic model to describe the laminar flow of a nonNewtonian power-law fluid over a flat plate. They stated that two widespread mistakes

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appear continuously in papers studying boundary layers involving the traditional twoparameter power-law model of non-Newtonian fluids: (a) failure to recognize that a length scale is associated with the power-law correlation and (b) unrealistic physical results, introduced by the traditional power-law correlation, that viscosity either vanishes or becomes infinite within the limit of a large or small shear rate, respectively. They removed the singularities at the leading edge of the flat plate and numerically solved the boundary layer equations using a simple finite difference method formulation. Hirschhorn et al. [6] provided a detailed analysis of a magneto-hydrodynamic boundary layer slip flow and heat transfer of a power-law fluid over a flat plate.

14.2.3.3

Laminar boundary layer flow over a moving stretched sheet

Investigations of boundary layer flow and heat transfer of viscous fluids over a stretched flat sheet are important in many manufacturing processes, such as polymer extrusion, drawing of copper wires, continuous stretching of plastic films and artificial fibers, hot rolling, wire drawing, glass-fiber, metal extrusion, and metal spinning. Sakiadis [19] initiated the study of the boundary layer flow over a stretched surface moving with a constant velocity and formulated a boundary-layer equation for twodimensional and axisymmetric flows. Although the geometry is similar to a flat plate, since the boundary conditions of the flow over a stretched flat sheet are different from that on a flat plate, the Blasius solution of the laminar boundary layer equations over a flat plate are not valid in this case. Andersson and Kumaran [20] analyzed the nonNewtonian fluid flow over a moving plane sheet with its surface velocity proportional to the distance from the slit raised to an arbitrary power m. Their numerical results showed that the boundary layer thickness decreases monotonically with increasing values of power-law index n.

14.2.3.4

Laminar boundary layer flow over a porous surface

Savins [21] presented a broad overview of non-Newtonian boundary layer flows followed by detailed analyses pertaining to models of porous media such as sand packs and matrices of uniformly packed spheres and woven screen, as well as alundum plugs, sandstone cores, porous metal disks, sintered glasses, and compressed glass wool. A model porous medium is typically unique in geometric morphology, and there are formidable problems in precisely defining the flow conditions existing within any particular porous structure. Abnormal increases in flow resistance that resemble a shear-thickening response is observed in flow of fluids through porous media. This general type of behavior has been observed in porous media flow experiments involving a variety of dilute to moderately concentrated solutions of high-molecular-weight polymers. Flow destabilization or premature departure from laminar-like behavior in dilute polymeric solutions have also been observed in flow experiments with porous media. Geometry-dependent flow behavior in porous media has been detected and seems to be characteristic of certain micellar systems.

424

14.2.3.5

Applications of Heat, Mass and Fluid Boundary Layers

Instabilities in non-Newtonian laminar boundary layers

Pearson [22] provided a detailed review of instability in laminar non-Newtonian boundary layers. Scenarios in which essentially non-Newtonian effects are responsible for the instability were separated from cases in which a mere modification of Newtonian analyses is sufficient. Unidirectional confined flow examples such as circular Couette flow (to study Taylor vortices) and cellular convection flow (to study Bernard instabilities) were followed by unconfined flow examples such as jets and filament drawings to explain scenarios in which modifications to Newtonian analysis is sufficient. To explain the dominance of instabilities due to purely non-Newtonian effects, scenarios such as melt fracture, Couette flow and fluidized beds were explained in detail. Pearson [22] concluded that “there are far too many parametric functions needed to describe the constitutive relations for any particular material, which give rise at best to a bewildering number of approximate dimensionless groups characterizing any particular flow” and recommended that further research be undertaken to better understand some of these characteristics of instability.

14.2.4 Heat transfer in non-Newtonian laminar boundary layers When a fluid and the immersed surface are at different temperatures, heat transfer occurs. If the heat transfer rate is small in relation to the thermal capacity of the flowing stream, its temperature remains constant. The surface may remain at a constant temperature, the heat flux at the surface may be maintained constant, or surface conditions may be between these two limits. Because the temperature gradient is highest near the surface and the temperature of the fluid stream is approached asymptotically, a thermal boundary layer may be postulated that covers the region close to the surface and in which the whole of the temperature gradient is assumed to lie. Thus, a momentum and a thermal boundary layer will develop simultaneously whenever the fluid stream and the immersed surface are at different temperatures. The momentum and energy equations are coupled, because the physical properties of non-Newtonian fluids are normally temperature and shear-rate dependent. The resulting governing equations for momentum and heat transfer require numerical solutions. However, if the physical properties of the fluid do not vary significantly over the relevant temperature interval, there is little interaction between the two boundary layers and they may both be assumed to develop independently of one another. The physical properties other than apparent viscosity may be taken as constant for commonly encountered non-Newtonian fluids. In general, the thermal and momentum boundary layers will not correspond. Metzner [2,23] provided an extensive overview of some of the critical aspects of heat transfer in a non-Newtonian laminar boundary layer and the mathematical formulation to describe its features. Specific studies related to thermodynamics and entropy generation in non-Newtonian laminar boundary layers are discussed in the following subsection.

Overview of non-Newtonian boundary layer flows and heat transfer

14.2.4.1

425

Thermodynamic-entropy generation in non-Newtonian boundary layers

As discussed in the previous section, the thermodynamic properties of non-Newtonian fluid systems play an important role in the understanding of the system’s overall behavior. One of the popular analyses undertaken in this direction is the exergy analysis. Exergy analysis relies on the laws of thermodynamics to establish the theoretical limit of an ideal or reversible operation and the extent to which the operation of the given system departs from the ideal. The departure is measured by the calculated quantity called destroyed exergy or irreversibility. This quantity is proportional to the generated entropy according to the well-established Gouy–Stodola theorem. The minimization of entropy generation requires the use of more than thermodynamics—fluid mechanics, heat and mass transfer, materials, constraints, and geometry are also needed in order to establish the relationships between the physical configuration and the destruction of exergy [24]. In the field of heat transfer, the entropy generation minimization method reveals the inherent competition between heat transfer and fluid flow irreversibilities in the optimization of devices subjected to overall constraints. Therefore, the main objective of this method is determining possible ways of minimizing entropy generation. The method used to achieve this purpose is known as entropy generation minimization or thermodynamic optimization. Thermodynamic irreversibility associated with the flow system provides insight into frictional and heat transfer losses in the system. Moreover, the entropy generation is associated with the thermodynamic irreversibilities occurring in the system. Consequently, thermodynamic irreversibility can be quantified through entropy calculations. The second-law analysis of a non-Newtonian fluid over a horizontal surface and through pipes and several other geometries has many significant applications in thermal engineering and industries. One of the fundamental problems of the engineering processes is improving thermal systems during convection in any fluid. The second-law analysis is one of the best tools for improving the performance of the engineering processes. It investigates the irreversibility due to fluid flow and heat transfer in terms of the entropy generation rate [25]. Bejan [26] provided an extensive overview of the application of second-law analysis for the study of entropy generation and how it affects the thermodynamic characteristics of a non-Newtonian laminar boundary layer. Second-law thermodynamic analysis was applied to the study of non-Newtonian fluids with variable thermophysical properties in a circular channel [27]. The study revealed that global entropy generation increases with power-law index and Brinkman number, and where the thermophysical property variation effect causes a decrease in entropy generation. Entropy analysis of mixed convective magnetohydrodynamic flow of a viscoelastic fluid over a stretching surface is considered in [28], showing that both the local and the average entropy generation number increases with increase in the viscoelastic and magnetic parameters. A decrease in average entropy generation is, however, observed and is due to mixed convection. Entropy generation of a non-Newtonian nanofluid from a stretching surface is presented in [29], where the dimensionless Deborah number, thermophoresis number, Eckert number, and Brinkman

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Applications of Heat, Mass and Fluid Boundary Layers

number all have the same increasing effect on the entropy generation number. Another particularly interesting study of entropy generation analysis of non-Newtonian fluid flow over micropatterned surface, considering the effect of slip at the surface, is presented in [30]. The study revealed that entropy generation number decreases with an increasing slip coefficient for shear-thinning and shear-thickening fluids and Newtonian fluids. This study provides some physical insights into the dynamics of boundary layer flows over micropatterned surfaces. Entropy generation serves as a practical tool for the optimization of non-Newtonian normal and nanofluid flows [31]. The study of the non-Newtonian Eyring–Powell nanofluid over a stretching surface shows that entropy generation is an increasing function of the relevant physical parameters such as the Nusselt number. The interested reader may consult other entropy generation works on non-Newtonian fluid flow and heat transfer [32–39].

14.2.5 Non-Newtonian nanofluid boundary layer transfer Nanofluids may have the potential to significantly increase heat transfer rates in a variety of areas such as industrial cooling applications, nuclear reactors, the transportation industry, micro-electromechanical systems, electronics and instrumentation, and biomedical applications. Nanofluids have also been found to possess enhanced thermophysical properties such as thermal conductivity, thermal diffusivity, viscosity, and convective heat transfer coefficients compared to those of base fluids such as oil or water. There is an ever-increasing interest in understanding the behavior of nanofluids, some of which can behave as non-Newtonian fluids due to their microstructure or higher volume fraction loading of nanoparticles. Nanofluids can behave as both Newtonian and non-Newtonian fluids depending on particle loading and higher volume fractions. Heat transfer is directly related to a fluid’s thermophysical properties. Buongiorno [40] presents an excellent review of some of the recent advances made in this direction at MIT. Nanofluids are engineered colloids made of a base fluid and nanoparticles 1–100 nm in size. Nanofluids have higher thermal conductivity and single-phase heat transfer coefficients than their base fluids. In particular, the heat transfer coefficient increases appear to surpass the thermal-conductivity effect, and cannot be predicted by traditional pure-fluid correlations such as the Dittus–Boelter’s correlation. In the nanofluid literature, this behavior is generally attributed to thermal dispersion and intensified turbulence, which are caused by nanoparticle motion. To test the validity of this assumption, Buongiorno [40] considered seven slip mechanisms that can produce a relative velocity between the nanoparticles and the base fluid. These are inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity. Buongiorno concluded that of these seven only Brownian diffusion and thermophoresis are important slip mechanisms in nanofluids. Brownian motion occurs in a direction from high to low nanoparticle concentrations, whereas thermophoresis occurs in the direction from hot to cold. The authors in [40] also proposed an alternative explanation for the abnormal heat transfer coefficient increases. The nanofluid properties may vary significantly within the boundary layer

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because of the effects of the temperature gradient and thermophoresis. For a heated fluid, these effects can result in a significant decrease in viscosity within the boundary layer, thus leading to heat transfer enhancement. The physics and understanding of nanofluids continues to be a multidisciplinary effort. Mathematical formulations and computational tools, including magnetohydrodynamic equations, are being considered to solve several of the research problems related to slip flow in non-Newtonian nanofluids [41,42]. This is still nascent, given that the mechanisms governing the enhancements of nanofluids in normal base fluids have yet to be well elucidated. Nonetheless, there are notable theoretical works in this regard, and the reader may further consult the open literature.

14.2.5.1

Extensions of the Merk–Chao–Fagbenle method to non-Newtonian fluids

Series-based methodologies continue to attract research interests. The Merk–Chao– Fagbenle (MCF) method is one such method that has equally found utility for research into non-Newtonian fluids. Lin and Chern [43] were perhaps the first to extend the MCF method to non-Newtonian fluid flow. Later, Kim [44] extended the MCF method to the study of power-law fluids with applications to a wedge and circular cylinder, considering a step change in surface temperature distribution. Chang et al. [45], at the University of Toledo, Ohio, utilized the methodology to study natural convection from two-dimensional and axisymmetric bodies of arbitrary contours, concluding that the technique provides a general, accurate, and relatively simple way to analyze transport phenomena in laminar boundary layers of power-law fluids. Kim and Esseniyi [46] studied forced convection of power-law fluids over a rotating nonisothermal body, remarking that, in addition to obtaining good results, they found that the role of rotation parameter in dilatant fluid flows is less significant than in pseudoplastic flows. Meissner [47], in his Master’s thesis, which later resulted in the published work [48], considered a power-law fluid flow over a flat plate, horizontal circular cylinder, and sphere. The work was the first to apply the MCF method to mixed convection power-law fluids, yielding very meaningful results. Howell et al. [49] also utilized the method, for a power-law fluid considering nonlinear velocity and linearly stretching surface velocity, with freestream velocity taken to be zero. The authors concluded that the MCF method is useful for solving difficult transport problems, where simple transformations and universal functions are used to solve the fundamental differential equations, regardless of geometry. Rao et al. [50] considered the utility of the MCF method with injection and suction at a moving wall, indicating that the convergence of the MCF method is excellent. Last, Shokouhmand and Soleimani [51] considered the effect of viscous dissipation of a power-law fluid using the MCF method. They remarked that, due to the difficulty of finding similarity solutions for non-Newtonian flows over a surface in the presence of injection or suction (there is no similarity solution for most cases),1 the MCF method is particularly appropriate for solving these 1 Similarity solutions for non-Newtonian fluid have been discussed in S.Y. Lee and W.F. Ames, AICHE

Journal, 700–708, 1966.

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problems when the transformation to a simple ordinary differential equation is not possible. In another recent work [52], the authors remark that the thermal boundary layer for pseudoplastic fluids is thicker than that of dilatant fluids, further establishing also that a direct relationship exists between dimensionless temperature and injection parameter or increase in heat generation parameter. With the exception of the authors of [46] and [52] who extended their works to consider dilatant and pseudoplastic fluids, it appears that all studies using the MCF method involved power-law or Ostwald–de Waele fluids. Though power-law fluids constitute the simplest and most useful type of non-Newtonian fluids, many research opportunities arise, and it is instructive to extend the methodology to other types of non-Newtonian fluids (e.g., Sisko, Casson, micropolar, and Jeffrey fluids) to better and further characterize the difficult transport problems therein. Furthermore, the very disciplined MCF method is yet to be employed to exposing the underlying transport phenomena of non-Newtonian nanofluid boundary layer transfer.

14.3 A note on current research status and applications of non-Newtonian fluids This section covers a selection of the current applications of research on nonNewtonian laminar boundary layer and heat transfer. As mentioned in the earlier sections of this chapter, non-Newtonian laminar boundary layer flows are observed in several domains, including but not limited to biological systems, chemical and process engineering systems, geoscience systems, transportation systems, and food and pharmaceutical processing systems. A prominent research example for each of these research domains is described in what follows.

14.3.1 Biological/biomedical systems: vascular fluid dynamics There is considerable evidence that vascular fluid dynamics plays an important role in the development and progression of arterial stenosis, one of the most wide-spread human diseases, that leads to the malfunction of the cardiovascular system. Although the exact mechanisms responsible for the initiation of this phenomenon are not clearly known, it has been established that once a mild stenosis is developed, the resulting flow disorder influences the development of the disease and arterial deformity and changes the regional blood rheology [53]. The assumption of the Newtonian behavior of blood is acceptable for high shear-rate flow, that is, the case of flow through larger arteries. It is not, however, valid when the shear-rate is low, as for the flow in smaller arteries and in the downstream of the stenosis. It has been pointed out that in some disease conditions, blood exhibits remarkable non-Newtonian properties. Thurston [54] has shown conclusively that blood, being a suspension of enumerable cells, possesses significant viscoelastic properties. In most of the investigations relevant to the domain under discussion, the flow is mainly considered in cylindrical pipes of uniform cross-section area. However, it is

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Figure 14.6 Snapshots showing instantaneous RBC distribution from representative simulations for nonstenosed (left column) and 84% stenosed (right column) vessels [56].

well-known that blood vessels bifurcate at frequent intervals and the diameter of the vessels varies with the distance. Mandal [55] has developed a mathematical model in order to study the notable characteristics of non-Newtonian blood flow through flexible tapered arteries in the presence of stenosis subject to the pulsatile pressure gradient. The apparent viscosity of blood is observed to increase by several folds when compared to non-stenosed vessels [56]. An asymmetric distribution of the red blood cells, caused by geometric focusing in stenosed vessels, is observed to play a major role in the enhancement, as seen in Fig. 14.6. With new biomedical applications related to the targeted delivery of drugs, minimally-invasive surgeries, and post-operative healing procedures becoming a critical area of research, several studies are being carried out in this direction [57,58].

14.3.2 Chemical systems: pharmaceutical products The processing of many pharmaceutical products involves non-Newtonian fluids. Xanthan gum and Carbopol are two common additives that cause a fluid to have shear-thinning properties. These fluids have unique mixing properties that have been studied extensively in experiments. When yield-stress shear-thinning fluids are mixed in stirred tanks, a well-known phenomenon that occurs is the formation of well-mixed zones around the impellers (which constitutes boundary layer flow) while the remainder of the fluid remains relatively stagnant. These well-mixed zones, commonly known as caverns, have been characterized using computational fluid dynamics (CFD) for various impellers [59]. These CFD tools can be used effectively to understand some of these complex nonNewtonian flow phenomena. Having a validated model for non-Newtonian fluids is extremely valuable for predicting the behavior of these fluids in larger and/or more complex equipment that cannot be easily validated. Because cavern sizes and nonNewtonian behavior can change when scaling up in ways that cannot be correlated simply to dimensionless numbers, the implementation of validated CFD models may prove to be the most effective way to scale up processes with shear-thinning fluids.

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Computational fluid dynamics may also help in the selection of novel approaches to designing scale-down experiments. Simulations could be used to identify experimental fluid properties with similar flow behavior on small scales, as for large-scale equipment. The most meaningful experiments for process scale-downs may not necessarily use fluids with physical properties identical to those of the final material. With the help of validated CFD simulations, fluids can be selected to recreate the flow patterns in the laboratory that will be formed at large scales.

14.3.3 Food processing systems: processing of tomato ketchup Ketchups are time-independent, non-Newtonian fluids that show a small thixotropy (Bottiglieri et al., 1991). Tomato ketchup obtains its v iscosity from naturally occurring pectic substances in fruits. Their rheological behavior is important during handling, storage, processing and transport of concentra ted suspensi ons in industry. Other factors such as enzymatic degradations, pectin fraction protein interaction, pulp content, Tomato ketchups is a time-independent, non-Newtonian fluid that shows a small thixotropy. Tomato ketchup obtains its viscosity from naturally occurring pectic substances in fruits. The rheological behavior of these fruits is important during handling, storage, processing, and transport of concentrated suspensions in industry [60]. Other factors such as enzymatic degradations, pectin–protein interaction, pulp content, homogenization process, and concentration may also affect the consistency of tomato products. Knowledge of the rheological properties of fluid and semisolid foodstuffs is important to the design of flow processes in quality control, storage, and processing stability, and in understanding and designing texture. Usually, viscosity is considered an important physical property related to the quality of food products. Therefore, reliable and accurate rheological data are necessary for designing and optimizing various food-processing equipment such as pumps, piping, heat exchangers, evaporators, sterilizers, filters, and mixers. Similar challenges exist in designing process flows for plants to produce toothpastes, shampoos, baby food, and several other products that are non-Newtonian in nature.

14.3.4 Geosciences: drilling muds During drilling operations, drilling muds are pumped from a surface mud tank through the drill-pipe (several kilometers in length), through nozzles in the rotating drill-bit, and back to the mud tank through the annular space between the wellbore wall and the drill pipe. Drilling muds have several functions: to support the wellbore wall and prevent its collapse; to prevent ingress of formation fluids (gas and liquid) into the wellbore; to transport rock cuttings to the surface; to minimize settling of the cuttings if circulation is interrupted; to clear the workface; to cool the drill-bit; and to lubricate the drill string [61]. A schematic of this is shown in Fig. 14.7. The underlying challenge to fluid dynamicists has been to calculate the flow-field within the drillstring–wellbore annulus, a situation usually idealized as one of steady isothermal, fully developed laminar flow of a shear-thinning liquid (modeled as a gen-

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Figure 14.7 A schematic of a drillstring–wellbore interaction, in which drill muds are used.

eralized Newtonian fluid) through an annulus consisting of an outer cylinder and an inner cylinder that may be offset (i.e., eccentric) and rotating. Escudier et al. [62] have employed a finite volume method-based numerical scheme to model the fully developed laminar flow of an inelastic shear-thinning power-law fluid through an eccentric annulus with inner cylinder rotation. The authors in [38] used a model generally used for Newtonian flows but with a viscosity term that is coupled to the local shear rate.

