An Introduction to Fluid Dynamics 9780521663960, 0521663962

First published in 1967, Professor Batchelor's classic work is still one of the foremost texts on fluid dynamics. H

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An Introduction to Fluid Dynamics
 9780521663960, 0521663962

Table of contents :
Cover......Page 1
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 15
Conventions and Notation......Page 20
1.1 Solids, liquids and gases......Page 21
1.2 The continuum hypothesis......Page 24
1.3 Volume forces and surface forces acting on a fluid......Page 27
Representation of surface forces by the stress tensor......Page 29
The stress tensor in a fluid at rest......Page 32
1.4 Mechanical equilibrium of a fluid......Page 34
A body 'floating' in fluid at rest......Page 36
Fluid at rest under gravity......Page 38
1.5 Classical thermodynamics......Page 40
1.6 Transport phenomena......Page 48
The linear relation between flux and the gradient of a scalar intensity......Page 50
The equations for diffusion and heat conduction in isotropic media at rest......Page 52
Molecular transport of momentum in a fluid......Page 56
1.7 The distinctive properties of gases......Page 57
A perfect gas in equilibrium......Page 58
Departures from the perfect-gas laws......Page 65
Transport coefficients in a perfect gas......Page 67
Other manifestations of departure from equilibrium of a perfect gas......Page 70
1.8 The distinctive properties of liquids......Page 73
Equilibrium properties......Page 75
Transport coefficients......Page 77
Surface tension......Page 80
Equilibrium shape of a boundary between two stationary fluids......Page 83
Transition relations at a material boundary......Page 88
2.1 Specification of the flow field......Page 91
Differentiation following the motion of the fluid......Page 92
2.2 Conservation of mass......Page 93
Use of a stream function to satisfy the mass-conservation equation......Page 95
2.3 Analysis of the relative motion near a point......Page 99
Simple shearing motion......Page 103
2.4 Expression for the velocity distribution with specified rate of expansion and vorticity......Page 104
2.5 Singularities in the rate of expansion. Sources and sinks......Page 108
2.6 The vorticity distribution......Page 112
Line vortices......Page 113
Sheet vortices......Page 116
2.7 Velocity distributions with zero rate of expansion and zero vorticity......Page 119
Conditions for [nabla]ø to be determined uniquely......Page 122
Irrotational solenoidal flow near a stagnation point......Page 125
The complex potential for irrotational solenoidai flow in two dimensions......Page 126
2.8 Irrotational solenoidal flow in doubly-connected regions of space......Page 128
Conditions for [nabla]ø to be determined uniquely......Page 132
Asymptotic expressions for u[sub(e)] and u[sub(v)]......Page 134
The behaviour of ø at large distances......Page 137
Conditions for [nabla]ø to be determined uniquely......Page 139
The expression of ø as a power series......Page 140
Irrotational solenoidai flow due to a rigid body in translational motion......Page 142
2.10 Two-dimensional flow fields extending to infinity......Page 144
Irrotational solenoidal flow due to a rigid body in translational motion......Page 148
3.1 Material integrals in a moving fluid......Page 151
Rates of change of material integrals......Page 153
Conservation laws for a fluid in motion......Page 155
3.2 The equation of motion......Page 157
Use of the momentum equation in integral form......Page 158
Equation of motion relative to moving axes......Page 159
Mechanical definition of pressure in a moving fluid......Page 161
The relation between deviatoric stress and rate-of-strain for a Newtonian fluid......Page 162
The Navier-Stokes equation......Page 167
Conditions on the velocity and stress at a material boundary......Page 168
3.4 Changes in the internal energy of a fluid in motion......Page 171
3.5 Bernoulli's theorem for steady flow of a frictionless non-conducting fluid......Page 176
Special forms of Bernoulli's theorem......Page 181
Constancy of H across a transition region in one-dimensional steady flow......Page 183
3.6 The complete set of equations governing fluid flow......Page 184
Isentropic flow......Page 185
Conditions for the velocity distribution to be approximately solenoidal......Page 187
3.7 Concluding remarks to chapters 1, 2 and 3......Page 191
4.1 Introduction......Page 194
Modification of the pressure to allow for the effect of the body force......Page 196
4.2 Steady unidirectional flow......Page 199
Poiseuille flow......Page 200
Two-dimensional flow......Page 202
A model of a paint-brush......Page 203
A remark on stability......Page 205
4.3 Unsteady unidirectional flow......Page 206
The smoothing-out of a discontinuity in velocity at a plane......Page 207
Plane boundary moved suddenly in a fluid at rest......Page 209
One rigid boundary moved suddenly and one held stationary......Page 210
Flow due to an oscillating plane boundary......Page 211
Starting flow in a pipe......Page 213
4.4 The Ekman layer at a boundary in a rotating fluid......Page 215
The layer at a free surface......Page 217
The layer at a rigid plane boundary......Page 219
4.5 Flow with circular streamlines......Page 221
4.6 The steady jet from a point source of momentum......Page 225
4.7 Dynamical similarity and the Reynolds number......Page 231
Other dimensionless parameters having dynamical significance......Page 235
4.8 Flow fields in which inertia forces are negligible......Page 236
Flow in slowly-varying channels......Page 237
Lubrication theory......Page 239
The Hele-Shaw cell......Page 242
Percolation through porous media......Page 243
Two-dimensional flow in a corner......Page 244
Uniqueness and minimum dissipation theorems......Page 247
4.9 Flow due to a moving body at small Reynolds number......Page 249
A rigid sphere......Page 250
A spherical drop of a different fluid......Page 255
A body of arbitrary shape......Page 258
4.10 Oseen's improvement of the equation for flow due to moving bodies at small Reynolds number......Page 260
A rigid sphere......Page 261
A rigid circular cylinder......Page 264
4.11 The viscosity of a dilute suspension of small particles......Page 266
The flow due to a sphere embedded in a pure straining motion......Page 268
The increased rate of dissipation in an incompressible suspension......Page 270
The effective expansion viscosity of a liquid containing gas bubbles......Page 273
4.12 Changes in the flow due to moving bodies as R increases from 1 to about 100......Page 275
5.1 Introduction......Page 284
5.2 Vorticity dynamics......Page 286
The Intensification of vorticity by extension of vortex-lines......Page 290
5.3 Kelvin's circulation theorem and vorticity laws for an inviscid fluid......Page 293
The persistence of irrotationality......Page 296