14.3.5 Transportation systems: transport of crude oil emulsions Water-in-crude oil emulsions often show a shear-thinning non-Newtonian behavior, meaning that the viscosity reduces as the shear rate increases. The influence of shear rate on the viscosity is observed to increase as the concentration of the dispersed phase increases [63]. The effective viscosity of an emulsion can greatly exceed either the crude or the water single-phase viscosities. The apparent viscosity of these mixtures depends on many factors such as the viscosity of the oil and of the water, water content, temperature, droplet size distribution, amount of solids in the crude oil, and shear rate. Wax content also appears to contribute to the stabilization of emulsions. At temperatures below the wax appearance temperature, wax crystals precipitate and interact with the oil–water interface. As a result, waxy crude-oil shows high emulsion stability at low temperatures. The experiments with several different crude-oil mixtures illustrate the complexity of the pipe flow of oil–water emulsions and the challenges in understanding the coupling of surface chemistry and fluid flow [63].

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14.4 Future directions One of the major open questions regarding non-Newtonian fluids addresses issues of physics, chemistry and engineering more than the mathematics involved. Which equation should be used to model a given fluid? Wilson [63] provided a short list of open mathematical problems related to non-Newtonian fluids and reported that there are several more application-oriented fluid flow problems involving non-Newtonian fluids that require attention. With the availability of high performance computing resources, DNS is becoming one of the popular tools used by researchers to understand some of the critical non-Newtonian boundary layer phenomena. Even though DNS is primarily being used in transitional and turbulent flow scenarios, the DNS codes are generally validated against laminar flow scenarios for which analytical and/or experimental results are available. As discussed in several of the previous sections, there is tremendous scope for pursuing research on non-Newtonian laminar boundary layers as applicable to a variety of applications relevant to the modern-day industry. However, due to the uncertainty associated with the question of which equations should be used for a given application, constructing a generalized predictive mathematical model that captures the complex phenomena underlying the observed flow features is at least challenging, if not impossible.

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[31] M.M. Bhatti, T. Abbas, M.M. Rashidi, Entropy generation as a practical tool of optimisation for non-Newtonian nanofluid flow through a permeable stretching surface using SLM, Journal of Computational Design and Engineering 4 (2017) 21–28. [32] A. Kahraman, Entropy generation due to non-Newtonian fluid flow in annular pipe with relative rotation: constant viscosity case, Journal of Theoretical and Applied Mechanics 46 (1) (2008) 69–83. [33] C. Ozalp, Entropy generation for nonisothermal fluid flow: variable thermal conductivity and viscosity case, Advances in Mechanical Engineering 2013 (2013) 797894, https://doi. org/10.1155/2013/797894, 9 pages. [34] M. Zakaria, A.M. Moshen, Entropy generation on MHD viscoelastic nanofluid over a stretching surface, IOSR Journal of Applied Physics, (IOSR-JAP) 8 (5 Ver. III) (Sep.–Oct. 2016) 42–50, e-ISSN: 2278-4861, www.iosrjournals.org. [35] O.D. Makinde, Irreversibility analysis for gravity driven non-Newtonian liquid film along an inclined isothermal plate, Available at http://www.ictp.it/~pub_off, last accessed: 2017-7-5. [36] K. Yang, D. Zhang, Y. Xie, G. Xie, Heat transfer and entropy generation of non-Newtonian laminar flow in microchannels with four flow control structures, Entropy 18 (2016) 302, https://doi.org/10.3390/e18080302. [37] M.M. Bhati, T. Abbas, M.M. Rashidi, Numerical study of entropy generation with nonlinear thermal radiation on magnetohydrodynamics non-Newtonian nanofluid through a porous shrinking sheet, Journal of Magnetics 21 (3) (2016) 468–475. [38] R.S.R. Gorla, A. Chamkha, W.A. Khan, P.V.S.N. Murthy, Second law analysis for combined convection in non-Newtonian fluids over a vertical wedge embedded in a porous medium, Journal of Porous Media 15 (2) (2012) 187–196. [39] A. Malvandi, D.D. Ganji, F. Hedayati, M.H. Kaffash, M. Jamshidi, Series solution of entropy generation toward an isothermal flat plate, Thermal Science 16 (5) (2012) 1289–1295. [40] J. Buongiorno, Convective transport in nanofluids, Journal of Heat Transfer 128 (2006) 240. [41] R. Ellahi, The effects of MHD and temperature dependent viscosity on the flow of nonNewtonian nanofluid in a pipe: analytical solutions, Applied Mathematical Modelling 37 (2013) 1451–1467. [42] M. Goyal, R. Bhargava, Boundary layer flow and heat transfer of viscoelastic nanofluids past a stretching sheet with partial slip conditions, Applied Nanoscience 4 (2013) 761–767. [43] F.N. Lin, S.Y. Chern, Laminar boundary layer flow of non-Newtonian fluid, IJHMT 22 (1979) 1323–1329. [44] H.W. Kim, D.R. Jeng, K.J. Dewitt, Momentum and heat transfer in power-law fluid over two-dimensional and axisymmetric bodies, IJHMT 26 (1983) 245–259. [45] T.A. Cheng, D.R. Jeng, K.J. Dewitt, Natural convection to power-law fluids from twodimensional or axisymmetric bodies of arbitrary contour, IJHMT 31 (3) (1988) 615–624. [46] H.W. Kim, A.J. Esseniyi, Forced convection of power-law fluids flow over a rotating nonisothermal body, Journal of Thermophysics and Heat Transfer 7 (4) (1993) 581–587. [47] D.L. Meissner, Master’s Thesis, University of Toledo, Toledo, Ohio, 1992. [48] D.L. Meissner, D.R. Jeng, K.J. Dewitt, Mixed convection to power-law fluids from twodimensional or axisymmetric bodies, IJHMT 37 (10) (1994) 1475–1485. [49] T.G. Howell, D.R. Jeng, K.J. Dewitt, Momentum and heat transfer on a continuous moving surface in a power law fluid, IJHMT 40 (8) (1997) 1853–1861. [50] J.H. Rao, D.R. Jeng, K.J. Dewitt, Momentum and heat transfer in a power-law fluid with arbitrary injection/suction at a moving wall, IJHMT 42 (1999) 2837–2847.

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[51] H. Shokouhmand, M. Soleimani, The effect of viscous dissipation on temperature profile of a power-law fluid flow over a moving surface with arbitrary injection/suction, Energy Conversion and Management 52 (2011) 171–179. [52] H. Radina, A.R.S. Nazar, Temperature profile of a power-law fluid over a moving wall with arbitrary injection/suction and internal heat generation/absorption, Journal of Heat and Mass Transfer Research (2017), https://doi.org/10.22075/JHMTR.2017.519. [53] D. Liepsch, An introduction to biofluid mechanics-basic models and applications, Journal of Biomechanics 35 (2002) 415–435. [54] G.B. Thurston, Viscoelasticity of human blood, Biophysical Journal 12 (1972) 1205–1217. [55] P.K. Mandal, An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis, International Journal of Non-Linear Mechanics 40 (2005) 151–164. [56] K. Vahidkhah, P. Balogh, P. Bagchi, Flow of red blood cells in stenosed microvessels, Scientific Reports 6 (2016) 28194. [57] C. Bertolotti, V. Deplano, J. Fuseri, P. Dupouy, Numerical and experimental models of post-operative realistic flows in stenosed coronary bypasses, Journal of Biomechanics 39 (2001) 1049–1064. [58] B. Ene-Iordache, A. Remuzzi, Disturbed flow in radial-cephalic arteriovenous fistulae for haemodialysis: low and oscillating shear stress locates the sites of stenosis, Nephrology, Dialysis, Transplantation 27 (Jan. 2012) 358–368. [59] J. Kukura, P.C. Arratia, E.S. Szalai, K.J. Bittorf, F.J. Muzzio, Understanding pharmaceutical flows, Pharmaceutical Technology (2002). [60] A. Koocheki, A. Ghandi, S.M.A. Razavi, S.A. Mortazavi, T. Vasiljevic, The rheological properties of ketchup as a function of different hydrocolloids and temperature, International Journal of Food Science & Technology 44 (2009) 596–602. [61] M.P. Escudier, P.J. Oliveira, F.T. Pinho, Fully developed laminar flow of purely viscous non-Newtonian liquids through annuli, including the effects of eccentricity and innercylinder rotation, International Journal of Heat and Fluid Flow 23 (2002) 52–73. [62] J. Plasencia, B. Pettersen, O.J. Nydal, Pipe flow of water-in-crude oil emulsions: effective viscosity, inversion point and droplet size distribution, Journal of Petroleum Science and Engineering 101 (2013) 35–43. [63] H.J. Wilson, Open mathematical problems regarding non-Newtonian fluids, Nonlinearity 25 (2012) R45–R51.

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Climate change in developing nations of the world

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Leye M. Amooa , R. Layi Fagbenleb a Stevens Institute of Technology, Hoboken, NJ, United States, b Mechanical Engineering Department and Center for Petroleum, Energy Economics Law, CPEEL (A John D. and Catherine T. McArthur Center of Excellence), University of Ibadan, Ibadan, Oyo State, Nigeria

15.1 Introduction From an unprecedented rise in global temperatures to the reported melting of the ice sheets in Antarctica [2] and rising sea levels, there is an increased awareness that our world is rapidly changing, and scientists have underpinned these events as a result of climate change. Since the late 19th century there has been a steady increase in the surface temperature of the Earth. The change is largely due to the excessive production of CO2 released into the atmosphere. The rise has been more rapid since circa 2001. As a consequence, the oceans have grown warmer by absorbing the increased heat on the land surface. The oceans are bearing the brunt of our industrial revolution. Our oceans have become 30% more toxic in the last century with the oceans absorbing more CO2 . Data from NASA’s Gravity Recovery and Climate show that ice masses in Greenland and Antarctica are thinning and are melting at rates faster than desired. Also, the amount of spring snow has decreased in the last five decades. The number of extremely hot days and cold days is also on the increase. Consequently, there are places which are experiencing significant periods of drought, such as Cape Town, South Africa, and others with heavy rainfall, such as Lagos, Nigeria. This is understood from the perspective that as temperature increases, water will become scarce, rivers will dry up, and springs and aquifers will become contaminated by rising seas. Fig. 15.1 shows the emissions contribution to the climate for low, middle, and high-income countries. However, while much attention is focused on the effects of climatic variations in the developed world, this chapter will explore how climate change is shaping the fate of developing nations. The justification for urgent intervention to combat climate change in developing nations of the world is that they are some of the hardest hit by climatic variations. An assessment by risk consultancy firm Verisk Maplecroft reveals that the economies of African and Southeast Asian countries will bear the major impact of rising temperatures over the next 30 years [92]. In countries like Ecuador, India, and Nigeria, local agriculture is chiefly dependent on the natural precipitation and temperature cycles with limited access to irrigation. In fact, climate changes are already negatively affecting the agriculture. Furthermore, unlike most developed nations, these countries also must grapple with inadequate capital for mechanized farming. The Food & Agriculture Organization (FAO) of the United Nations reports that if left unmitApplications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00023-2 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Figure 15.1 Contributions of low, middle, and high-income countries to global emissions 1960–2050 [1].

igated, the effects of current climate change in developing nations could plunge as many as 122 million people (including women and children) into extreme poverty by the year 2030 [3]. This chapter is broadly divided into two parts. The first part discusses fundamentals and definitions important to the discussion of climate change. It addresses climate changes as they apply to some key sectors of the global economy, the role and integration of renewable energy, agriculture and conflict in some regions of the world. The second part of this chapter is concerned with climate changes in selected developing countries. The selection of the countries is perhaps random, but focuses on some key nations such as China and Nigeria.

15.2 Climate change Climate change refers to the gradual and persistent alteration in the statistical distribution of weather patterns of extensive areas of the Earth over long periods. Climate occurs as a direct or indirect effect of a plethora of factors, including widespread changes in ecosystems, changes in atmospheric conditions, subsurface geological conditions, and human activity. Climate change is a disruptive menace. Approximately 97% of peer-reviewed climate research concludes that GHGs from burning fossil fuels, industrial farming/activities, and deforestation (all human caused activities) are the main causes of anthropogenic climate change [75,76]. In fact, the authors in [75] brought to light the significant flaws in methodologies and incomplete physics used by the 3% of scientists whose research states otherwise. Nonetheless, humans, intentionally or unintentionally, are causing changes to the planet. Climatology in another regard is the scientific study of variations in weather patterns over long periods and the underlying causes of these changes. Many people use the terms “Climate Change” and “Global Warming” interchangeably, albeit incor-

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Figure 15.2 Impact of the Earth system on human lives [6].

rectly. Climate changes are responsible for pronounced costly flooding, the killing of wildlife, causing human respiratory problems, causing more problems of waste disposal (which is already a big mess and will only become bigger), more rains, more damaging hurricanes, and higher temperatures.

15.2.1 Global warming Global warming refers to a rise in atmospheric temperatures across the world as a result of harmful human activity such as the production of carbon dioxide (CO2 ) and other greenhouse gases (GHGs), and the release of environmental pollutants. Global warming is a serious issue with far-reaching effects on the environment and terrestrial ecosystems. Some effects of global warming include rapid desertification, change in annual precipitation, and a rise in global temperatures. In the fourth quarter of 2018, reports published by National Geographic predicted that 2019 may be the hottest year to date [4]. Interestingly, temperatures in Europe reached a peak of 49.5°C (or 121°F) in June/July of 2019 whereas the US state of Alaska experienced temperatures of 32.2°C (or 90°F), highest in recorded Alaskan history. Several nations or comity of nations have enacted policies to regulate the causative factors of global warming – especially those with direct human influence, e.g., the Paris Climate Accord [5]. Global warming is caused by a mix of natural and human factors (see Fig. 15.2). Some natural factors that influence global warming include: • Movement of the Sun which leads to more solar radiation in certain regions • Naturally-produced greenhouse gases • Volcanic eruptions. Greenhouse gas contribution by human activity includes:

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• Industrial activity • Mining activities (e.g., exploration & production of crude oil, gold, etc.) • Deforestation.

15.2.2 Difference between “climate change” and “global warming” Briefly, the central difference between the two terms is that global warming is an aspect of climate change. Climate change is a broader term encompassing global warming, which only borders on temperature variations. Climate change includes variations in temperature, rainfall, and wind patterns. In a nutshell, it is possible for a climate change to occur without global warming, but global warming does not occur in the absence of climate change. The interested reader may consult the literature for a more penetrating understanding of the differences.

15.3

Anthropogenic influences on climate change

Anthropogenic influence (or human impact) is a significant component of climate change. The effects of human activity on the immediate environment are widespread and profound. Extensive scientific research from all over the world has pointed to human activity as one of the leading causes of climate change [7]. In the 20th century, there has been increased awareness concerning how anthropogenic factors, e.g., industrial activity, deforestation, mining, and land use changes, to name a few, affect climate of several regions of the world.

15.3.1 Industrial activity Industries utilize a variety of raw materials in fluid and solid forms to produce new products. The byproducts of chemical reactions in these industries are mostly toxic waste matter (effluents) in fluid or solid forms, e.g., carbon dioxide (CO2 ), carbon monoxide, nitrous oxides (NOx), soot, and various sulfates. CO2 , for example, causes seas to be more acidic. Industrial effluents can have adverse effects on marine and terrestrial ecosystems, cause odors, and alter atmospheric conditions. Despite governmental efforts via environmental regulatory bodies, these effluents from industries continue to contaminate the environment on a large scale in developing countries. Besides effluents, direct industrial processes can also contribute to global warming – a significant contribution of climate change. A good example is the issue of gas flaring done by petrochemical companies in Southern Nigeria [8]. Gas flaring is the combustion of unwanted natural gas at petroleum refineries and production facilities. The burned gas releases several air pollutants due to incomplete combustion of hydrocarbons. When these pollutants are released into the atmosphere, they contribute to the “Greenhouse effect.”

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Figure 15.3 Annual rates of global deforestation in 1990–2014 [11].

15.3.2 Deforestation Deforestation refers to the clearing out of a stand of trees or forest for human use. Forests cover about 30% of the Earth’s landmass, and are being cleared daily for urbanization, industrial, or commercial use. Forests help to mitigate climate change by absorbing CO2 (an important greenhouse gas) from the environment and releasing oxygen. There is a delicate balance of CO2 release and consumption which directly affects the Earth’s temperature. Scientific research shows that carbon dioxide is responsible for about 20% of the global greenhouse effect [9]. The felling of trees inversely increases the saturation of CO2 in the atmosphere. Also deforestation exposes the subsoil, making it prone to erosion and destroys critical wildlife habitat, e.g., the unabated deforestation [9] of the tropical rainforests of South America. Sustained deforestation leads to desertification. According to the World Wildlife Forum, about 18.3 million acres of forest is lost every year [10] due to deforestation. Fig. 15.3 graphically illustrates the annual rates of global deforestation in the period 1990–2014.

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Greenhouse gases (GHGs)

A greenhouse gas is any gas which absorbs infrared radiation or heat emanating from the Earth and prevents it from being released back into space. Even slight variations in the concentration of these gases over time can significantly affect global temperature and weather patterns. Some important GHGs are CO2 , water vapor, ozone, chlorofluorocarbons (CFCs), and methane gas. Greenhouse gases induce the “greenhouse effect,” which causes a net increase in the temperature of the Earth’s atmosphere – also known as global warming. It should be remarked that GHGs do not entirely serve negative purposes. In fact, some quantities of greenhouse gases are ever present in our environment and help regulate the Earth’s temperature by preventing extremely low temperatures that would be unsustainable for food production and survival of marine and land ecosystems. The current high levels of GHGs in the environment have been linked to human influence. Effluents from industries, emissions from cars and other means of transportation, coupled with a dearth of recycling facilities for nonbiodegradable materials, are some of the challenges experienced in both advanced and developing countries.

15.4.1 Major greenhouse gases Carbon dioxide – Carbon dioxide (CO2 ) is an atmospheric gas that forms a critical part of several biological processes. It is an important gas in photosynthesis, a process in which it is utilized by plants and green algae. CO2 is produced naturally or as a result of human industrial activity in the form of combustion of fossil fuel and industrial production. It is also one of the most significant GHGs directly responsible for global warming. Its significance stems, in part, from its high stability, availability in high amounts and its high anthropogenic emission rates [12]. Methane – Methane (CH4 ) is the second largest greenhouse contributor after CO2 . It is produced in vast amounts, as a byproduct of natural gas extraction. While it is second to carbon dioxide as a greenhouse gas, the effects over a hundred-year period are expected to be 34 times that of CO2 [13]. This can be attributed to the continued rise in industrial processes, such as natural gas exploration, that have the gas as a byproduct. A 2010 estimate by NASA’s Goddard Institute shows that methane production has gone up by 150% since the preindustrial age [14]. Nitrous oxide – Nitrous oxides (NOx ) are produced naturally in rain forests but can also be emitted by industrial actions in the synthesis of nitric acid and the manufacture of nylon. Like methane, their contribution to the greenhouse effect has also increased in the industrial age with atmospheric levels rising by 16% [15]. Ozone – Ozone pollution is also linked to the burning of fossil fuels. While preindustrial ages also had a small presence of ground-level ozone, concentrations have peaked in the industrial age [16]. The impact of ozone on climate change is only regionally measurable, but assessments of its potency by the Intergovernmental Panel on Climate

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Figure 15.4 Greenhouse effect illustrated [22].

Change reveal that the radiative forcing of ground-level ozone is about a quarter of that of CO2 [17]. Chlorofluorocarbons – CFCs are not naturally occurring compounds and were first synthesized to aid refrigeration. They have since been used as propellants in aerosols. Their detrimental effects on the ozone layer and their tremendous capacity to trap heat, and thus contribute to greenhouse effects, have led to strong regulations against their use [18]. While the use of CFCs is being phased out, their long lifespans in the atmosphere will ensure their effects continue well into the present century [19]. Hydrofluorocarbons (incl. HCFCs and HFCs) were first synthesized to serve as an alternative to ozone layer depleting substances such as CFCs. However, there has been an unprecedented growth in the amount of HFC emission with growth pegged at around 10–15% per year. The damage done by this class of compounds to the climate is also immense. The most abundant HFC is about 1400 times more damaging than CO2 [20]. The Kigali amendment to the Montreal Protocol provides for phasing out the use of HFCs starting from January 2019 [21].

15.4.2 The greenhouse effect The Greenhouse effect is the gradual warming of the lower atmosphere and the Earth’s surface when greenhouse gases are introduced into the atmosphere. Under ideal conditions, a delicate balance of temperature is maintained by a combination of solar radiation (radiant energy emitted by the Sun) and terrestrial radiation (electromagnetic energy emitted by naturally-occurring radioactive elements). GHGs upset this

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balance by trapping infrared radiation coming from the Earth’s surface and preventing it from being released back into space, which increases the ambient temperature. The greenhouse effect is illustrated in Fig. 15.4. Two major indexes for quantifying the impact of greenhouse gases are the Global Warming Potential (GWP) and Atmospheric Lifetime and are discussed briefly in what follows.

15.4.2.1

Global warming potential (GWP)

Global Warming Potential is a measure of how much heat (infrared radiation) a unit of greenhouse gas will trap in the atmosphere over a specified period relative to what will be trapped by the same amount of carbon dioxide (CO2 ). GWP can be estimated over a specific timeframe, ranging from a few decades to several centuries. The lengthy period required for accurate estimation is ostensibly from the residence time of most greenhouse gases.

15.4.2.2

Atmospheric lifetime

Atmospheric lifetime refers to the duration of time a greenhouse gas remains in the atmosphere before being decomposed by chemical processes. This implies that greenhouse gases with higher atmospheric lifetimes will exert a higher degree of warming than another gas with a shorter atmospheric lifetime assuming the same GWP. Besides water vapor, which has a residence time of 9 days, all other GHGs take several years to completely decompose from the atmosphere.

15.5

Earth’s energy budget

The “Earth’s energy budget” is a measure of how much energy flows in and out of the Earth’s climate over a given period, the ratio of energy inflow to outflow, and the resulting effects on climatic conditions. The Earth’s energy budget correlates to Newton’s first law of thermodynamics which posits that “energy can neither be created or destroyed, but only transformed from one form to another.” Thus, this implies that the total heat energy absorbed by the atmosphere and that released into the atmosphere are of equal value. The Earth’s energy budget is an essential index used to determine the degree of global warming and predict future temperature changes that affect climate change. Using an energy budget framework, Loeb et al. showed that top-of-atmosphere radiations in the short wave are most likely to influence temperature increase after the global warming hiatus [23].

15.6

Some climate change trends

In recent decades, climate change has received significant attention across the globe, as the universal trends resulting from global warming are evident and overwhelming. As

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noted earlier, developing countries are more susceptible to widespread adverse effects of climatic change than their developed counterparts. Developing nations also tend to be ill-prepared to combat the economic effects that may be caused by global warming. In developing countries like Ecuador, Brazil, and Indonesia, climate change effects such as gradual loss of rainforests, and in northern Africa, famine due to lower precipitation levels and drought have been observed. Since low-income populations in these developing countries depend almost solely on agricultural proceeds for sustenance, they are expected to bear most of the economic brunt of frequent climatic variations [24]. Besides the effects on food security, climatic variations subject developing countries, particularly those within the African continent, to periods of water scarcity and exacerbation of disease outbreaks. These are projected to become even more likely as the urban population in these regions grow [25]. Also, tropical forests are being lost at a rate of 15 million hectares per year due to various human activities. Tropical forests cover 7% of land on the Earth. These forests store and capture more carbon than any other habitat on land and cool our planet providing food and medicine. Their ongoing loss is at our peril. Together, these trends suggest various proactive measures to address the climate menace are warranted for developing nations. In later sections, more will be discussed.

15.6.1 Modeling of climate change Modeling of climate change is an attempt to simulate the rate, extent, and effects of climatic variations in a given region over a specific period. Climate change modeling helps climatologists, scientists, and researchers to understand the physical, biological, and chemical processes that take place during any climate change event. Modeling of climate change trends necessitates the development of a hydroecological model. The hydroecological model is used to reduce, capture, and quantify tangible parameters that influence climatic variations over time. Climate change modeling enables scientists to quantify climate change developments with a degree of precision. The findings may be made available to the public or through relevant regulatory authorities. Such data and reports may provide directions that may be useful to alter the course of climate variations. Global climate change models (or GCMs) are the most robust tools for modeling climatic variations as they can capture the main climatic change components in three dimensions. GCMs use experiments to deduce the real-life evolution of climate change aspects such as the rate of melting of ice, sea level, precipitation, etc. Each GCM follows a specified structure and format based on the parameter to be tested and available climate change data of the studied region. Computer-aided design plays an essential role in mapping out climate change trends over time. The information obtained from GCMs may provide insights concerning past, present, and projected climatic variations in the studied region.

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15.7 Climate change and the transport sector The transport sector has had a significant impact on climate change in the 21st century. The EIA [26] reports that vehicular transport accounts for 32% of global energy consumption which mostly runs on gasoline and diesel [27]. Diesel engines have better fuel economy than gasoline engines but are far dirtier than spark-ignition gasoline engines. The transport sector is the most prominent contributor to annual global emissions, accounting for about 28% of all greenhouse emissions in 2016 [28]. Road, rail, sea, and air transport vehicles running on fossil fuels emit greenhouse gases and particulate matter which contribute to a warmer planet. Towards curbing the emissions from the transportation sector, renewable energy technologies have a significant role to play. Renewable transportation (electrification, hydrogen fuel cells, etc.) initiatives are being employed to accelerate the rate of transition. In 2018, the global number of electric passenger vehicles increased by 63% as compared to 2017 with more cities transitioning to electric buses. So far, however, renewable energies and policies towards this transition are insufficient and underwhelming [77]. The automobile and marine transport sectors are major emitters and discussed in what follows.

15.7.1 Automobiles Vehicles that utilize renewable sources of energy such as solar and biogas are being promoted in the Western world. For example, the production of electric cars is fast gaining traction in the EU as Europe moves away from fossil fuels. However, the same cannot be said for many countries in South America, Asia, and Africa, where these technologies are mostly nonexistent. Thus, the planet still faces the challenge of utilizing fossil fuels for transport in such a way that the carbon footprint continues to increase. In a study to estimate the impact of the automobile industry on climate change and future trends, Mamalis et al. found that the emission of greenhouse gases (especially CO2 ) will only increase in the coming years [29]. This projection was made based on the most conservative estimates of the growth of the world-fleet of automobiles indicating that the world-fleet will have tripled by 2050.

15.7.2 Marine transport We highlight marine transport as it is seldom discussed. Water transport accounts for 14% of all transport worldwide, moving essential goods between countries. Shipping of goods across the Atlantic, Pacific, and Indian oceans contributes in no small measure to annual global emissions as oil tankers, ships, and boats primarily run on fossil fuels. Prospects of a turnaround in the contribution of marine transport to climate change are also bleak. The world’s fleet of carriers has continued to grow, growing by 3.5% between 2015 and 2016 [30]. The International Maritime Organization estimates that the number of greenhouse gases emitted as a direct result of marine operations could grow by about 50–250% in the coming years if urgent actions to reverse the trend are not taken [31].

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Figure 15.5 Emissions and climate change vulnerability in different regions [33].

15.8 Climate change and the industrial sector While industrialization improves the overall quality of life of people in both technologically advanced and developing countries of the world, it is not without some adverse effects on the environment. Rapid industrialization in countries like the USA, the UK, France, and China has led to a corresponding increase in the emission of greenhouse gases which contribute to a warmer world [32]. Similarly, production activities in the developing world, e.g., in the petroleum-rich region of West Africa and parts of South America release toxic substances into the environment which also contributes to global warming. Fig. 15.5 illustrates emissions and climate change vulnerability in different regions.

15.9 Climate change mitigation and adaptation A CCA program is a series of activities aimed at modifying human and natural systems in the face of current and projected effects of climate change. CCA programs can be implemented at the state level in terms of policies, environmental regulations,

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and advocacy to reduce or eliminate the causative factors of global warming. In this section, a few studies in the literature are reviewed. Researchers posit that at the current rate of environmental pollution, there will be sustained and widespread effects of climate change in every nation of the world. However, while governments set up policies, programs, and regulations to curb human influences that contribute to global warming, they also need to make adjustments and plan towards adapting to the effects of climate change. It has been stated and emphasized that the effects of climate change will lead to unforeseen economic, social, and environmental effects requiring mitigation and adaptation efforts to combat the inevitable menace [78]. Water in particular has been described as the major medium through which the effects of climate change will be experienced, exacerbating the irregularity of distribution of water [78]. The effect of the changing climate on dwindling water resources in Africa has also been evaluated in [79] calling for further research into climate change by African scientists. Also, Nigeria’s capital city Abuja, established in 1976 and with a population estimate of approximately 2 million, has a growing water deficit partly due to climate changes. This has necessitated the need to source for an additional water resource and to raise awareness to change consumption and consumer behavior as two recommended approaches [80]. The water problem has also been discussed for Ethiopia [81]. A much broader discussion of climate change and water has been presented in [78]. Disaster risk reduction in climate change refers to an immediate human response to the effects of climatic change which threaten the environment, livelihood, and lifestyle of people in affected regions. In developing countries like those of northern Africa, climate change may have hazardous effects on the economy and population. According to a report by the World Health Organization (WHO), some of the most impacted people are women and children [34]. A qualitative research analysis in semiarid regions of Ghana on how climate variability is currently been addressed as insight on how they may address it in the future is presented in [82] showing that gender plays a role in decisions towards climate change adaptation along with the need to focus on both climatic and nonclimatic stressors that might produce a particular adaptation practice in a given locality. It has been well established that developing nations will bear the major brunt of climate change effects where progress in this regard is limited by human, financial, and technical resource ability [83]. Also, in the work of [83] applied to the island nation of Trinidad and Tobago, the author proposed the need for coastal sectors to adopt policies that will lead to “climate proofing” of activities for sustainability and other policies whereby if well-grounded would lead to better for the nation. A socio-economic analysis to identify coping mechanisms and strategies to Niger is considered in [84]. It was identified that improving rural road networks or adopting new crop varieties could be a worthwhile coping strategy where the benefits greatly exceed their cost implications. Similarly, extending irrigated areas could usefully contribute to broader policies to combat the effects of the climate. The implications of bioenergy in a climate changing world is examined in [85] where it was shown that biomass, as a renewable energy resource, provides twin solutions to improve food security and social development, and provides a substantial contribution to keeping temperature rise below 2°C. In another study considering Nigeria, climate change

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mitigation through renewable energy utilization using a discourse analysis approach is presented and several policy prescriptions are recommended [86]. The impacts of current and future climate scenarios are evaluated for selected farms in Burkina Faso [87], showing that some current climate interventions considered are insufficient in fully addressing the complex issue of climate and agriculture. In summary, climate change mitigation and adaptation strategies form a vital discourse which has been and continues to be the topic of many meetings by both governmental and nongovernmental bodies. A lack of enabling legislation is one of the most important limitations of climate change mitigation and adaptation, especially in developing countries. An example is the poor recycling culture on the African continent [35]. The lack of the political will to implement policies that shape climate change mitigation and adaptation, and adequate investments will lead to dire consequences for the average citizen.

15.10

Climate change and conflict

There is a strong link between climate change and geopolitical and socioeconomic stability. Climate change has also been closely linked to a likely increase in conflict in many African countries as people battle more intensely for the limited resource of fertile land and water. In both developed and developing countries of the world, internal conflicts may arise due to the pressures of global warming on the environment. Climate change has a considerable impact on the quality of life, economy, health, the socioeconomic, and political structure of affected countries. Some of the most vulnerable regions are the countries of North Africa, where climate change causes widespread poverty and the depletion of critical natural resources needed for survival like water and food [36]. Also, developing countries where the mainstay of the economy is hinged on one or more resources known to be causative factors of global warming may degenerate into conflict when there is a policy change to combat climate change. For example, increased pressure from the developed world to curb the effects of climate change can impact the economy of a developing country with fossil fuels as its mainstays like Venezuela and Nigeria [37]. The lack of a diversified economy, a decline in agricultural activities due to temperature variance, and resultant unemployment can create stresses in the polity. According to UNESCO [38], countries in the “Horn of Africa” are some of the most geopolitically impacted by conflict due to climate change.

15.11 Climate change and agriculture Climate change is a menace that has been knocking on our doors for quite some time. There is a unanimous agreement that climate change is affecting the world. Countering climate change is a pressing concern for developing nations, even more so in developing countries with a large population like China and Nigeria. Also, African

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nations are extremely vulnerable to the potential impacts of climate change compared with other continents, though the continents contribution of greenhouse gases is the smallest. African countries contribute just 3.8% to the world’s greenhouse gas emissions (China alone contributes 23%, while the US produces 19%). Yet the developing continent of Africa will bear the brunt of the effects of global warming. Droughts, heat waves, and floods are now frequent occurrences on the African continent and other developing regions of the world. Agriculture is a mainstay of many developing economies and the livelihood of many in the developing world. Climate change caused by human activities continues to have a massive impact on planet Earth’s ecosystem, influencing both physical and socio-economic activities. The impacts of climate change in Africa continues to increase. Over the last 50 years, animal populations, for example, have decreased by about 60% due to climate changes. Rapid population increases also continue to place undue stress on food production yield. The physics and science of climate change and crop physiology suggests climate change is an immense threat to the sustainability of global food production [94]. It is still not clearly understood, for example, how the spatial and temporal variation of rainfall, surface and ground water transport for irrigation, evaporation rates, and other factors influence different climatic regions. Agriculture is certain to be one industry that would be significantly affected by climate change. Perceptions of farmers and a regression analysis was performed for the Malaysian agricultural sector in [95] presenting barriers that hinder adaptation, showing that no current specific policy will aid in limiting climate change effects, while establishing policy measures that might be more effective in tackling climate issues. One key takeaway from this work is the need to improve farmers’ understanding on climate change adaptation to ensure sustainability of produce, their livelihood, and environment. Researchers in Malaysia explored several adaptation strategies for small-scale fishermen such as managing fishermen’s climate change knowledge, involving fishermen in climate change adaptation planning, and managing risks with routine fishing practices [96]. In addition to this, funding provisions for fishermen is also important according to a statistical study of perspectives of fishermen in Oyo State, Nigeria [97]. The impacts of current and future climate scenarios are evaluated for selected farms in Burkina Faso [98], showing that some current climate interventions considered are insufficient in fully addressing the complex issue of climate and agriculture. Climate change effects on cocoyam farmers in southeast Nigeria is studied in [88] identifying eight major challenges on of which is the poor access to information and resources to effectively mitigate and/or adapt to the effects of climate change. Considering Benin in Nigeria, Bayesian statistics is employed in [89], showing that pineapple, maize, groundnuts, cassava, and cowpeas will face harmful effects with an average yield reduction in the range of 11–33% by 2050, whereas sorghum, yam, cotton, and rice will benefit from climate change with an average yield gain of 10–39%, largely due to an increase in temperature. Climate change and food security concerns in Tanzania have been reported in [99], concluding that current agricultural practices cannot ensure long-term food security since they rely on increasingly erratic rainfall. The work called for more multidisciplinary research to enhance and develop potential coping and adap-

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tation strategies that have positive impacts on food security. Similar conclusions are found for Ethiopia [100]. A mathematical analysis of climate change and crop choice in Zambia is analyzed in [101], concluding that large scale interventions are needed to help farmers cope with the necessary adaptation strategies. In another analysis on climate change impact to maize yield and biomass in Ghana, the researchers show that there is an increase of 57% and 59% of maize yield and aboveground biomass, respectively, under different scenarios, due to the positive effect of CO2 and reduced water stress due to lower atmospheric water demand during crop growth period. It was also noted that water, which is also under threat due to a changing climate, is the limiting factor for maize production [102]. For a more detailed picture on the impact of climate change for agriculture and farmers in Africa, the reader is referred to the thorough works in [103–105]. In many ways, agriculture and agricultural production must innovate to adapt to changing climatic conditions. Solving climate change issues in Africa means solving some of the many economic issues that have plagued and continue to plague many of the nations on the continent. In [106], the authors conclude that lack of structural transformation in Africa continues to limit the capacity to respond to climate change. In another research work [107], the authors concluded that for local institutions to adapt to climate changes is to ultimately deal with many developmental challenges.

15.12 The role and integration of renewable energy technologies Renewable energy technologies will play a vital role in mitigating the effects of climate change. It is predicted that sustainable sources of energy such as solar and wind will become the mainstay as the world moves away from overreliance on fossil fuels which contribute to global warming. The continued burning of fossil fuels releases several greenhouse gases such as water vapor, CO2 , and chlorofluorocarbons (CFCs) into the atmosphere leading to problems like ozone layer depletion [39], and rise in global temperatures, as all mentioned earlier. Sustainable sources of energy (or renewables) such as solar, wind, hydro, hydrogen, and geothermal hold the key to vast amounts of energy which can replace fossil fuels. Hydrogen, in particular, has a high energy storage density per unit mass. Hydrogen can be stored in underground caverns and its economic benefits (overhead) compared to battery technology is compelling. According to the NREL, hydrogen produced from natural gas has a significant carbon footprint of about 8.62 tons of CO2 for each ton of produced hydrogen. Other technological challenges of hydrogen are close to being solved. As of 2018, renewable technologies now make up more than one-third of the global installed power capacities, and 26% of global electricity produced. Solar photovoltaic (PV) technology leads in renewable capacity additions as of 2018. Solar had 55% new additions, wind had 28%, and hydropower had 11% in 2018. Together, China leads the way in installed capacity of renewable energy, specifically, hydropower, solar PV, and wind. China ac-

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counted for 32% of all global renewable investment in 2018 [77]. Even oil-rich Nigeria continues to make effort towards renewable energy [90]. The now popular nighttime NASA composite photograph [91] which shows the continent of Africa in darkness continues to be a stark reminder that as of 2016, the electrification rate is 42% according to World Bank estimates. This is partly due to most of the power utilities being effectively bankrupt, unable to generate enough revenue to cover their expenses. Notably Africa and Southeast Asia are two regions of the world with low electrification rates. Beyond the climate benefits, there is significant value in the diversity of electrical power generation sources in terms of the various customers and the grid. The growth of clean energy offers a comprehensive solution both to developed and developing nations. The electrical energy sector continues to dominate the modern energy transition through its uptake of renewable forms of energy, most notably solar and wind. The heating and cooling sector, however, and the transportation sector are still lagging in their transition to cleaner forms of energy [77]. Many nations in the developing world have the alternative energy resource potential to end their fossil fuel dependence, and China in particular continues to demonstrate the transition away from carbonaceous fuels. A significant hurdle to integrating renewables on a large scale is cost constraints. Despite the fact that renewable energy costs are falling drastically and are in some regards cheaper than fossils such as coal, the cost may still be prohibitive for several developing nations. The renewable energy industry in developing nations will require more enabling policies and increased funding for indigenous research if they are to replace fossil fuels. BP’s 2018 energy outlook suggests renewable energy will account for about half of the global increase in power generation through 2040 [93]. The future energy landscape is moving towards being cleaner and more distributed. A key driver towards this effort will be in electrifying transportation and decarbonize the environment significantly.