5.4 The source of vorticity in motions generated from rest......Page 297
(a) Flow along plane and circular walls with suction through the wall......Page 302
(b) Flow toward a 'stagnation points' at a rigid boundary......Page 305
(c) Centrifugal flow due to a rotating disk......Page 310
5.6 Steady two-dimensional flow in a converging or diverging channel......Page 314
Purely convergent flow......Page 317
Purely divergent flow......Page 318
Solutions showing both outflow and inflow......Page 321
5.7 Boundary layers......Page 322
5.8 The boundary layer on a flat plate......Page 328
5.9 The effects of acceleration and deceleration of the external stream......Page 334
The similarity solution for an external stream velocity proportional to x[sup(m)]......Page 336
Calculation of the steady boundary layer on a body moving through fluid......Page 338
Growth of the boundary layer in initially irrotational flow......Page 341
5.10 Separation of the boundary layer......Page 345
5.11 The flow due to bodies moving steadily through fluid......Page 351
Flow without separation......Page 352
Flow with separation......Page 357
Narrow jets......Page 363
Free shear layers......Page 366
Wakes......Page 368
5.13 Oscillatory boundary layers......Page 373
The damping force on an oscillating body......Page 375
Steady streaming due to an oscillatory boundary layer......Page 378
Applications of the theory of steady streaming......Page 381
The boundary layer at a free surface......Page 384
The drag on a spherical gas bubble rising steadily through liquid......Page 412
The attenuation of gravity waves......Page 415
The force on a regular array of bodies in a stream......Page 417
The effect of a sudden enlargement of a pipe......Page 418
6.1 The role of the theory of flow of an inviscid fluid......Page 423
6.2 General properties of irrotational flow......Page 425
Integration of the equation of motion......Page 427
Expressions for the kinetic energy in terms of surface integrals......Page 428
Positions of a maximum of q and a minimum of p......Page 429
6.3 Steady flow: some applications of Bernoulli's theorem and the momentum theorem......Page 431
Efflux from a circular orifice in an open vessel......Page 432
Flow over a weir......Page 436
Jet of liquid impinging on a plane wall......Page 437
Irrotational flow which may be made steady by choice of rotating axes......Page 441
6.4 General features of irrotational flow due to a moving rigid body......Page 443
The velocity at large distances from the body......Page 444
The kinetic energy of the fluid......Page 447
The force on a body in translational motion......Page 449
The acceleration reaction......Page 452
6.5 Use of the complex potential for irrotational flow in two dimensions......Page 454
Flow fields obtained by special choice of the function w(z)......Page 455
Conformal transformation of the plane of flow......Page 458
Transformation of a boundary into an infinite straight line......Page 463
Transformation of a closed boundary into a circle......Page 465
The circle theorem......Page 467
6.6 Two-dimensional irrotational flow due to a moving cylinder with circulation......Page 468
A circular cylinder......Page 469
An elliptic cylinder in translational motion......Page 472
The force and moment on a cylinder in steady translational motion......Page 478
The practical requirements of aerofoils......Page 480
The generation of circulation round an aerofoil and the basis for Joukowski's hypothesis......Page 483
Aerofoils obtained by transformation of a circle......Page 486
Joukowski aerofoils......Page 489
Generalities......Page 494
A moving sphere......Page 497
Ellipsoids of revolution......Page 500
Body shapes obtained from source singularities on the axis of symmetry......Page 503
Semi-infinite bodies......Page 505
Slender bodies of revolution......Page 508
Slender bodies in two dimensions......Page 511
Thin aerofoils in two dimensions......Page 512
6.10 Impulsive motion of a fluid......Page 516
Impact of a body on a free surface of liquid......Page 518
6.11 Large gas bubbles in liquid......Page 519
A spherical-cap bubble rising through liquid under gravity......Page 520
A bubble rising in a vertical tube......Page 522
A spherical expanding bubble......Page 524
6.12 Cavitation in a liquid......Page 526
Examples of cavity formation in unsteady flow......Page 530
Collapse of a transient cavity......Page 531
Steady-state cavities......Page 536
6.13 Free-streamline theory, and steady jets and cavities......Page 538
Jet emerging from an orifice in two dimensions......Page 540
Two-dimensional flow past a flat plate with a cavity at ambient pressure......Page 542
Steady-state cavities attached to bodies held in a stream of liquid......Page 547
7.1 Introduction......Page 552
The self-induced movement of a line vortex......Page 554
The instability of a sheet vortex......Page 556
7.2 Flow in unbounded fluid at rest at infinity......Page 562
The resultant force impulse required to generate the motion......Page 563
The total kinetic energy of the fluid......Page 565
Flow with circular vortex-lines......Page 566
Vortex rings......Page 567
7.3 Two-dimensional flow in unbounded fluid at rest at infinity......Page 572
Integral invariants of the vorticity distribution......Page 573
Motion of a group of point vortices......Page 575
Steady motions......Page 577
7.4 Steady two-dimensional flow with vorticity throughout the fluid......Page 581
Uniform vorticity in a region bounded externally......Page 583
Fluid in rigid rotation at infinity......Page 584
Fluid in simple shearing motion at infinity......Page 586
7.5 Steady axisymmetric flow with swirl......Page 588
The effect of a change of cross-section of a tube on a stream of rotating fluid......Page 591
The effect of a change of external velocity on an isolated vortex......Page 595
The restoring effect of Coriolis forces......Page 600
Steady flow at small Rossby number......Page 602
Propagation of waves in a rotating fluid......Page 604
Flow due to a body moving along the axis of rotation......Page 609
7.7 Motion in a thin layer on a rotating sphere......Page 612
Geostrophic flow......Page 616
Flow over uneven ground......Page 618
Planetary waves......Page 622
General features of the flow past lifting bodies in three dimensions......Page 625
Wings of large aspect ratio, and 'lifting-line' theory......Page 628
The trailing vortex system far downstream......Page 634
Highly swept wings......Page 636
(a) Dry air at a pressure of one atmosphere......Page 639
(c) Pure water......Page 640
(e) Surface tension between two fluids......Page 642
2 Expressions for some common vector differential quantities in orthogonal curvilinear co-ordinate systems......Page 643
Publications referred to in the text......Page 649
C......Page 654
E......Page 655
I......Page 656
P......Page 657
S......Page 658
V......Page 659
Z......Page 660
Plates......Page 385