Part 2 – Climate changes in selected developing nations

According to the World Economic Forum and Global Cities Institute, in 100 years the world’s biggest cities will be in Africa. The city of Lagos in Nigeria is projected to be the largest city in the world with 88.3 million people while Kinshasa in Democratic Republic of Congo will be the second biggest city with 83.5 million people. Dar es Salam in Tanzania will be in third place with 73.7 million inhabitants. The growth of African cities is driven by urbanization, with more than 70% of Africans being under the age of 30. Africa’s population is expected to grow to more than 2 billion by the year 2050, whereby more than 80% of that increase will occur in cities. Several of the proposed solutions to the menace of global warming are expected to be favorable in terms of their economic and political impacts. A brief discussion of the dynamics of climate changes in selected developing countries is in what follows.

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Figure 15.6 The 10 with the largest reductions and increases of CO2 [43].

15.13

Climate changes in China

Located in Eastern Asia, China is known as the largest country on the Asian continent. It is currently the most populated country in the world [41] and covers most of the landmass in Eastern Asia, which is approximately one-fourteenth of the total land mass on the Earth. Over the last few years, China has been greatly affected by climate change. This positioning is largely due to human activities, greenhouse gases, and fossil fuel, especially since the country is one of the largest emitters of CO2 [42]. Fig. 15.6 depicts the ten countries with the largest reductions and increases in CO2 in millions of tons for the year 2017, whereas Fig. 15.7 depicts energy consumption by energy source in China between the years 200 and 2017. Like other countries affected by global warming, China has in the last century noticed significant changes in surface temperature. The most significant changes have been the rise of average temperature from 0.5°C to 0.8°C, on the same level as the global temperature of 0.8°C [44]. The eastern, western and northern regions of China have experienced the most significant changes in temperature, while the south, though affected, was not as consequential. It is expected that the average temperature will continue to rise if adequate measures are not taken to mitigate it. The most profound effect of climate change has been on the health of Chinese citizens, especially with the increased mortality rate from extreme weather events and changes in air and water quality, as well as alterations impacting the evolution of infectious diseases. Cities such as Beijing and Shanghai have been most affected, as a result of the extreme weather conditions, particularly heat waves. This has been attributed to cardiovascular and respiratory diseases, notably in residents above 60 years. This trend is likely to continue since there is a paucity of information available on the constant changes of climate and its impacts such as thermal extremes, which would affect preexisting health status and population mortality [45]. The unpredictable temperature changes are leading to unfavorable conditions which also affect food production

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and supply. According to researchers, the yield from harvests is expected to decrease by 8% in developing countries in South Asia and Africa experiencing climate change by 2050. This has proven to be accurate [46], with China having to meet up with grain demands through import. The trend might likely continue if the gap between the country’s food demand and national food production widens. In 2007, China became one of the largest emitters of CO2 . In efforts to combat climate change, the Chinese government adopted several strategies to reduce emissions. The measures and strategies have proven quite effective since 2008. In 2016, the Chinese government approved a climate change five-year plan, which is currently ongoing. The key focus of the plan was to improve the quality of life by environmental protection of its citizens, through the reduction of CO2 emissions, energy and water consumption [48]. Other strategies in the plan include a focus on improving the quality of air and new regulations in place to substitute China’s place in the world’s emission landscape from pollution-based industrialization to cleaner, sustainable, and more environmentally friendly nation. It is also important to note that in 2017 [47], after 3 years of no emissions growth, there was a global increase in the emissions of CO2 and, while other countries failed to mitigate the effects of this rise, China rose to the occasion, ahead of predictions that there will be an increase in its CO2 emissions by 2030. China announced in 2015 that it intends to drastically reduce its quota of CO2 and other greenhouse gases before then. Unfortunately, experts believe China will experience the height of climate change by 2025 or sooner. After the implementation of the five-year plan between 2011–2015 by the Chinese government, a low carbon emission model was introduced through a reduction in the production of fossil fuel and more focus on renewable energy. As a result of heavy investment in renewable energy, China has become one of the leading producers of all kinds of renewable energy including wind, solar, and even hydro energy [50]. This reduction in fossil fuel capacity led to a decrease in industrial energy production, thereby fostering an environment of cleaner energy. As a result, China’s CO2 emissions have grown at a much slower rate, when compared to the older climate model, which would have seen a much higher rate of CO2 emissions. The significant strides China has been able to make is largely due to environmental authoritarianism, which allows a few agencies to enforce laws to ensure the success of these policies. Furthermore, this system means there is nonparticipation from businesses and citizens, allowing for smooth execution of policies. Some of these strong-arm approaches include refusal of bank loans to polluting companies, cutting off power to shut down these businesses to achieve a reduction in fossil fuel and the introduction of emissions trading. As a result of all of the above, China is likely to see an improvement in its environment, as these policies continue to be enforced [51].

15.14

Climate changes in Malaysia

Malaysia is a country located in Southeast Asia, north of the equator. With an estimated population of 32 million, it covers 126,854 km2 and is made up of Peninsular

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Figure 15.7 Annual energy consumption by energy source in China from 2000–2017 [49].

Figure 15.8 Trend of CO2 emission by sector in Malaysia (1990–2030) [53].

Malaysia (40%) and Malaysian Borneo (60%) [52]. Like other developing countries, the climate in Malaysia is changing in line with global climate change. The country experiences tropical rainfall characterized by high temperatures and rainfalls all year round. The debate on climate change in Malaysia is still somewhat muted, despite evidence of impacts such as more frequent floods and rising sea levels and temperatures. Malaysia has been greatly impacted by climate change in the last few decades, as the country is one of the biggest contributors of CO2 and the 26th largest emitter of GHGs in the world. CO2 emissions by sector in Malaysia are shown in Fig. 15.8. The average temperature of Malaysia has increased from 0.6°C to 1.2°C in the last half-

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century and will further rise to 2.0°C by 2050 if swift and effective policies are not put in place to decrease the effects of these climate changes [54]. Like other Asian countries, such as Thailand and the Philippines, Malaysia has experienced severe rainfall and rise in sea levels and river flows, and this trend is likely to continue and perhaps even increase. The climate changes pose a threat to the population’s health, especially communities located in coastal regions, which are more likely to be impacted by flooding as a result of the rise in sea level. Specifically, it is projected that 85,800 people will be at risk of river floods every year by 2030. It is expected that people will also be affected by the rise in mean annual temperature, causing heat-related medical challenges. Children, the chronically ill, and the elderly are more at risk of being affected by these conditions. It is also projected that there will be a severe impact on health and rise in mortality rate caused by drowning, disease outbreak, and vector distribution. If these high emissions continue, senior citizens (65+ years and above) are at risk, and mortality rate could be as highs as 45 deaths per 100,000 by 2080. In addition, experts fear that there will be a long-term effect of flooding such as post-traumatic stress [55]. Reduced agricultural productions is another outcome of climate change. The unreliability of the weather, high temperatures, and floods cause a breakdown in food systems, leading to food insecurity. The most important region for agricultural production in Malaysia is the Cameron Highlands, which experiences high rainfall and as a result, agricultural activities thrive in this region. Unfortunately, due to rainfall, this region is at risk of soil erosion and landslide impact. In order to combat this, farmers will need to employ the use of an irrigation system, to diminish potential soil loss [56]. Climate factors could cause a decline in the yield of rice, a staple food in Malaysia. This could be a drastic decrease from 13% to 80%. In addition, cash crops, especially oil palm, rubber, and cocoa, would also decline from 10% to 30% due to the negative impacts of climate change. This is certainly a cause for worry for the Malaysian government, as the population is expected to rise significantly in the next few decades and would provoke an increased demand for food. While policies have been implemented to decrease the effect of climate change, Malaysia would also need to intensify its production of food or be at risk of famine [57]. To combat climate change, the Malaysian government has in the last few years set up policies to address the menace that is considered to be a global issue. These policies serve as directives for the general public, government agencies, and NGOs on how to manage issues related to climate change. These policies are: (a) The National Policy on Environment First enacted in 2002, this policy is focused on ways to develop the economy through the sustainability of natural resources. The main aim of this policy is to create a clean, safe, and healthy environment for present and future indigenes, as well as conserve the country’s natural resources. (b) National Green Technology Policy Implemented in 2009, this policy was established to influence the environment and reduce climate change through mandates for sustainable energy (for instance, fewer emissions of CO2 ), sustainable transport and building practices, as well as better waste management.

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(c) National Renewable Energy Policy and Action Plan This policy was established in 2010, with the aim of reducing the emissions of greenhouse gases and pollution of the environment, to lessen the effects of climate change. To do this, the government set up the enabling infrastructure to promote renewable energy and a cleaner environment. (d) Renewable Energy Act 2011 This act was created to promote the longevity of renewable energy investment among individuals and companies [58].

15.15

Climate changes in Nigeria

Referred to as the giant of Africa, Nigeria is the most populated country in Africa, and the 7th most populated country in the world, with a population of approximately 200 million. It is located in the humid tropics and constitutes a land area of 923,850 km2 [59]. As one of Africa’s biggest exporters of oil, Nigeria is plagued with meeting energy demands and addressing climate changes. These variable weather conditions have led to floods (as a result of off-season rains) and scorching hot and dry spells, affecting food production and distribution in the last few decades. Since 1901, Nigeria has seen an increase in temperature. This increase in temperature trend was steady till the late 1960s, but in the early 1970s, there was a spike in temperatures, which has continued till now. In the period from 1901 to 2005, the mean air temperature was 26.6°C. The average mean temperature increase was 1.1°C, significantly higher than the global mean temperature increase of 0.84°C [60]. If the situation remains as it is, climatic variability will continue and the mean annual temperature could rise as high as 4.9°C by 2100. However, if stringent and effective measures are taken to reduce emissions, the nation may only see a temperature rise not higher than 1.4 C. These possible fluctuations in climate will affect the population, as an average of approximately 548,300 people is projected to either be displaced by flood or lose life or property between 2070 and 2100, as a result of the rise in sea level. On the other hand, if emissions reductions and better drainages are constructed, an average of approximately 300 people will be affected. Another cause for concern is how the reduction of fossil fuel (one of the ways of combating climate change) will affect the Nigerian economy. Fossil fuel emissions in Nigeria as compared to other nations are depicted in Fig. 15.9. Though Fig. 15.9 may suggest low emissions compared to other nations, the expected population increase and urbanization will alter the magnitude of Nigeria’s emissions in the coming years. At present, the bulk of Nigeria’s revenues are earned from the oil sector and this includes 90% of foreign revenue (obtained from foreign exchange), 95% of export earnings, and 80% of the government remuneration. However, in the last four years or thereabout, Nigeria has made efforts to diversify its economy and shift focus from oil exportation, with particular preference being given to natural gas and agriculture. Despite these efforts, Nigeria has nonetheless been significantly affected by climate change, and this effect continues. Some of these impacts include rising temperatures,

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Figure 15.9 Climate change contributions by Nigeria relative to other nations [61].

extreme weather conditions, and rising sea levels. Their consequences are heat stress, flood, ultraviolet radiation and certain diseases dependent on climates such as malaria, typhoid, diarrhea, and cholera. The northern and southern parts of Nigeria are greatly affected, and one of the biggest threats posed by climate change is variability, which is a detriment to subsistence farming. In efforts to overcome these changes, farmers are developing new and alternative agricultural practices, which so far seems to be working. While the effects of climate change can be seen in virtually all parts of the country, it should be remarked that Nigeria, like most developing nations, is not prepared for its impacts. By perhaps share stroke of luck, the country has not experienced major natural disasters, despite severe sea level rise along its coasts. Though people have been affected by floods and high temperatures, one cannot accurately give a figure as a result of the country’s lack of data collection and data management. Realizing the impact that climate change can have on the environment and the economy, the Nigerian government established structures such as the National Adaptation Strategy and Plan of Action on Climate Change for Nigeria (NASPA-CCN) to combat what has now become a global issue. Furthermore, a special unit was created to research, manage, and support climate change activities within the country. In addition, the government has also established a National Climate Change Policy, as well as the Nationally Appropriate Mitigation Action (NAMA) program. The vision of NASPA-CCN and NAMA is to create an environment that fosters climate change adaptation and sustainable growth of the Nigerian economy, as well as decrease susceptibility to the effects of climate change. The action plan includes strategies for agriculture, freshwater resources, coastal water resources, forests, biodi-

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versity, sanitation, human settlements and housing, energy, transportation, industry and commerce, disaster, migration and security, vulnerable groups, and education [62].

15.16 Climate changes in Brazil Located in South America, Brazil has a population of 200 million and is the 5th largest emitter of greenhouse gases in the world as a result of agricultural activities (including land use change and forestry), making the country the largest contributor of GHG emissions. This positioning for Brazil is largely due to having one of the largest ecosystems and forests in the world. This is called Amazon. Agriculture is an integral part of Brazil’s economy. In 2016, it made up 5.5% of the country’s Gross Domestic Product (GDP) and also provided jobs for 18 million people. While great strides have been made in the sector in the last few years, climate changes such as flooding and droughts pose a huge risk to the progress that has been made. Brazil has about five million farms and 85% of them are family owned small farms, who have limited or no resources to combat the onslaught of climate change. This ultimately affects their annual yields [63]. Although Brazil is currently able to meet food demands as a result of its robust food production, this may be greatly diminished by climate change in the future. A decline in food production will invariably cause food insecurity and may even lead to famine. By 2030, Brazil may lose about 11 million hectares of land. Fig. 15.10 illustrates deforestation in the Amazon between the years 2004 and 2018. Fish farmers are also at risk of loss as a result of climate change as a result of elevated ocean temperature, which leads to current change and could cause a decline in maximum fish catch. As expected, one of the impacts of climate change in Brazil is climate-sensitive diseases. Typically, high temperature and increased flooding, allows such diseases to thrive. Coastal regions and communities by the Amazon basin are particularly at risk of suffering these health-related conditions as a result of heightened flooding caused by high sea level. Malaria is currently the most prevalent disease in the central, northern, western, and Amazon regions of Brazil. In 2014, the government disclosed that there over 150,000 cases of malaria. By 2070, it is projected that as a result of climate change 126–168 million people will be at risk. An increase in vector-borne diseases is expected as well, as the temperature rises in the coming years, as some of these diseases thrive in regions with high temperature. For instance, when there was an outbreak of Zika virus in 2015, it was largely spread as a result of weather conditions that favored it. There is also a genuine concern for heat-related medical conditions. Chronically ill people, children, and the elderly are at risk. These conditions may also affect the economy and human health, leading to reduced productivity, which could also lead to food insecurity and malnutrition. Also raising concern for Brazil is the tourism sector, which could be impacted by high temperatures and floods. In 2010, the southeastern region of the Amazon suffered from drought, which also affected river transportation, severely impacting the GDP for

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Figure 15.10 Deforestation in the Amazon from 2004–2018 [64].

that year. Being one of the top countries with high GHG emissions, Brazil has a pivotal role to play in combating climate change. This is a positioning that Brazil has gladly embraced, with the creation of the Inter-Ministerial Committee on Climate Change (CIM) and the National Policy of Climate Change (PNMC), which promote low carbon agriculture and renewable energy. Managed by Casa Civil, CIM comprises the Brazil Climate Change forum (FBMC), as well as 17 federal organizations. The CIM was created in 2007 with the aim of executing, monitoring, and managing the National Plan on Climate Change. While the FBMC was created to inform and educate the public on climate change issues, the PNMC was established in 2009 to substantially reduce greenhouse gas emissions in Brazil by 2020. Some of the strategies within this policy include promoting alternative energy sources, decreasing deforestation in the Amazon and Cerrado by 80% and 40%, respectively, and promoting afforestation at municipal, state, and federal levels. To execute this policy, Brazil needed to create the Climate Fund, which allocates resources such as grants annually to mitigate climate change and also promote adaptation. Also, Brazil is also a member of the Clean Development Mechanism (CDM) under the Kyoto Protocol. Through this, Brazil can earn certified emission credits which can be traded and used to meet up with its emission targets. Together with the initiatives mentioned above, the Brazilian government has also made efforts to promote a sustainable environment through the use of renewable and efficient energy use [65].

15.17 Climate changes in India India is located on the Asian continent and is a densely populated country with over 1.3 billion people. In the last few years, the Indian government has made major strides in the economy by diversifying. This has led to an increase in goods and services production, in other to increase the GDP. Some of these are tied to energy consumption, hence the emission of greenhouse gases. CO2 contributions by India in the period 1980–2019 is shown in Fig. 15.11.

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Figure 15.11 CO2 contributions by India (1980–2019) [66].

The most prominent season in India is the summer monsoon, which occurs annually between June and September and makes up 80% of the country’s precipitation. The precipitation in India differs in different regions. For instance, in the northeast, the precipitation may be as high as 11,000 mm, while in the northwest there is a stark contrast, with the region experiencing 130 mm or less. Protecting the country from the low temperatures of the north is the Himalayan mountain range. The annual average temperature in different regions of the country varies. In the south, the average mean temperature is 28.8°C, while in regions like Kashmir and Jammu, it’s as low as 2.4°C. India also experiences extreme weather conditions which could be as low as −4.5°C and high as 51°C, depending on the region. As might be expected, this unpredictability and changes in climate affect the Indian citizens directly or indirectly. For instance, in 2015, the country saw a heatwave that caused 2300 deaths. Again, in the following year, the country experienced a drought in its northern region and this led to a heightened temperature of 40°C. The implications of this were: there was a rise of heat-related conditions including stroke and exhaustion, which influenced the country’s mortality rate. India is also experiencing a rise in sea level as a result of heavy rainfall in major cities like Delhi and along its coasts. This puts these regions at risk of flood. In 2013, Uttarakhand in the northern area experienced a fatal flood that claimed over 5,500 lives and affected 4,200 villages. In addition to deaths caused by flood, there are also deaths caused by climate-sensitive diseases such as diarrhea and cholera. In the next few years, the spread of diarrhea is expected to increase by 21% as a result of favorable weather conditions for the pathogen. Currently, diarrhea is a major detriment to child mortality in the country, with 300,000 deaths per year. This figure is expected to rise as the diseases become more widespread. A large percentage of Indians are directly or indirectly dependent on agriculture for their livelihood. Some of them lose a large sum of their livelihood as a result of climate change. India’s most popular crops are rice, wheat, and millet. With the heightened temperatures likely to cause droughts, India’s agricultural sector is projected to lose US$7 billion by 2030. Experts say a temperature increase above 1°C will greatly

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impact crops by reducing harvests and even quality. The projections by 2050 indicate a 50% reduction in wheat yield as a result of a severe increase in temperature [67]. However, India embarked on climate change adaptation fairly early with awareness and adaptation strategies. In 2002, at the United Nations Framework Convention on Climate Change held in Delhi, India was champion on a joint declaration on the impact of global warming on the world. The National Action Plan on Climate Change (NAPCC) was introduced in 2008. This plan detailed how India would combat and adapt to climate change. The core focus of the plan includes sustainable energy production and efficiency, housing, ecosystem preservation of Himalayas, afforestation and adaptive agricultural practices. Ministries with different objectives have been tasked with the responsibilities to create execution plans, timelines and key performance indicators, which will be presented to the Council on Climate Change appointed by the Indian Prime Minister. The Council will then evaluate, ascertain and develop a report on each ministry’s progress [68].

15.18

Climate changes in South Africa

Located in southern Africa, South Africa is a middle-income developing country and has the third-largest economy on the continent. Despite this positioning, the country faces a high rate of unemployment and poverty. Since its independence in 1994, the country has tried to tackle these social ills. South Africa is one of the largest emitters of greenhouse gas. In 2000, it was estimated that the country emitted 461 million tonnes. Much of its greenhouse emissions are as a result of human activities including but not limited to agriculture, waste, energy supply and consumption, as well as industrial processes. From 1994 to 2000, emissions increased by 28%. This largely due to industrialization. For the same period, agricultural-related emissions also increased by 9% while emissions from waste declined by 43%. As a result of its diverse topography, the country has a similar climate to deserts and subtropical zones. Climate change is projected to increase temperature, rainfalls, and sea levels. This could impact the population as regards food and water insecurity, as well as changes in the ecosystem. It has been observed that surface air temperature has changed over time since 1950 and has sometimes surpassed the global mean temperature. Also rising is the sea level along the country’s coasts, though they vary by regions. In the east coast, the sea level sees an increase of 2.74 mm per year, while the west coast sees 1.877 mm per year and the south 1.47 mm. Though the real impact of this remains unclear, it is expected that there will be a high tendency for flood as the sea level continues to rise annually. It is expected that in the coming years, the emission of greenhouse gases will increase within the country and along its coasts. As the effects of climate change continue to increase, it has been projected that the surface air temperature will likely rise to 6°C–7°C within the country and 3°C–4°C along its coasts within the next 50 years [70]. Annual temperature trends of South Africa are depicted in Fig. 15.12.

Figure 15.12 South African annual temperature trends [69].

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The impact of climate change in South Africa will mostly be felt on agriculture, affecting crop yields as a result of the shortage of water supply, as well as water quality. Large-scale farmers are able to overcome this impact through irrigation. Small-scale farmers, on the other hand, remain vulnerable without irrigation infrastructure. However, current projections indicate that there will be an expansion of areas suitable for cultivation, which, of course, would improve food production. The South African government has made strides in combating climate change, especially since it was discovered that over 80% of its greenhouse gas emissions were as a result of energy production and consumption. To this end, there have been interventions such as strategies to ensure efficient energy supply, long term mitigation policies, as well as another policy in response to climate change. In addition, the Climate Support Program (CSP) was created to ensure the creation, execution, and review of policies. Furthermore, South Africa’s Climate Change Bill (later Climate Change Act) made major strides in 2018, with research and consultation taking place in regions in the country. This will support the progress of the national climate change legal framework. The Climate Change Bill is a well-detailed response to the threat of climate change within the country [71].

15.19

Climate changes in Ecuador

Boarded by Peru, Colombia, and the Pacific Ocean, Ecuador is a South American country located on the west coast and known for running through the equator. Hence, its name “Ecuador” (Spanish for equator) [72]. As a result of its unique positioning and topography, Ecuador experiences varied shifts in weather in different regions, such that inhabitants with high altitude may observe variants in temperatures that may go beyond 20°C during the day and at night. Just like other countries located in the tropics, precipitation also varies in Ecuador. Wet trade winds from the Amazon basin and tropical Atlantic have a huge effect on the eastern slope. Similarly, in the northwestern region, precipitation is created as a result of moisture being transferred from the eastern Pacific. In contrast, the southwest slopes are impacted by the Peru Current, which receives air from South America’s west coast and brings to Ecuador’s north coast. Due to these variables, the west slope has lower maximum annual precipitation of (2000–2500 mm), while the east side has (400–800 mm). The climate change in Ecuador is in line with global climate change assessments, which projected an elevation in temperature. These projections have proven to be true, as Ecuador has seen a major change in temperature in the last five decades. This increase in temperature has been projected to continue. By 2100, Ecuador is likely to see a surface air temperature rise from 2.2°C to 3.0°C. These changes in temperature will vary in different regions. Eastern lowlands will experience higher maximum and minimum temperatures, while on the northern coast, there will be a significant decrease in temperature. Due to this, cities in the highlands are likely to experience a decline in water supply, as well as increases in temperature, which may cause heat stress. Other anticipated impacts are an increase in heat-related conditions, pollution, migration, increase in energy demand, decrease in food production and inflation. Similar to

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temperature, the seal level of Ecuador is also projected to rise as global temperatures increase. Experts say that by the end of the century, there will be 1–3 m rise and maybe even a potential surge of 5 m. Based on the country’s topography, this change in sea level could affect 20% of the population assuming 3 m rise, and 12% of the population assuming 1 m rise. Some of the impacts of this could include loss of coastal and recreational amenities, loss of beaches, and loss of fish breeding areas in rural areas. In addition, it could also impact agriculture, particularly shrimp and rice, as areas for cultivation could be destroyed as a result of excess water. Other projections include altering of the ecosystem and floods that huge impact on life, property, and the Ecuadorian economy. For instance, Ecuador felt the brunt of climate change between 1997 and 1998, when it experienced a flood that cost the country over US$3 billion in losses, 11% of the country’s GDP for both years. The Ecuadorian government will need to make it a point of priority to develop specific and not generic climate change strategies that will combat these changes in the environment [73]. To that end, the government has made efforts to protect the environment for so many years. Internationally, Ecuador signed The United Nations Framework Convention on Climate Change in 1994, embarked on Clean Development Mechanism ratified by the Kyoto protocol in 1999, and signed the Climate Change Paris Agreement in 2016. Locally, Ecuador developed National Environmental Policy, National Development Plan, and National Climate Change Strategy. While Ecuador may have strong legislation, it is also important to note that addressing climate change would require major or total transformations in the way of life of the indigenes, in order to cope with and effectively combat climate change [74].

15.20 Conclusion To survive climate change, we have to adapt to our environment, which implies that we have to evolve from business as usual or the way we have always done things. Without a doubt, climate change has a significant impact on developing countries and will continue to do so unless significant steps are employed. Without several steps and initiatives, developing countries may find themselves at risk of health-related conditions and loss of life and property, thereby impacting their respective GDP. To mitigate the effects of climate change, countries will need to take drastic, but necessary steps to reduce the emission of greenhouse gases and encourage adaptation. Developed countries have a role to play as if they intensify efforts to reduce greenhouse gases emission, developing countries are more likely to follow suit. Developing countries with rapidly developing economies such as China and India will need to make further reductions in greenhouse gases the focus, rather than adaptation to climate change. Another direction to reducing emissions of greenhouse gases is through the continued adoption of renewable and sustainable energy technologies. Some other actions may also include increasing energy efficiency, improving agricultural practices, and reducing deforestation. While bodies such as the European Union (EU) have made efforts to ensure the reduction of these emissions through schemes

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like Clean Development Mechanisms (CDM), individual countries will need to ensure the sustainable production of goods, to greatly reduce climate change, not for protectionism but for a cleaner, friendlier environment. Perhaps, the key for developing nations is innovations, stretching from basic and applied university research to dissemination of information in the market place; this will be key in developing local technologies to meet the challenges of a carbon-constrained world. Another area that has been severely overlooked is adequate funding to support all countries coping with the effects of climate change. While the Adaptation Fund (established in 2010) has committed over half a billion US dollars to climate adaptation in developing countries, more funds in this regard need to be established, to enable developing countries adequately cope with climate change as, according to projections, it is only likely to get worse if adequate measures are not taken.

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Some mathematical background of fluid mechanics

A

Felix Ilesanmi Alaoa,b , Samson Babatundec , Zounaki Ongodiebid a Department of Mathematical Sciences, University of Texas, Richardson, TX, United States, b Federal University of Technology, Akure, Nigeria, c Department of Mathematics, University of Texas, Dallas, TX, United States , d Department of Mathematics and Computer Science, Niger Delta University, Bayelsa, Nigeria

We review the following mathematical definitions which are essential in the differential formulation of the basic laws of fluid mechanics. − → (a) Velocity Vector V . Let u, v, and w be the velocity components in the x, y and − → z directions, respectively. The vector V is given by − → V = ui + vj + wk.

(A.1)

(b) Velocity Derivative. The derivative of the velocity vector with respect to any one of the three independent variables is given by − → ∂V ∂u ∂v ∂w = i+ j+ k. ∂x ∂x ∂x ∂x

(A.2)

(c) The Operator ∇. In Cartesian coordinates the operator ∇ is a vector defined as ∇≡

∂ ∂ ∂ i+ j+ k. ∂x ∂x ∂x

(A.3)

In cylindrical coordinates this operator takes the following form: ∇≡

1 ∂ 1 ∂ ∂ ir + iθ + iφ . ∂r r ∂θ r sin θ ∂φ

(A.4)

− → (d) Divergence of a Vector. The divergence of a vector V is a scalar defined as − → − → ∂u ∂v ∂w + + . div. V ≡ ∇. V = ∂x ∂y ∂z

(A.5)

(e) Derivative of the Divergence. The derivative of the divergence with respect to any one of the three independent variables is given by

∂  − ∂ ∂u ∂v ∂w → ∇. V = + + . (A.6) ∂x ∂x ∂x ∂y ∂z

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Some mathematical background of fluid mechanics

The right-hand side of (A.6) represents the divergence of the derivative of the − → vector V . Thus (A.6) can be rewritten as ∂ ∂  − → ∇. V = ∇. (ui + vj + wk) , ∂x ∂x

(A.7)

− → ∂  − ∂V → ∇. V = ∇. . ∂x ∂x

(A.8)

or

(f) Gradient of Scalar. The gradient of a scalar, such as temperature T , is a vector given by GradT = ∇.T =

∂T ∂T ∂T i+ j+ k. ∂x ∂y ∂z

(A.9)

(g) Total Differential and Total Derivative. We consider a variable of the flow field designated by the symbol f . This is a scalar quantity such as temperature T , pressure p, density ρ, or velocity component u. In general, this quantity is a function of the four independent variables x, y, z, and t. Thus in Cartesian coordinates we write f = f (x, y, z, t) .

(A.10)

The total differential of f is the total change in f resulting from changes in x, y, z, and t. Thus, using (A.9), df =

∂f ∂f ∂f ∂f dx + dy + dz + dt. ∂x ∂y ∂z ∂t

(A.11)

Dividing through by t, we have df Df ∂f dx ∂f dy ∂f dz ∂f = = + + + . dt Dt ∂x dt ∂y dt ∂z dt ∂t

(A.12)

However, dx dy dz = u, = v, = w. dt dt dt

(A.13)

By putting Eq. (A.12) into (A.11), we have df Df ∂f ∂f ∂f ∂f = =u +v +w + , dt Dt ∂x ∂y ∂z ∂t

(A.14)

Df where df dt is called the total derivative, which is also written as Dt . It represents the change in f which results from changes in the four independent variables. It is also referred to as the substantial derivative. Note that the first three

Some mathematical background of fluid mechanics

475

terms on the right-hand side are associated with motion. They are referred to as the convective derivative. The last term represents changes in f with respect to time and is called the local derivative. Thus, u

∂f ∂f ∂f +v +w = convective derivative, ∂x ∂y ∂z

∂f = local derivative. ∂t

(A.15)

(A.16)

Letting f = u and substituting into Eq. (A.13), we have ∂u ∂u ∂u ∂u ∂u Du = =u +v +w + . ∂t Dt ∂x ∂y ∂z ∂t

(A.17)

We rewrite (A.16) in another form as u

∂u ∂u ∂u +v +w = convective acceleration in the x-direction, ∂x ∂y ∂z

(A.18)

where ∂u = local acceleration. ∂t

(A.19)

Similarly, (A.13) can be applied to the y and z directions to obtain the corresponding total acceleration in these directions. The three components of the total acceleration in the cylindrical coordinates r, θ , z are v2 dvr ∂vr ∂vr Dvr vθ ∂vr ∂vr = = vr + − θ + vz + , dt Dt ∂r r ∂θ r ∂z ∂t

(A.20)

dvz Dvz ∂vθ ∂vθ vθ ∂vθ v θ vr ∂vθ = = vr + + + vz + , dt Dt ∂r r ∂θ r ∂z ∂t

(A.21)

dvz Dvz ∂vz vθ ∂vz vθ ∂vz ∂vz ∂vz = = vr + + + vz + . dt Dt ∂r r ∂θ r ∂z ∂z ∂t

(A.22)

Another example of the total derivative is obtained by setting f = T in (A.13) to obtain the total temperature derivative dT DT ∂T ∂T ∂T ∂T = =u +v +w + . dt Dt ∂x ∂y ∂z ∂t

(A.23)

(i) Gauss’ Theorem. The integral of a derivative equals the net value of the function (whose derivative is being integrated) over the boundary of the domain of integration.

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Some mathematical background of fluid mechanics

In one-dimension, this is precisely the fundamental theorem of calculus: 

b

f (x) dx = F (b) − F (a),

(A.24)

a

where F (x) is a function such that F  (x) = f (x), i.e., F is the antiderivative or primitive of f . Thus Eq. (A.24) above results in 

b

F  (x) dx = F (b) − F (a).

(A.25)

a

(ii) Divergence Theorem The basic idea of Eq. (A.25) in three dimensions is referred to as Gauss’ theorem or the divergence theorem. Theorem A.1 (Gauss or divergence theorem). For any smooth vector field F over a region $t ⊂ R3 with a smooth boundary ∂$t ,   ∇ · F dv = F · ndA. (A.26) $t

∂$t

Remark. (a) The term “smooth” in the definition means “sufficiently” differentiable that any desired operation associated with differentiation or integration can be easily justified. (b) A “vector field” is a vector whose components are functions of the spatial coordinate and possibly also time.

Transport theorem Transport theorem is essential in the derivation of equations of fluid dynamics and other transport processes. (i) Leibnitz’s Formula Leibnitz’s formula is of the form:  b(t)  b(t) d ∂f ∂b ∂a f (x, t) dx = dx + f (b, t) − f (a, t) , dt a(t) ∂t ∂t ∂t a(t) ∂b where ∂a ∂t and ∂t are the speeds with which the respective boundary points are moving. Notice that we have denoted the parameter by  t  since our transport theorems will be in the context of time-dependent problems. (ii) General Transport Theorem (GTT) Similar to the fundamental theorem of calculus, Leibnitz’s formula possesses higher-dimensional analogues. For the three-dimensional case, we have the following:

Theorem A.2 (General Transport Theorem). Let F be a smooth vector (or scalar) field on a region $t whose boundary is ∂$t , and let W be the velocity field of the

Some mathematical background of fluid mechanics

time-dependent movement of ∂$t . Then    ∂F d F (x, t)dv = F W · ndA. dv + dt $t $t ∂t ∂$t

477

(A.27)

(iii) Reynolds Transport Theorem (RTT) Reynolds’ transport theorem is the most widely encountered corollary of the general transport theorem in fluid dynamics. It is stated as follows:    D ∂ (x, t)dv = U · ndA. (A.28) dv + Dt $t $t ∂t ∂$t Suppose we define  F (x(t), y(t), z(t), t) =

φdv,

(A.29)

$t

where φ is a scalar. Now the general transport theorem, when applied to the scalar φ, becomes    d ∂φ φdv = φW · ndA. (A.30) dv + dt $t $t ∂t ∂$t Also, since $t is a fluid domain, we can replace W in (A.30) with U to have    d ∂φ φdv = φU · ndA. (A.31) dv + dt $t $t ∂t ∂$t From (A.31), it is evident that  d dF ∂F ∂F ∂F ∂F φdv = = +u +v +w dt $t dt ∂t ∂x ∂y ∂w DF = Dt  D = φdv, Dt $t which is the same as Eq. (A.31). This implies that for a fluid element traveling with the fluid velocity (u, v, w)T , we can replace the ordinary derivative with the substantial derivative, that is, d D = . dt Dt

Conservation of mass (continuity equation) The principle of conservation of mass states that, provided there are no sources or sinks within the control volume, the mass of fluid flowing from one region of the

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Some mathematical background of fluid mechanics

Figure A.1 Control volume and control surface.

control volume to another remains the same. Simply put, matter cannot be created or destroyed but changes from one form of matter to another. The principle is applied to fixed volumes, known as control volumes (or surfaces), as shown in Fig. A.1. Consider an arbitrary volume v having mass m. Then the average density is ρ = m v . Now for an arbitrary small volume, that is, taking the limit as v approaches zero, ρ = lim

v→0

m dm = , v dv

dm = ρdv. Integrating the equation above results in  m = ρdv.

(A.32)

Now by the principle of conservation of mass as stated above, m = constant. This implies that  d dm ρdv = 0 = dt dt $t   ∂ρ ⇒ ρW · ndA = 0 (by GTT) dv + $t ∂t ∂$t   ∂ρ ⇒ ρU · ndA = 0 (by RTT) dv + $ ∂t ∂$  t  t ∂ρ ⇒ ∇ · ρU dv = 0, dv + $t ∂t $t by the divergence theorem,  ∂ρ + ∇ · ρU dv = 0. ⇒ $t ∂t

Some mathematical background of fluid mechanics

479

Since the arbitrary volume dv = 0, ∂ρ + ∇ · ρU = 0. ∂t

(A.33)

Eq. (A.33) is the differential form of the continuity equation valid for both compressible and incompressible fluids.

Alternative forms of the continuity equation Expanding equation (A.33) gives ∂ρ ∂(ρu) ∂(ρv) ∂(ρw) + + + =0 ∂t ∂x ∂y ∂z ⇒



∂ρ ∂u ∂v ∂w ∂ρ ∂ρ ∂ρ +ρ + + +u +v +w =0 ∂t ∂x ∂y ∂z ∂x ∂y ∂z

⇒

∂ρ + ρ(∇ · U ) + (U · ∇)ρ = 0 ∂t

or

∂ρ + (U · ∇)ρ + ρ(∇ · U ) = 0 ∂t

Dρ + ρ(∇ · U ) = 0. Dt If the fluid is incompressible, i.e., has constant density, the above equation reduces ⇒

to ∇ · U = 0. Eq. (A.34) is the continuity equation for an incompressible fluid. In 3D rectangular coordinates, we have ∂u ∂v ∂w + + = 0. ∂x ∂y ∂z In cylindrical coordinates (r, φ, z), 1 ∂ 1 ∂uφ ∂uz (rur ) + + = 0, r ∂r r ∂φ ∂z or ur ∂ur 1 ∂uφ ∂uz + + + = 0. r ∂r r ∂φ ∂z And in spherical coordinates (r, θ , φ), 1 ∂  2  1 ∂uφ 1 ∂ r ur + = 0. (uθ sin θ ) + 2 r sin θ ∂θ r sin θ ∂φ r ∂r

(A.34)

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Some mathematical background of fluid mechanics

Balance of momentum We remark that for any given material region or control volume, {time rate of change of momentum of the control volume} = {sum of forces acting on the control volume}. From Newton’s second law of motion, d(mu) F = ma = m du dt = dt (since m is constant; and mu is the momentum). Now d (ρu) F = , v dt which implies that force per unit volume equals time rate of change of momentum per unit volume. In our present derivation, we shall consider a component at a time. Thus,  D D(mu) ρudv = Dt Dt $t which is equivalent to the x-component momentum change. Therefore, the complete velocity vector momentum change is given as D Dt

 ρU dv = {time rate of change of momentum vector}. $t

Sum of forces Here, we will consider two main types of forces: (i) Body forces are forces which act on the entire region $t and are denoted by  FB dv, and $t

(ii) Surface forces are force which act on the surface ∂$t of the region $t and are denoted by  FS dA. ∂$t

Thus the word “problem” given at the beginning of this subsection can be replaced by D Dt



 ρU dv = $t

 FB dv +

$t

FS dA.

(A.35)

∂$t

The first term on the right-hand side of (A.35) is easy to handle since, in a fluid, the most common body force arising in practice is the buoyant force due to gravitational acceleration; typically, FB = ρg where g is the gravitational acceleration, often taken

Some mathematical background of fluid mechanics

481

as a constant. We know that there are other body forces such as electromagnetic forces and forces due to rotational effects. With reference to Reynolds transport theorem,    D ∂ (x, t)dv = U · ndA, (A.36) dv + Dt $t $t ∂t ∂$t where  is a vector field in general. But as mentioned earlier, for the componentwise derivation of the momentum equation,  will be replaced by the scalar φ which in our present analysis is φ = ρu, i.e., the x-component of the momentum per unit volume. Thus Eq. (A.36) becomes    D ∂ρu ρudv = ρuU.ndA dv + Dt $t $t ∂t ∂$t 



$t

∂ρu dv + ∂t

$t

∂ρu dv + ∇ · (ρuU )dv. ∂t

=  =

∇ · (ρuU )dv $t

(A.37)

Observe that the second term in Eq. (A.37), i.e., ∇ · (ρuU ) = U · ∇ (ρu) + ρu∇ · U,

but ∇ · U = 0 (continuityequation)

thus ∇ · (ρuU ) = U · ∇ (ρu) and so for constant density we have ∇ · (ρuU ) = ρU · ∇u.