Citation preview

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AN INTRODUCTION TOFLUID DYNAMICS

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AN INTRODUCTION TO T TTTT\ F A^ vJ A A^#

Tj^\/^T A IkM¥£^C! U A X il A!L JLYA A v*/ O

BY

G . K . B A T C H E L O R , F.R.S. Professor of Applied Malhanalks

in the University of Cambridge

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi, Dubai, Tokyo. Mexico City Cambridge University Press 32 Avenue of the A mericas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521663960 © Cambridge University Press 1967, 1973, 2000

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1967 Reprinted 1970 First paperback edition 1973 Reprinted 1974, 1977, 1979, 1980, 1981, 1983, 1985, 1987, 1988, 1990, 1992, 1994, 1996, 1998 First Cambridge Mathematical Library Edition 2000 14th printing 2010 A catalog recordfor this publication is available from the British Library.

ISBN 978-0-521-66396-0 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

v

CONTENTS

page xiii

Preface Conventions and Notation

xviii

Chapter 1. The Physical Properties of Fluids

I. I

Solids, liquids and gases

I

I .2

The continuum hypothesis

4

I ;J

Volume forces and surface forces acting on a fluid

7

Representation of surface forces by the stress tensor, 9 The stress tensor in a fluid at rest, I 2

1.4

Mechanical equilibrium of a fluid

I4

A body 'floating' in fluid at rest, 16 Fluid at rest under gravity, 18

I. 5

Classical thermodynamics

I.6

Transport phenomena

20

The linear relation between flux and the gradient of a scalar intensity, 30 The equations for diffusion and heat conduction in isotropic media at rest, 32 Molecular transport of momentum in a fluid, 36