(A.38)

Substituting (A.38) into (A.39) results in   ∂u D ρudv = ρ + ρU · ∇udv Dt $t $t ∂t  Du = ρ dv. $t Dt

(A.39)

Remark. By enforcing constant density and definition of the substantial derivative on the right-hand side of (A.39), we have thus succeeded in interchanging differentiation (i.e., total derivative) with integration over the fluid element $t . It follows that the y and z components of momentum are   D Dv ρvdv = ρ dv Dt $t $t Dt and D Dt



 ρwdv =

$t

ρ $t

Dw dv, Dt

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Some mathematical background of fluid mechanics

respectively. Hence we have    DU ρ FB dv + FS dA. dv = Dt $t $t ∂$t

(A.40)

Surface forces FS To obtain mathematical consistency of (A.40), FS must be a vector since ρU and FB are vectors. Thus there must exist a matrix, say T, such that FS = T · n,

(A.41)

where n is the outward unit normal vector to the surface ∂$t . Using (A.41) in (A.40), we have    DU ρ FB dv + T · ndA, dv = Dt $t $t ∂$t which results in  DU ρ dv − FB − ∇ · Tdv = 0. Dt $t

(A.42)

Note that Gauss’s theorem has been applied to the surface integral. Since $t is an arbitrary fluid element that must be greater than zero, from Eq. (A.42) we have that ρ

DU − FB − ∇ · T = 0. Dt

(A.43)

Eq. (A.43) above is a general momentum balance equation valid at all points of any fluid flow. A lot of interesting information is hidden in the surface-force term in the general momentum equation (A.43), therefore, there’s a need for further analysis.

Further analysis of FS There are two types of forces associated with a fluid surface: (i) Normal forces, (ii) Tangential forces. Note that since FS is a 3D vector, T must be a 3 × 3 matrix with a total of nine elements. From Newton’s law of viscosity, we have that τ = μ du dy , which is a 1D flow in the x direction changing only in the y direction. Consider the 3D sketch in Fig. A.2. Extending the analogy of Newton’s law of viscosity to 3D, imagining another fluid element in contact with the above figure placed on top of it, and sliding past the y plane in the x direction, the above formulation is Newton’s 1D viscosity formulation. Thus in this 3D situation, it will be τyx = μ

∂u . ∂y

Some mathematical background of fluid mechanics

483

Figure A.2 Schematic of viscous stresses and pressure acting on a fluid element.

Similar argument will result in τxy = μ

∂v , ∂x

τxz = μ

∂w , ∂x

τyz = μ

∂w , ∂y

and so on. Remark. To avoid discontinuity, we assume the matrix T is symmetric, i.e., τxy = τyx ,

τxz= τzx ,

and τyz= τzy .

But, due to τxy = τyx , this implies ∂v ∂u = , ∂x ∂y which accounts for an irrotational flow. Yet most flow scenarios encountered in practice are rotational. Therefore, the shear stresses acting on the surface of a 3D fluid element are redefined as follows:

∂u ∂v τxy = μ + = τyx , ∂y ∂x τxz = μ τyz = μ

∂u ∂w + ∂z ∂x ∂v ∂w + ∂z ∂y

= τzx ,

= τzy ,

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Some mathematical background of fluid mechanics

which in compact form are:   τxy = μ uy + vx , τxz = μ (uz + wx ) ,

(A.44)

  τyz = μ vz + wy .

Normal viscous stresses and pressure The normal viscous stresses are different from pressure (which is always compressive) and are denoted by τxx , τ yy , and τ zz . By analogy, τxx acts in the x − direction and on the x-face of the fluid element; similar interpretations hold for both τyy and τzz . Again we must have

∂u ∂u τxx = μ + = 2μux , ∂x ∂x

∂v ∂v + = 2μv y , τyy = μ ∂y ∂y

∂w ∂w (A.45) + = 2μw z τzz = μ ∂z ∂z to maintain consistency with (A.49).

Construction of the matrix T Observe that since the compressive forces of pressure P , which are scalar quantities, act in directions opposite to the respective outward unit normal to the fluid element, they have negative signs attached to them all as shown in the figure. Thus we have the required matrix as follows: ⎤ ⎡ τxy τxz −P + τxx ⎦ τyx −P + τyy τyz (A.46) T=⎣ τzx τzy −P + τzz ⎡

−P =⎣ 0 0

0 −P 0

⎡

⎤ ⎡ 0 τxx 0 ⎦ + ⎣ τyx τzx −P

⎤ ⎡ 1 0 0 τxx = P ⎣ 0 1 0 ⎦ + ⎣ τyx 0 0 1 τzx

τxy τyy τzy

τxy τyy τzy

⎤ τxz τyz ⎦ τzz

⎤ τxz τyz ⎦ τzz

= −P I + τ, where I is the identity matrix and τ is often termed the viscous stress tensor.

(A.47)

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485

The Navier–Stokes equations Having obtained the matrix T, we can now compute the divergence of the matrix T. Thus, ⎡ ⎤ τxy τxz −P + τxx ⎦. τyx −P + τyy τyz ∇ ·T = ∇ ·⎣ τzx τzy −P + τzz Note: We are familiar with obtaining the divergence of a vector, but we are presently faced with evaluating the divergence of a matrix T. Thus we will view the matrix as three column vectors to each of which the operator ∇· can be applied independently, with each application yielding a component of a vector. Thus

∂τyx ∂τzx ∂P ∂τxx + + + e1 ∇ ·T = − ∂x ∂x ∂y ∂z

∂P ∂τxy ∂yy ∂τzy + − + + + e2 ∂y ∂x ∂y ∂z

∂P ∂τxz ∂τyz ∂τzz + − + + + e3 , ∂z ∂x ∂y ∂z

(A.48)

where e1 , e2 , e3 are included just to keep track of each component. Now substituting Eq. (A.48) into Eq. (A.41) and considering the x-component, we have ρ

∂τyx ∂P ∂τxx ∂τzx Du =− + + + + FB,x , Dt ∂x ∂x ∂y ∂z

(A.49)

where FB,x denotes the x-component of the body force vector FB . Now using Eqs. (A.43) and (A.45) in Eq. (A.49); the second, third, and fourth terms of the righthand side become   2μuxx + μ uy + vx y + μ (uz + wx ) z ⇒

μuxx + μuxx + μuyy + μvxy + μuzz + μwxz

⇒

μ(uxx + uyy + uzz ) + μ(uxx + vxy + wxz )

⇒

  μ(uxx + uyy + uzz ) + μ ux + vy + wz x,

(A.50)

assuming that the velocity components are sufficiently smooth to allow interchange of the order of partial differentiation. Apparently, the second term of (A.50) vanishes since (∇ · U = 0). Hence Eq. (A.49) collapses to ρ

∂P Du =− + μ∇ 2 u + FB,x ; Dt ∂x

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similarly, the y and z components will yield ρ

∂P Dv =− + μ∇ 2 v + FB,y , Dt ∂y

Dw ∂P =− + μ∇ 2 w + FB,z . Dt ∂z In vector form, the momentum equation is given as ρ

ρ

DU = −∇P + μ∇ 2 U + ρg Dt

(A.51)

i.e., FB = ρg. Since in our present analysis ρ is constant, we can divide (A.51) by ρ to have DU −∇P = + ν∇ 2 U + g, Dt ρ

(A.52)

where ν = μρ is the kinematic viscosity. Equation (A.52) in expanded form of the x-component is given as ∂u ∂u ∂u ∂u 1 +u +v +w = − Px + ν∇ 2 u + g,x . ∂t ∂x ∂y ∂z ρ

(A.53)

In (A.53), ∂u ∂t is referred to as local acceleration, ∂u ∂u u ∂x + v ∂u ∂y + w ∂z is referred to as convective − ρ1 Px is known as pressure force term, ν∇ 2 u is referred to as viscous force term, and

acceleration,

g,x , the gravitational force term. Most often, the sum of the first two terms ∂u ∂u + v ∂u u ∂x ∂y + w ∂z is known as the total acceleration or inertial term.

∂u ∂t

+

Note: In Eq. (A.52) there are four unknowns: three velocity components (u, v, w) and the pressure term P . Thus we need four equations to be able to solve the system. But Eq. (A.52) is a coupled system of three nonlinear PDEs, therefore, a fourth equation is required to closed the system. Hence it is wise at this point to include the continuity equation ux + vy + wz = 0. Hence the Navier–Stokes equations in component form become: ux + vy + wz = 0, 1 ut + uux + vuy + wuz = − Px + v∇ 2 u + g,x , ρ 1 v + uvx + vvy + wvz = − Py + v∇ 2 v + g,y , ρ

Some mathematical background of fluid mechanics

487

1 wt + uwx + vwy + wwz = − Pz + v∇ 2 w + g,z . ρ Observe that the Navier–Stokes equations given above do not account for compressibility effect. They are only applicable to an incompressible fluid. Therefore, to capture compressibility effects in the Navier–Stokes equations, let us reconsider the matrix (A.46) which is reproduced here to ease our analysis ⎤ ⎡ τxy τxz −P + τxx ⎦. τyx −P + τyy τyz (A.54) T=⎣ τzx τzy −P + τzz From the knowledge of stress analyses, let the principal stresses of the above matrix (A.54) be P1 , P2 , and P3 which are the diagonal entries of (A.54). Thus we have P1 = −P + τxx = −P + 2μux , P2 = −P + τyy = −P + 2μvy ,

(A.55)

P3 = −P + τzz = −P + 2μwz . It follows that P1 + P2 + P3 = −3P + 2μ(ux + vy + wz ) = −3P + 2μ (∇ · U ) , in which for an incompressible fluid (i.e., when ∇ · U = 0), P1 + P2 + P3 = −3P .

(A.56)

In the case of a compressible fluid, we have the additional effect of the rate of dilation ∇ ·U having equal effect in all directions. It has been suggested that this effect is accounted for by adding to the right-hand side of (A.55) the quantity β(∇ ·U ) where β is a constant. Thus we have: P1 = −P + τxx = −P + 2μux + β (∇ · U ) , P2 = −P + τyy = −P + 2μvy + β (∇ · U ) , P3 = −P + τzz = −P + 2μwz + β (∇ · U ) . Adding together the equations in Eq. (A.57) results in P1 + P2 + P3 = −3P + 2μ (∇ · U ) + 3β(∇ · U ) and by (A.56), P1 + P2 + P3 = −3P , and for ∇ · U = 0, this implies 2μ + 3β = 0,

(A.57)

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Some mathematical background of fluid mechanics

or 2 β = − μ. 3

(A.58)

Substituting (A.58) into (A.57) results in 2 P1 = −P + τxx = −P + 2μux − μ (∇ · U ) , 3 2 P2 = −P + τyy = −P + 2μvy − μ (∇ · U ) , 3

(A.59)

2 P3 = −P + τzz = −P + 2μwz − μ (∇ · U ) . 3 Observe that the additional term due to compressibility effect is on the diagonal entries of the matrix (A.54). Thus in the operation (A.48), we only need to include the terms ∂ ∂ ∂ (∇ · U ), − 23 μ ∂y (∇ · U ), and − 23 μ ∂z (∇ · U ) to the x, y, and z components, − 23 μ ∂x respectively. Hence in (A.50), the we have for the x-coordinate 2 ∂ μ(uxx + uyy + uzz ) + μ(ux + vy + wz )x − μ (∇ · U ) 3 ∂x ∂ 2 ∂ = μ(uxx + uyy + uzz ) + μ (∇ · U ) − μ (∇ · U ) ∂x 3 ∂x 1 ∂ = μ(uxx + uyy + uzz ) + μ (∇ · U ) . 3 ∂x Similar arguments hold for both the y and z coordinates. Thus, the set of equations for the compressible mass conservation, as well as the compressible Navier–Stokes equations, are: ∂ρ + ∇ · ρU = 0, ∂t DU μ = −∇P + μ∇ 2 U + ∇ (∇ · U ) + ρg. Dt 3 The interested reader may refer to the references in [1–8] for more depth. ρ

(A.60)

(A.61)

References [1] P.H. Oosthuizen, W.E. Carscallen, Introduction to Compressible Fluid Flow, 2nd edition, CRC Press (Taylor & Francis Group), New York, 2013. [2] F.I. Alao, S.B. Folarin, Adomian approximation approach to thermal radiation with heat transfer effect on compressible boundary layer flow on a wedge, Ghana Mining Journal (2013) 70–77.

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[3] F.I. Alao, S.B. Folarin, Similarity solution of the influence of the thermal radiation and heat transfer on steady compressible boundary layer flow, Open Journal of Fluid Dynamics (2013) 82–85. [4] T.L. Bergman, A.S. Larine, F.P. Incropera, D.P. Dewitt, Introduction to Heat Transfer, 6th edition, John Wiley & Sons, USA, 2011. [5] D.F. Young, B.R. Munson, T.H. Okiishi, W.W. Huebsch, A Brief Introduction to Fluid Mechanics, 5th edition, John Wiley & Sons, Inc., USA, 2010. [6] J.H. Lienhard IV, J.H. Lienhard V, A Heat Transfer Textbook, 3rd edition, Phlogiston Press, Cambridge, Massachusetts, USA, 2008. [7] W.M. Rohsenow, J.P. Hartnett, Y.I. Cho (Eds.), Handbook of Heat Transfer, 3rd edition, McGraw-Hill, USA, 1998. [8] L. Rosenhead (Ed.), Laminar Boundary Layers, 3rd edition, Dover Publication, USA, 1963.

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Some fundamentals of fluid mechanics

B

Zounaki Ongodiebi Department of Mathematics and Computer Science, Niger Delta University, Bayelsa, Nigeria

B.1 Types of fluid flow Looking at a flowing fluid, either at open surface flows or channel flows, different flow patterns are seen. In this appendix, we present some of these flow patterns, as well as their mathematical descriptions. (i) Steady and unsteady flows. In such flows, the fluid characteristics such as velocity, pressure, density, etc., at any point within the fluid domain do not change with time. Mathematically,    ∂u  ∂v  ∂w  = 0, = 0, = 0, ∂t x1 ,y1 ,z1 ∂t x1 ,y1 ,z1 ∂t x1 ,y1 ,z1   ∂ρ  ∂P  = 0, = 0, ∂t  ∂t  x1 ,y1 ,z1

x1 ,y1 ,z1

whereas in the case of an unsteady flow, the fluid characteristics stated above are functions of time, i.e., they vary with time. Thus we have:    ∂u  ∂v  ∂w  =  0, =  0, = 0, ∂t x1 ,y1 ,z1 ∂t x1 ,y1 ,z1 ∂t x1 ,y1 ,z1   ∂ρ  ∂P  = 0, = 0, ∂t  ∂t  x1 ,y1 ,z1

x1 ,y1 ,z1

where (x1 , y1 , z1 ) is a fixed location in the flow field where measurements of these variables were taken with respect to time. (ii) Uniform and nonuniform flows. In a given flow field, if the velocity at any particular time does not change with respect to space, then the flow is referred to as a uniform flow. Mathematically,  ∂V  = 0, ∂S t=constant where ∂V is change in velocity and ∂S is the displacement in any direction. Uniform flows can be observed in flows through a straight pipe with constant

492

Some fundamentals of fluid mechanics

diameter. On the other hand, in a nonuniform flow, the velocity at any given time changes with respect to space. Thus,  ∂V  = 0. ∂S t=constant Nonuniform flows can also be seen in pipe flows with nonuniform diameter or at pipe bend, etc. (iii) Rotational and irrotational flows. In rotational flows, the fluid particles in the flow direction rotate about their mass centers. Rotational flows can be seen in the motion of the liquid in a rotating tank or stirring a cup of coffee. Mathematically, ∇ × U = 0, where ∇ is the del operator, or nabla, U is the velocity vector, and ∇× is referred to as the curl or rot. In other words, a flow is said to be rational if the curl of the velocity vector is different from zero. For irrotational or nonrotational flows, the fluid particles, while moving in the direction of flow, do not rotate about their mass centers. In mathematical representation, a flow is said to be irrotational if and only if ∇ × U = 0. Remark. If a flow is characterized by being steady and irrotational, it is known as a potential flow.

Compressible and incompressible flows A fluid flow is said to be a compressible flow if the fluid density varies from point to point. In this case, the density of the fluid is not constant (i.e., ρ = constant). For example, the flow of gases through nozzles or orifices are regarded as a compressible flow. Conversely, if the density of the flowing fluid does not vary from point to point (i.e., ρ = constant), it is regarded as an incompressible flow. Flows of liquids are generally considered as incompressible flows. (i) Laminar and turbulent flows. A fluid flow is said to be laminar if the paths taken by individual particles do not cross one another. Each particle maintains the well-defined path. Laminar flows are smooth, regular, and simple, and as a result, the evolution of laminar flows with time can be predicted with a very high degree of accuracy. Laminar flow is also referred to as a streamline flow. Such flows are seen in ground water flows or flow through a capillary tube. In turbulent flows, the fluid particles move is a zig-zag manner. The path lines of individual particles cross one another. In fact, a turbulent flow is irregular, chaotic, and rotational. The flow being chaotic means that the evolution of a turbulent flow with time cannot be predicted with a very high degree of accuracy for a longer time scale. Such flows are seen in high-velocity flows and are common in nature.

Some fundamentals of fluid mechanics

493

(ii) Viscous and inviscid or nonviscous flows. A flow is said to be a viscous flow if the effect of viscosity is significant, while a flow for which the influence of viscosity is considered negligible and is therefore not taken into account is known as a nonviscous flow. (Note that viscous flows are associated with laminar flows, while inviscid flows are characterized by turbulent flows.) (iii) One-, two-, and three-dimensional flows. If the fluid velocity is a function of time and one space coordinate, it is known as a one-dimensional flow. Mathematically, u = f (x, t),

v = 0,

w = 0,

where u, v, and w are the velocity components along the x, y, and z directions, respectively. For a two-dimensional flow, the fluid velocity is a function of time and two space coordinates. In this case, u = f1 (x, y, t),

v = f2 (x, y, t),

w = 0.

In a three-dimensional flow, the fluid velocity is a function of time and three space coordinates so that u = f1 (x, y, t),

v = f2 (x, y, t),

w = f3 (x, y, t) .

B.2 Flow visualization In an experimental study of a nonstationary fluid, flow visualization is an absolute necessity. The emergence of computational fluid dynamics (CFD) has made it possible for researchers to visualize even 3D, time-dependent fluid flows which were a mirage in our laboratory experiments, thus emphasizing the key features of the flow physics in a situation of interest. In this subsection, we wish to describe the following: pathlines, streamlines, stream tube, and streaklines, as well as their mathematical representations and physical interpretations.

B.2.1 Pathlines Definition. A pathline is a trajectory traced by the single fluid particle during its flow. It gives a traveling history of the fluid element over a specified time period. A pathline can intersect itself at a different time (see Fig. B.1). A family of pathlines represents the trajectories of different particles, in Fig. B.1, say, P1 , P 2 , P3 , and P4 as shown in the above figure. Mathematically, the equation of a pathline is represented by dx = u, dt

dy = v, dt

and

dz = w, dt

which accounts for the trajectory of a particle in the x, y, and z directions, respectively.

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Some fundamentals of fluid mechanics

Figure B.1 Pathlines.

Figure B.2 Streamlines.

Figure B.3 Stream tube.

B.2.2 Streamlines Definition. A streamline is an imaginary curve such that a tangent drawn on this curve represents the direction of instantaneous velocity at that point. Streamlines cannot intersect because a fluid particle cannot have two different velocities at the same time (see Fig. B.2). Streamlines are the solutions of the differential equations: dx dy dz = = . u v w

B.2.3 Stream tube Definition. A stream tube is a set of streamlines drawn within the fluid domain to form a closed loop. The resultant shape is an imaginary tube. It is constructed such that the velocity vector is tangent to the surface at any point (see Fig. B.3).

Properties of a stream tube 1. The stream tube is bounded on all sides by streamlines.

Some fundamentals of fluid mechanics

495

2. There is no flow across the imaginary walls of the stream tube. It can flow only through the ends. 3. The flow through the stream tube is unidirectional.

B.2.4 Streakline his is a curve which gives an instantaneous picture of the location of the fluid particles which have passed through a fixed point in the flow field. Streaklines can be created by injecting a dye into a liquid at a fixed point in the flow field so as to trace the subsequent position of fluid passing through it. Remark. In steady flow, there is no geometrical difference between a pathline, streamline, and streakline. They are identical or coincident if they originate at the same point. The references in [1–6] may be consulted further.

References [1] Andrew Sleigh, Lecture Notes: An Introduction to Fluid Mechanics, University of Leeds, 2001. [2] John J. Bloomer, Practical Fluid Mechanics For Engineering Applications, Marcel Dekker, Inc, 2000, 124. TA357.B59. [3] Frank Chorlton, Text Book of Fluid Dynamics, Van Nostrand Co, London, Princeton, N.J., 1979. [4] J.M. McDonough, Lectures in Elementary Fluid, University of Kentucky, Lexington, KY, 2009, KY 40506-0503. [5] R.K. Rajput, A Textbook of Fluid Mechanics in SI Units, S. Chand and Company LTD, New Delhi, 2004. [6] Robert S. Brodkey, The Phenomena of Fluid Motions, Dover Publications Inc., New York, 1995.

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Index

A Acoustic velocity, 56 Activation energy, 270, 273 Adiabatic flow, 63 Adiabatic wall temperature, 26 Advanced fluids, 48, 323, 333, 355, 408 Adverse pressure gradient, 47, 104 Ambient temperature, 444 Annual temperature trends, 462, 463 Application boundary layer, 48, 257, 333, 415 compressible flow, 65 nanoparticles, 350 opportunities, 328 Apropos boundary layer, 96 Atmospheric temperatures, 439 Atmospheric water, 451 Atomic energy, 345 Atomic motion, 24 Automotive applications, 350 Axisymmetric flows, 423 B Barcode nanoparticles, 352 Base fluid, 282, 283, 285–289, 326, 327, 329, 331, 334, 335, 337, 339, 350, 408, 409, 426 Bingham fluids, 13 Bioenergy, 448 Biological flows, 332 Biomedical applications, 426, 429 nanofluids, 352 Bisection method, 224, 245, 247–250, 253 Blasius, 91, 96, 234, 235, 239, 242–244, 256, 257 boundary layer, 42, 43 boundary value problem, 256 equation, 91, 234, 235, 240, 241, 243, 244, 246, 248–250, 253, 256–258 equation system, 245 flow entropy generation, 408

problem, 242 series, 104, 106, 107, 124 solution, 100 Boiler, 388–391 pipes, 390 pressure, 388 process, 389 Boiling liquid, 50 Boiling temperature, 388 Bottom dead center (BDC), 384 Boundary layer, 29, 30, 32–34, 40, 42, 43, 55, 85, 87, 95–97, 99, 119, 124, 127, 147, 152, 166, 167, 234, 244, 253, 257, 259, 263, 266, 282, 289, 332, 333, 407–409, 413, 414, 417, 418, 420, 424, 425 application, 257, 333, 415 flow, 33, 38, 42, 43, 48, 99, 103, 120, 121, 141, 150, 274, 275, 333, 342, 416, 417, 423, 429 flow influences, 413 flow problems, 30, 179 governing equations, 85 heat transfer, 49 in incompressible flows, 233 MCF procedure, 128 nonsimilarity, 99 separation, 46 slip flow, 423 stagnation point flow, 151 suction, 48 thickness, 43, 47, 55, 86, 234, 342, 423 velocity profiles, 96 Boundary surface, 34, 333, 342 Bounding surface, 25 Brownian motion, 426 Bulk flow, 29 fluid motion, 25 temperature, 219, 221, 224 viscosity, 9

498

Buoyancy, 34, 35, 152, 160, 165, 173, 174, 192 centrifugal, 201, 226, 228 centripetal, 199, 201, 228 effects, 34, 199, 201 flow, 25 force, 25, 35, 39, 151, 169, 195, 199, 201 parameter, 155, 173 term, 37 thermal, 167 C Carbon nanotube (CNT) nanofluids, 289, 323, 349 Cellular convection flow, 424 Centerline velocity, 77, 421 Chebyshev differentiation, 155–158, 181 Circular Couette flow, 424 cylinder, 98, 99, 101, 103, 106, 111, 120, 124, 427 ducts, 193 Clean Development Mechanism (CDM), 460, 466 Climate Support Program (CSP), 464 Coastal water resources, 458 Cogeneration efficiency, 348 Cold inlet temperature, 51 Cold surface, 44 Collocation points, 136, 145, 147, 180, 181, 183 Compressible flow, 56–58, 61, 62 application, 65 field, 63 gas, 63 fluid, 4, 60, 61 fluid flow, 58, 61, 63 fluid flow application, 65 Compressor, 392–395, 397, 402 outlet, 393 process, 395 Computational fluid dynamics (CFD), 429 Condenser, 50, 390–394, 404 heat from vapor compression, 402 heat transfer, 394 pressure, 388, 390, 392, 393 rejection heat rate, 394 temperatures, 394

Index

Conditioned space, 392, 398, 403, 405 Conduct heat, 18 Conduction, 219 Conduction heat, 18, 24 Conduction heat transfer, 24 Conductive fluids, 338 Conductors, 58, 191 Conservation, 2, 3, 99 energy, 2, 3, 15, 58, 62, 332, 340, 356, 402, 406 equations, 3, 201 laws, 195 mass, 2–6, 34, 37, 39, 58, 59, 67, 195, 400, 403 momentum, 3, 58, 59, 65 PDEs, 96 Constant entropy, 63 pressure, 28, 36, 55, 153, 196, 389, 396, 397, 401 pressure heat, 384 addition, 384, 386, 387, 395–397 capacity, 404 rejection, 395 temperature, 50, 149, 219, 424 velocity, 150, 423 viscosity, 11 volume, 28, 55, 63, 384, 385 volume heat, 384 addition, 385 rejection, 384 transfer, 387 Continuity equation, 3, 4, 17, 18, 31, 32, 40, 58–60, 67, 68, 70–72, 74, 75, 154, 195, 197, 263 Continuum flow, 286 Control surface, 5 Control volume, 3–6, 58–60, 62, 64, 67, 87 Convection flow, 259 free, 25, 151, 191, 195, 201, 203, 259 heat transfer, 25, 332, 337, 339 laminar flow, 39, 127 mass transfer, 337 nanofluids, 334 thermal, 332 Convective boundary layer, 333, 334 flow, 259, 268

Index

heat, 151, 350, 408 heat transfer, 193, 287, 323, 332, 334, 338, 348, 351, 426 boundary layer, 259 coefficient, 35, 38, 219, 283, 316 enhancement, 350 in nanofluids, 344 in porous media, 336 mass transfer, 33, 338 coefficient, 34 in nanofluids, 338 rate, 34 motion, 23 nanofluid, 333, 343 Conventional fluids, 282, 283, 286–288, 326, 343, 356 fluids application, 345 industrial fluids, 287, 351 liquid coolants, 351 nanofluids, 324 Conversion efficiency, 349 Couette flow, 424 Counterflow, 48, 51 Counterflow configuration, 49 Counterflow heat exchanger, 51 Coutte flow, 70 Crossflow, 50, 51, 78, 101 Crossflow heat exchangers, 48 Crossflow stream function, 193 Curved surface, 60 Cycle efficiency, 385, 389 Cylindrical nanoparticles, 327 pipes, 428 D Darcy flow, 265 Desiccant dehumidification, 399–402 Diesel cycle, 383–387 Diesel cycle efficiency, 386 Dilatant fluids, 418, 419, 427, 428 Dimensionless flow velocity, 253 form, 29, 32, 33, 86, 93, 141, 179, 261, 416 governing equations, 86 micropolar parameters, 155, 160 momentum thickness, 106 stream function, 149, 197 temperature, 40, 42, 154, 171, 428

499

variables, 32, 86, 97, 260, 266, 267, 272, 278, 417 velocity, 203 velocity parameter, 263 velocity profile, 40, 91 Direct numerical simulation (DNS), 422 Dirichlet boundary conditions, 235 Dirichlet boundary layer, 103 Dissimilar nanoparticles, 289 Dry bulb temperature, 399, 400 Duct, 60, 65, 78, 89, 193, 194, 202, 222 eccentricity, 228 elliptic, 192, 194, 195, 199, 201, 202, 221, 222, 225, 228 porous, 78 E Eckert number, 149, 155, 160–162, 168, 173, 179, 425 Effective medium theory (EMT), 288 Efficiency energy, 356 heat transfer, 284, 287, 337, 340 nanofluids, 286 thermal, 348, 395 trends, 391 Electrical current flow, 58 Electrical energy, 348, 353, 452 Electromagnetic energy, 443 Electromagnetic field, 338, 339 Elliptic duct, 192–195, 199, 201, 202, 221, 222, 225, 228 tube, 193, 219–221 Endoreversible thermodynamics, 406 Energy balance, 27, 49 balance equations, 49–51 boundary layer, 42 budget, 444 carrier, 349 conservation, 2, 3, 15, 58, 62, 332, 340, 356, 402, 406 consumption, 408, 446, 453, 460 conversion, 219, 282, 348 demands, 457, 464 efficiency, 356 efficient vehicles, 350 equation, 15, 37, 38, 58, 61, 62, 65, 266 flow rate, 28

500

forms, 383 harvesting, 323 heat, 165 loss, 406, 407 magnetic, 338 potential, 15 renewable forms, 348, 349, 452 resource potential, 452 solar, 286, 337, 349 sources, 402, 453, 460 spectrum, 349 storage, 323, 324, 349 storage systems, 335 system optimization, 406 thermal, 15, 23, 24, 33, 286, 326, 349 transfer, 24, 25, 193, 259 transport, 326 Engine efficiency, 350 Engine jacket heat, 402 Engineered fluids, 283 Enhanced thermal management, 351 Enhanced thermal transport, 283 Entropy changes, 28 constant, 63 generation, 128, 284, 316, 334, 337, 344, 348, 386, 387, 390, 391, 393, 397, 400, 401, 404, 406–409, 424–426 in boundary layer, 410 in boundary layer flow, 406–408 in convective boundary layer, 407 in convective boundary layer flows, 407 in convective heat transfer, 407 in porous media, 408 generation rate, 409, 425 increase, 28, 408 production, 18, 28, 407, 408 Environmental sustainability, 348 European Patent Office (EPO), 345 Evaporated water vapor, 402 Evaporating liquid water, 402 Evaporative cooling, 390, 402–405 Evaporative cooling process, 403 Exergetic efficiency, 284, 348, 408 Exhaust heat, 388, 394 F Favorable pressure gradient, 47 Finite temperature difference, 407

Index

Finned surfaces, 48 Flow applications, 329 behavior, 415 boundary layer, 33, 38, 42, 43, 48, 99, 103, 120, 121, 141, 150, 274, 275, 333, 342, 416, 417, 423, 429 bulk, 29 buoyancy, 25 characteristics, 343 compressible, 56–58, 61, 62 conditions, 193, 194, 199, 423 convection, 259 convective, 259, 268 curve, 419 destabilization, 423 direction, 48, 83, 96, 420 domain, 1, 85, 414 features, 432 field, 56, 57, 65, 74, 75, 96, 150 fluids, 423 free convection, 264, 265 friction, 398 gas, 56–58 geometries, 12 heat, 15, 16, 24, 28, 49, 282 heat exchangers, 51 heated, 192 incompressible, 11, 67, 69–71 instability, 104 isentropic, 63, 64 laminar, 20, 31, 38, 85, 89, 91, 335, 401, 422, 430–432 map, 193 mass, 57, 78 measurements, 328 nanofluids, 284, 337, 340, 341, 408, 409, 426 passages, 48, 335 patterns, 47, 430 porous media, 336, 423 potential, 39, 98, 99, 104, 127 pressure, 342 problems, 29, 100 properties, 159, 165 rate, 28, 50, 51, 71, 265, 336, 394 regimes, 192, 193, 337 regulators, 65 resistance, 341, 423

Index

reversal, 47 Reynolds numbers, 73 separation, 29, 46, 47, 123 situations, 332 system, 12, 165, 425 thermal energy, 27 transfer, 286 turbulent, 2, 19, 20, 38, 88, 91, 92, 335, 341, 432 unsteady, 4, 151 velocity, 57, 78, 86, 160, 165, 341, 342 vertical, 75 viscous, 2, 20, 29, 99 Flowing fluid particle, 15 fluids, 25, 56 gases, 56 nanofluids, 338, 340 stream, 424 water, 2 Fluids compressible, 4, 60, 61 conventional, 282, 283, 286–288, 326, 343, 356 flow, 423 heat transfer, 286, 289, 316, 329, 345, 389 incompressible, 60, 414 industrial, 327 motion, 9, 23 Newtonian, 9, 12, 13, 17, 85, 99, 332, 413, 415, 419, 421, 426, 431 viscosity, 328 viscous, 160, 165, 342, 343 Food production, 442, 450, 453, 454, 457, 459, 464 Forecast International (FI), 395 Forward stagnation point, 100, 107, 111, 112, 114, 120 Free boundary condition, 241 convection, 25, 151, 191, 195, 201, 203, 259 currents, 191, 195 effects, 38 flow, 263–265 heat transfer, 25, 193 velocity, 191 stream temperature, 96 stream velocity, 19, 40, 55, 87, 167, 427

501

Fresh water, 390 Fresh water resources, 458 Frictionless flow, 63 Fuel efficiency improvements, 387 Functionalized nanoparticles, 287 G Gas critical pressure, 63 flow, 56–58, 63 molecules, 19, 20 pressure, 57 turbine, 191, 388, 394, 395, 397 turbine engines, 394, 395, 397, 398 Geothermal energy, 349, 388 extraction, 337 generation, 340 utilization, 349 Geothermal reservoirs, 335 Global warming potential (GWP), 444 Gold nanoparticles, 349, 351 Governing equations, 2, 23, 30, 42, 43, 84, 86, 95, 96, 101, 138, 141, 142, 153, 172, 185, 187, 193, 194, 199, 200, 202, 221, 222 boundary layer, 85 dimensionless, 86 for natural convection, 37 Graphene nanofluids, 324 Graphite nanofluids, 353 Gravitational buoyancy, 230 Greenhouse gases (GHGs), 438, 439, 442–444, 446, 447, 450, 451, 453–455, 457, 459, 460, 462, 465 Gross Domestic Product (GDP), 459 Ground vehicle applications, 394 Ground water transport, 450 H Hagen Poiseuille flow, 72 Heat addition, 384, 386, 387, 397, 398 capacity, 49, 153, 171, 331 conduction, 18, 24 conductivity, 282, 345 convective, 151, 350, 408 diffusion, 407 dissipation, 339 dissipation capacity, 348 energy, 165

502

engines, 395 equation, 153, 234, 235, 254 exchange, 340 exchanger, 48–51, 329, 337, 340, 348, 350, 356, 389, 391, 397, 402, 430 handbooks, 50, 51 pipes, 391 tubes, 337 flow, 15, 16, 24, 28, 49, 282 flow rate, 28 flux, 24, 28, 99, 151, 193, 349, 353, 424 flux applications, 192 generation, 151, 357, 409, 428 generation absorbtion, 151 input, 385 local, 104 losses, 398 management problems, 351 pipe, 351 production, 351 rate, 28 recovery, 48 rejection, 384, 386, 387, 396, 397 removal, 400 resistance, 351 resistance coefficient, 28 sink, 316 solar, 402 source, 150, 388, 389, 395 stress, 458, 464 transfer application areas, 351 characteristics, 287 coefficient, 25, 26, 28, 49, 114, 120, 192, 264, 285, 334, 338, 341, 350, 351 efficiency, 284, 287, 337, 340 enhancements, 48, 285, 326, 332, 337, 427 fluids, 286, 289, 316, 329, 345, 389 liquids, 282 rate, 50, 337 thermodynamics, 27 wave, 450, 453, 461 Heated flow, 192 fluid, 427 vertical surface, 151

Index

Heating Newtonian, 334, 353 surface, 44 viscous, 123 Heightened temperatures, 461 Homotopy analysis method (HAM), 133, 140, 145, 147, 150, 151, 334, 409 Homotopy perturbation method (HPM), 409 Hybrid nanofluids, 287, 289, 316, 323, 332, 409 in heat transfer applications, 316 in turbulent flow, 316 thermal characteristics, 316 thermal conductivity, 316 thermophysical properties, 316 Hydrodynamic heating, 339 Hydrogen energy, 349 I Immersed surface, 424 Incompressible flow, 11, 67, 69–71 fluids, 60, 414 laminar boundary layer, 39, 127 laminar flow, 74 plane Poiseuille flow, 45 steady flow, 72 viscous fluid flow, 137 Industrial applications, 128, 353, 403 cooling applications, 426 energy production, 454 fluids, 327 Inefficiency, 327 Inelastic fluids, 421 Inflow, 25, 67 radial, 192 Injection, 74, 75, 79, 137, 150, 270, 384, 409, 427 Injection parameter, 428 Injection rates, 418 Inlet, 49, 50, 64, 90, 197, 396, 400, 404 conditions, 404 influences, 200 temperatures, 50, 51, 404 Innerlaminar shear stresses, 353 Instantaneous heat, 384 Instantaneous heat addition, 384 Instantaneous heat rejection, 384

Index

Interlaminar stresses, 353 Internal combustion engine (ICE), 383 Inviscid flow, 18, 32, 276 Inviscid fluids, 2 Irrotational motion, 275 Isentropic compression, 384–386, 391–393 compressor efficiency, 392 flow, 63, 64 Isothermal process, 28, 61, 62 sphere, 121 vertical plate, 38 J Jeffrey fluids, 428 Joule heating, 407 K Kinematic viscosity, 55, 69, 74 Kinetic energy, 23, 275, 327 L Laminar boundary layer, 19, 30, 43, 44, 85, 101, 233–235, 258, 341, 414, 418, 420, 421, 423, 427 boundary layer entropy generation, 407 boundary layer flow, 275, 422, 423 flow, 20, 31, 38, 40, 42, 85, 89, 91, 335, 401, 422, 430–432 fluid, 264 forced convection, 332 heat transfer, 192, 413 nanofluid flow, 340 pipe flow, 422 regime, 342 simulation, 422 velocity profile, 73 Laminarization, 356 Lanthanum nanoparticles, 349 Latent heat exchange, 25 Lattice-Boltzmann (LB) method, 343, 344 Linear momentum, 6 Linearly stretching surface velocity, 427 Liquid absorbents, 401 condensate, 400, 401 cooling, 351

503

cooling jacket, 393 crystals, 344, 414 desiccant, 399, 401 desiccant air conditioning, 399 flows, 344 fuel energy density, 350 metals, 351, 389 slippage, 342 water, 388, 399, 403, 404 water droplets, 390 Local heat, 104 Nusselt number, 154, 179 peripheral Nusselt number, 219 Sherwood number, 154 temperature, 49, 197 thermodynamic pressure, 10 velocity, 56 velocity distribution, 123 Local thermal equilibrium (LTE), 336, 408 Log mean temperature difference (LMTD), 49, 52 M Mach number, 19, 44, 56, 57, 276 Magnetic energy, 338 field, 12, 58, 153, 165, 173, 174, 259, 263, 264, 270, 273, 338–340 nanofluids, 338, 339, 351, 352 nanoparticles, 352 Magnetically controllable nanofluids, 339 Magnetite nanofluids, 409 Magnetohydrodynamics (MHD) convective heat transfer, 339 nanofluid flow, 409 Mainstay, 324, 342, 449–451 Mainstream velocity, 31, 103, 104, 112, 114, 120 Maritime applications, 388 Mass balance, 79 conservation, 2–6, 34, 37, 39, 58, 59, 67, 195, 400, 403 diffusivity, 34 energy transfers, 286 exchange, 34 flow, 57, 78 flow rate, 4, 28, 49, 67, 388, 392–394, 396, 400, 403

504

flux, 5, 32, 34, 60 product, 195 rate, 28 suction, 409 transfer, 33, 34, 58, 97, 98, 114, 117, 150–152, 161, 172, 259, 270, 282, 284, 288, 334, 337, 350, 399, 408, 413, 425 coefficient, 117 enhancements, 337, 338 in multiphase flows, 172 in nanofluids, 281 rates, 104, 114, 160, 287 Maximum cycle temperature, 389 mean Nusselt number, 228 temperature, 391, 396 thermal efficiency, 397 Meant time between failures (MTBF), 351 Merk-Chao-Fagbenle (MCF) method, 98, 99, 103, 104, 106, 107, 109–111, 120, 427, 428 methodology, 100, 108 procedure, 101, 103, 106, 108, 110, 111, 119–121, 123, 128 series, 107, 108, 114, 117–119, 121, 124, 125 Metallic nanoparticles, 349 Metallic oxide nanofluids, 332 Microchannel flow, 343 Microchannel heat sink, 351 Micropatterned surfaces, 426 Micropolar fluid, 150–152, 159, 160, 165, 172–174 fluid applications, 150 fluid boundary layer flow, 151 parameters, 155, 160, 165, 172–174 Minimum entropy production, 397 pressure, 104 temperatures, 464 Molecules gas, 19, 20 kinetic energy, 24 Momentum balance, 33 boundary layers, 424 conservation, 3, 58, 59, 65 conservation equation, 33, 35

Index

diffusion, 171 equation, 6, 16, 31, 34, 36, 37, 41, 58, 60, 68, 69, 72, 74, 80, 193, 195–197, 199, 201, 234 flux, 60, 87 integral, 87, 88 law, 93 thickness, 55, 87–89, 93, 104, 107 transport equations, 416, 417 Motion convective, 23 fluids, 9, 23 nanoparticles, 426 Newtonian fluids, 9, 11 thermal, 283 vertical, 25 viscous flows, 95 mRNA nanoparticles, 352 Multidisciplinary applications, 345 N Nanofluids, 345 aggregates, 330 applications, 282, 345, 349–352 biomedical applications, 352 boundary layer, 334, 341 boundary layer transport, 289 concentration, 348 convection, 334 convective boundary layers, 334, 343 heat transfer, 281, 332, 335, 338 mass transfer, 337, 338 transport, 337 conventional, 324 disposal, 355 efficiency, 286 enhance, 326 flow, 284, 337, 340, 341, 408, 409, 426 flow entropy generation, 408, 409 flow in magnetic fields, 281 flow in porous media, 337, 356 heat transfer, 284, 285, 334, 355 high applications heat flux, 352 in heat transfer, 345 in porous media, 337 in solar applications, 349 investigations, 284 literature, 354, 355 magnetic, 338, 339, 351, 352

Index

mass transfer applications, 352 mixture, 289 nanocomposite, 289 particle concentration, 332 physics, 284 potential applications, 281, 356 preparation, 287 preparation methods, 354 properties, 357 research, 283, 288, 316, 333, 344, 356 studies, 341, 348 sustainability, 287, 328 temperature, 327 thermal conductivity, 327, 328, 355 thermal conductivity enhancements, 327 thermal transport, 283 thermophysical properties, 324, 326, 330, 357 turbulent heat transfer, 342 types, 289, 323, 324, 338, 348, 353 viscosity, 328–330, 332, 354, 355, 357 viscosity models, 329 water, 332, 341 Nanoliquids, 343 Nanoparticles, 282–289, 326–332, 334, 336–341, 344, 348, 350–356, 426 application, 350 cylindrical, 327 magnetic, 352 motion, 426 thermal conductivity, 288 viscosity, 328 viscosity coefficients, 331 Nanotube nanofluids, 323 National renewable energy policy, 457 Nationally Appropriate Mitigation Action (NAMA), 458 Natural convection, 25, 34, 35, 37, 38, 192, 402, 427 flows, 34, 35, 38 heat transfer, 34, 39 problem, 37 Net reduction in heat transfer coefficient, 192 surface force, 8 volumetric outflow, 67 Newtonian, 151, 259, 286, 332, 343, 414, 424, 426 analyses, 424

505

behavior, 334, 414, 428 flows, 431 fluids, 9, 12, 13, 17, 85, 99, 332, 413, 415, 419, 421, 426, 431 heating, 334, 353 viscosity, 13 Nonhybrid nanofluids, 323 Nonisothermal, 97, 124 Nonisothermal cases, 125 Nonisothermal surfaces, 124, 125 Nonisothermality, 126 Nonlinear energy equation, 339 Nonlinear nonsimilar boundary layer, 188 Nonmaterial surfaces, 419 Nonmixing heat exchanger, 404 Nonsimilar boundary layer, 124, 177–179, 188 Nonsimilar flows, 99, 100, 103 Nusselt number, 34, 37, 113, 119, 161, 184, 185, 192, 219, 224, 261, 273, 285, 316, 323, 334, 337, 338, 340, 341, 354, 426 local, 154, 179 peripheral, 219 peripheral local, 193, 219–221, 224 O Optimal efficiency, 284, 408 Ordinary differential equation (ODE), 96, 234, 235, 256 Organic Rankine cycle (ORC), 353, 388 Otto cycle, 384–386 Otto cycle efficiency, 386 Outflow, 67 radial, 192 Outlet temperatures, 49–52 Oxide nanofluids, 341 P Parallel flow, 48, 72 Parallel flow configuration, 49 Parallel pipes, 48 Partial differential equation (PDE), 95, 234 Peak temperatures, 397 Pipes boiler, 390 cylindrical, 428 flow, 72, 73, 431 heat exchanger, 391 Platelets nanoparticles, 332

506

Poiseuille flow, 77 incompressible plane, 45 plane, 71 Porous duct, 78 media, 128, 150, 151, 265, 335, 336, 338, 408, 423 flow, 336, 423 nanofluid flow, 281 thermal conductivity, 337 surface, 48, 420, 423 vertical heated surface, 269 Potential applications, 324 energy, 15 flow, 39, 98, 99, 104, 127 future applications, 128 velocity distribution, 114 Prandtl Boundary Layer (PBL), 95 Prandtl number, 34, 37, 39, 114, 123, 149, 170, 171, 173, 179, 193, 197, 198, 201, 273, 275 Pressure boiler, 388 condenser, 388, 390, 392, 393 constant, 28, 36, 55, 153, 196, 389, 396, 397, 401 differences, 2 distribution, 83, 197, 200 drag coefficient, 115 drag data, 115 drop, 48, 65, 83, 265, 323, 336, 341, 350, 391, 393, 397, 421 flow, 342 forces, 36, 59, 68 gas, 57 gradient, 47, 70, 71, 75, 86, 100, 166, 179, 191, 196, 197, 276, 421 gradient boundary layer flow, 234 minimum, 104 reservoir, 56 thermodynamics, 3, 10 Property entropy, 387, 390, 397, 400, 404 Proton exchange membrane (PEM), 350 Pseudoplastic flows, 427 Pseudoplastic fluids, 428 Pulsatile pressure gradient, 429 Q Quantum thermodynamics, 406

Index

R Radial inflow, 192 outflow, 192 temperature distribution, 192 Radiative heat flux, 153, 170, 339 Radiative heat flux, thermal, 149, 153 Radiative heat transfer, 27 Radiative surface, 339 Random motions, 23, 24 Rankine cycle, 388–391, 394, 395 Rarefied gas flow, 1 Rational thermodynamics, 406 Recovery temperature, 26 Rectangular coordinate system, 3, 4, 10, 17 Regeneration energy, 402 Reheat, 397 Reheating, 391, 399 Relative velocity, 342, 426 Relative viscosity, 327 Renewable energy, 348, 438, 449, 451, 452, 454, 457, 460 Renewable energy resource, 448 Renewable energy technologies, 446, 451 Renewable hydrogen energy generation, 349 Reservoir temperature, 29 Resultant energy changes, 57 Retarded flows, 99 Reynolds number, 19, 30, 32, 38, 44, 46, 55, 77, 78, 84–86, 117, 118, 121, 137, 140, 179, 198, 201, 333, 336, 340, 341, 348 flow, 42 suction, 81 Rheopectic fluids, 419, 420 Rotational Rayleigh number, 193, 198, 199, 201, 203, 217, 221, 222, 226 Rotor conductors, 191 S Salient thermophysical properties, 326 Saturation temperature, 390 Secant method, 245–247, 249, 250, 253, 274 Secondary flow, 25, 192, 193, 205, 223 Sensible heating, 400 Sensible temperature depression, 403 Sherwood number, 116, 160, 161, 265, 268, 273, 337, 338

Index

Shooting method, 241, 242, 244, 246, 248, 249, 253, 256, 257, 261, 264, 268 Silicon nanoparticles, 349 Similarity solution, 99, 100, 234, 266, 268, 427 Similarity transformations, 30, 96, 151, 172 Skin friction, 42, 104, 107, 121, 123, 160, 179, 183–185, 273 Skin friction coefficient, 99, 159, 160, 261 Skin friction drag, 123, 274 Slip flow, 342, 427 Slip flow boundary layer, 423 Slip velocity, 19, 151, 334, 343 Solar cells, 348, 349 collector, 348, 349 collector efficiency, 349 energy, 286, 337, 349 heat, 402 paint, 350 photovoltaic (PV) technology, 451 radiation, 349, 439, 443 systems, 348, 349 thermal cooling, 399 thermal energy, 349, 388, 399 Solutal boundary layer thicknesses, 164, 167 Solutal buoyancy, 168 Solvothermal, 287 Spectral homotopy analysis method (SHAM), 133, 138, 140, 145–147 algorithm, 146, 147 approximations, 145 results, 145 solution, 137, 142 Spectral quasi-linearization method (SQLM), 151, 155, 159, 172 Spherical nanoparticles, 288, 327, 332 Standing water, 403 Static pressure, 120 Steepest temperature gradient, 24 Stokesian fluids, 13 Stream function, 80, 150, 154, 192, 193, 197, 198, 200, 201, 203–205, 207, 217, 222, 223, 260, 263, 266, 267 Stream velocity, 100 distribution, 107, 108, 121, 123 free, 19, 87, 167

507

Stretching surface, 128, 151, 172, 425, 426 Suction, 47, 78, 79, 99, 137, 150, 266, 270, 409, 427 boundary layer, 48 drag, 47 mass, 409 Reynolds number, 81 Superheat, 388, 391, 393 Superheated steam, 388 Superheating, 391 Superior thermal performance, 354 Superior thermal properties, 287 Supersonic flows, 26, 44 Superthermal conductivity, 351 Surface air temperature, 462 area, 4, 27, 48, 324, 337, 351 boundary, 34, 333, 342 boundary conditions, 100 chemistry, 431 concentration, 34 effects, 343 element, 103 forces, 8 heat flux, 337 heat transfer, 123, 159 heating, 44 mass transfer, 100, 159 porous, 48, 420, 423 roughness, 46 temperature, 25, 35, 152, 275, 337, 437, 453 temperature distribution, 427 treatments, 283 velocity, 423 vertical, 268, 274 Sustainability, nanofluids, 287, 328 Sustainable energy production, 462 Sustainable energy technologies, 465 T Temperature boundary layers, 338 bulk, 219, 221, 224 changes, 57, 62, 387, 444, 453 constant, 50, 149, 219, 424 dependence, 327 depression, 402–404

508

difference, 20, 23, 26, 37, 39, 127, 153, 219, 281, 402 dimensionless, 154, 171, 428 dimensionless form, 193 distribution, 114, 121, 123, 165, 201, 206, 209, 215, 227, 234 field, 37, 193, 203, 204, 222, 339 gradient, 24, 26, 27, 114, 155, 194, 199, 219, 220, 224, 270, 283, 327, 424, 427 increase, 457, 461 jump, 19, 20, 334, 342 local, 49, 197 maximum, 391, 396 nanofluids, 327 profiles, 42, 50, 96, 114, 170, 255 range, 112, 327 ratio, 340 reduction, 353 rise, 191, 457, 459 surface, 25, 35, 152, 275, 337, 437, 453 variations, 25, 440 Thermal boundary condition, 194 boundary layer, 33–35, 40, 42, 114, 116, 171, 424, 428 buoyancy, 167 buoyancy force, 151 buoyancy parameter, 160, 165–168 capacity, 424 characteristics, 192 condition, 170 conductivity, 1, 3, 24, 26, 38, 150, 153, 194, 196, 199, 259, 265, 271, 274, 275, 283, 284, 286, 288, 289, 316, 323, 326–332, 334, 337, 339, 350, 351, 354, 357, 426 conductivity enhancements, 328, 354 convection, 332 devices, 345 diffusion, 171 diffusivity, 33, 196, 283, 426 dispersion, 426 efficiency, 348, 395 energy, 15, 23, 24, 33, 286, 326, 349 flow, 27 storage, 352 transfers, 23

Index

enhancement, 286 entrance length, 192 entrance region, 40 equilibrium, 285, 336 expansion, 150, 153, 198, 330 expansion coefficient, 330, 331 fields, 151 load, 350 management for heat transfer, 323 motion, 283 nonequilibrium, 27, 336 oil nanofluid mixture, 348 performance, 287 plume, 35 properties, 33, 289, 330, 339 radiation, 26, 149, 151, 169, 170, 172, 408 radiative heat flux, 149, 153 resistance, 26, 351 solar, 399 systems, 326, 328 transport phenomena, 354 Thermic fluids, 283 Thermodynamics applications, 383 heat transfer, 27 pressure, 3, 10 Thermophoresis, 151, 268, 270, 283, 285, 341, 425, 426 effects, 284 parameter, 334 Thermophoretic velocity, 271, 284 Thermophysical properties, 281, 289, 316, 324, 326, 345, 425, 426 hybrid nanofluids, 316 nanofluids, 324, 326, 330, 357 Thixotropic fluids, 419, 420 Top dead center (TDC), 384 Turbulent boundary layer, 44, 46, 93, 97 eddy viscosity, 92 flow, 2, 19, 20, 38, 88, 91, 92, 335, 341, 432 heat transfer coefficient, 341 jets, 43 nanofluid flow, 281, 340 pipe flow, 92 profiles, 92, 93 regime, 323, 341

Index

U Unsteadiness parameter, 154, 160–164, 173, 174 Unsteady flow, 4, 151 heat equation, 244, 253, 255 nanofluid MHD flow, 408 V Vapor compression, 391–393, 405 Vapor compression efficiency, 403 Variable viscosity, 265 Variational iteration method (VIM), 409 Velocity boundary layer, 31, 41, 165 boundary layer thickness, 163 changes, 35 components, 31, 32, 40, 67, 80, 83, 86, 141, 149, 153, 165, 172, 194, 270 constant, 150, 423 dimensionless, 203 distribution, 103, 104, 106, 107, 109–111, 113, 114, 121, 165 field, 3, 6, 37, 259, 343 flow, 57, 78, 86, 160, 165, 341, 342 gradients, 18, 164, 165 local, 56 normal, 90 parallel, 90 profiles, 35, 42, 71–73, 78, 80, 84, 89, 96, 99, 108, 114, 162, 168, 421, 422 slip, 19, 342 squared term, 336 surface, 423 vector, 3 vertical, 77, 78 Vertical flow, 75 heated plate, 35 isothermal plate, 38 motion, 25 surface, 268, 274 velocity, 77, 78 velocity component, 75 Viscoelastic liquids, 414 Viscoplastic flows, 419 Viscoplastic fluids, 415, 419

509

Viscosity bulk, 9 coefficient, 9, 13, 331 constant, 11 fluids, 328 models, 289, 329 nanofluids, 328–330, 332, 354, 355, 357 nanoparticles, 328 Newtonian, 13 ratio, 418 term, 431 value, 329 Viscous boundary layers, 40 dissipation, 17, 18, 99, 151, 162, 168, 200, 259, 264, 284, 338, 407, 408, 427 flow, 2, 20, 29, 99 fluids, 160, 165, 342, 343 fluids heat transfer, 423 heat generation, 18 heating, 123 Volumetric flow rate, 67 W Water base fluid, 408 body, 390 consumption, 454 content, 431 cooled condensers, 390 desalination processes, 324 flow rate, 341 flowing, 2 insecurity, 462 liquid, 388, 403, 404 loss, 391 nanofluids, 332, 341 purification, 353 purification processes, 391 quality, 453, 464 resource, 448 scarcity, 445 spray, 402 temperatures, 390 transport, 353, 446 transports momentum, 286 vapor, 350, 399, 401, 442, 444, 451 World Health Organization (WHO), 448

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WOODHEAD PUBLISHING SERIES IN ENERGY Presents advances in boundary layer theory, emphasizing industrial applications to better develop solutions for energy efficiency and sustainability. •

Presents up-to-date research on boundary layers with very practical applications across a diverse mix of industries

•

Includes mathematical analysis to provide detailed explanation and clarity

•

Provides discussions on global energy and environmental sustainability

Applications of Heat, Mass and Fluid Boundary Layers edited by Prof. Fagbenle, Dr. Amoo, Aliu and Falana brings together the latest research boundary layers where there have been remarkable advancements over the last 5 years. This book highlights relevant concepts and solutions to energy issues and environmental sustainability by combining comprehension of fundamental theory on boundary layers with real-world industrial applications from, among others, the thermal, nuclear, and chemical industries. The editors and their team of expert contributors discuss many core themes, including the advanced heat transfer fluids and boundary layer analysis, physics of fluid motion and viscous flow, thermodynamics and transport phenomena, alongside key methods of analysis such as the Merk–Chao–Fagbenle method. The main focus is to assess the role and influence of boundary layers on energy related technologies and to offer a close look at current global energy-related problems. This book’s multidisciplinary coverage will give engineers, scientists, researchers, and graduate students in the areas of heat, mass, fluid flow and transfer a thorough understanding of the technicalities, methods, and applications of boundary layers, with a unified approach to energy, climate change, and a sustainable future. About the Editors Prof. Fagbenle is an adjunct professor of mechanical engineering at the Center for Petroleum, Energy Economics and Law, University of Ibadan, Nigeria. He is also the former head of the Mechanical Engineering Department at the University of Ibadan, Nigeria. He has taught engineering courses at several schools, including the University of Illinois (Urbana-Champaign, IL, USA), Iowa State University (Ames, IA, USA), Kwame Nkrumah University of Science and Technology (Kumasi, Ghana), South Bank University (London, UK) and Covenant University (Ota, Ogun State, Nigeria). He is a fellow of the Nigerian Society of Engineers (NSE), the Solar Energy Society of Nigeria (SESN), and the Nigerian Association for Energy Economics (NAEE), and a member of the American Society of Mechanical Engineers (ASME). His areas of research are laminar boundary layers, energy, exergy, climate change, and energy policies for a more sustainable future. Dr. Amoo was a doctoral boundary layer student of Professor Fagbenle at the prestigious University of Ibadan, Nigeria. His industrial experience has mostly been in the aerospace and defense industry, which includes companies such as Northrop Grumman, Raytheon, Lockheed Martin, and UTC-Pratt & Whitney. Other industrial experiences include those obtained at Toyota Motor Manufacturing and Harley-Davidson. He is a member of the American Society of Mechanical Engineers (ASME), the National Society of Black Engineers (NSBE), and the American Institute of Aeronautics and Astronautics (AIAA). His areas of research are energy, exergy, nanofluids, and laminar boundary layer convective heat transfer. Dr. Aliu was a doctoral boundary layer student of Prof. Fagbenle. He is currently with the Mechanical Engineering Department at the University of Benin, Benin City, Nigeria, where he teaches various engineering courses and conducts research in the field of thermo-fluids. Dr. Falana was a doctoral boundary layer student of Prof. Fagbenle and is currently with the Mechanical Engineering Department at the prestigious University of Ibadan, Ibadan, Nigeria, where he teaches both under- and post-graduate engineering courses. He is an avid researcher in theoretical thermo-fluids, in general, and boundary layer convective heat transfer, in particular. ISBN 978-0-12-817949-9

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