I.7

The distinctive properties of gases

37

A perfect gas in equilibrium, 38 Departures from the perfect-gas laws, 45 Transport coefficients in a perfect gas, 47 Other manifestations of departure from equilibrium of a perfect gas, so

I .8

The distinctive properties of liquids

53

Equilibrium properties, 55 Transport coefficients, 57

I ·9

Conditions at a boundary between two media

6o

Surface tension, 6o Equilibrium shape of a boundary between two stationary fluids, 63 Transition relations at a material boundary, 68

Chapter 2. Kinematics of the Flow Field 2. I

Specification of the flow field Differentiation following the motion of the fluid, 72

2.2

Conservation of mass

73

Use of a stream function to satisfy the mass-conservation equation, 75

2.3

Analysis of the relative motion near a point Simple shearing motion, 83

79

Contents

VI

page 84

2.4

Expression for the velocity distribution with specified rate of expansion and vorticity

2.5

Singularities in the rate of expansion. Sources and sinks

88

2.6

The vorticity distribution

92

Line vortices, 93 Sheet vortices, 96

2.7

Velocity distributions with zero rate of expansion and zero vorticity

99

Conditions for vq, to be determined uniquely, IO:Z Irrotational solenoidal fl.ow near a stagnation point, IOS The complex potential for irrotational solenoidal fl.ow in two dimensions, Io6

2.8

ro8

Irrotational solenoidal flow in doubly-connected regions of space Conditions for vq, to be determined uniquely, II:Z

2.9

Three-dimensional flow fields extending to infinity Asymptotic expressions for u. and u., II4 The behaviour of¢' at large distances, II7 Conditions for vq, to be determined uniquely, II9 The expression of¢' as a power series, I:ZO Irrotational solenoidal fl.ow due to a rigid body in translational motion, 122

2.10 Two-dimensional flow fields extending to infinity

124

Irrotational solenoidal Row due to a rigid body in translational motion, 128

Chapter 3. Equations Governing the Motion of a Fluid 3·1

Material integrals in a moving fl u id

1 31

Rates of change of material integrals, IJ3 Conservation laws for a fluid in motion, I35

3·2

The equation. of motion

1 37

Use of the momentum equation in integral form, IJ8 Equation of motion relative to moving axes, 139

3·3

The expression for the stress tensor

141

Mechanical definition of pressure in a moving fluid, 14I The relation between deviatoric stress and rate-of-strain for a Newtonian fluid, 1 42 The Navier-Stokes equation, I47 Conditions on the velocity and stress at a material boundary, I48

3·4

Changes in the internal energy of a fluid in motion

151

3·5

Bernoulli's theorem for steady flow of a frictionless nonconducting fluid

15 6

Special forms of Bernoulli's theorem, I6I Constancy of H across a transition region in one-dimensional steady Row, I63

3 .6

The complete set of equations governing fluid flow Isentropic Row, I6S Conditions for the velocity distribution to be approximately solenoidal, I67

3·7

Concluding remarks to chapters r, 2 and 3

1 71

Contents Chapter 4. Flow of 4· I

Uniform Incompressible Viscous Fluid

page 174

Introduction Modification of

4.2

a

VII

t he

pressure to allow for the effect of the body force, I76

Steady unidirectional flow

179

Poiseuille flow, ISo Tubes of non-circular cross-section, 182 Two-dimensional flow, I82

A model of a paint-brush, IS3 A remark on stability, ISS

4-3

Unsteady unidirectional flow

186

The smoothing-out of a discontinuity in velocity at a plane, 1S7 Plane boundary moved suddenly in a fluid at rest, IS9 One rigid boundary moved suddenly and one held stationary, I90 Flow due to an oscillating plane boundary, I9I Starting flow in a pipe, I93

4·4

The Ekman layer at a boundary in a rotating fluid

195

The layer at a free surface, 197 The layer at a rigid plane boundary, 199

4·5

Flow with circular streamlines

201

4.6

The steady jet from a point source of momentum

205

4·7

Dynamical similarity and the Reynolds number

2II

Other dimensionless parameters having dynamical significance, 2I 5

4.8

Flow fields in which inertia forces are negligible

216

Flow in slowly-varying channels, 2I7 Lubrication theory, 2I9 The Hele-Shaw cell, 222 Percolation through porous media, 223 Two-dimensional flow in a corner, 224 Uniqueness and minimum dissipation theorems, 227

4·9

Flow due to a moving body at small Reynolds number

229

A rigid sphere, 230 A spherical drop of a different fluid, 235 A body of arbitrary shape, 238

4.10 Oseen's improvement of the equation for flow due to moving bodies at small Reynolds number

240

A rigid sphere, 24I A rigid circular cylinder, 244

4.II The viscosity of a dilute suspension of small particles The flow due to a sphere embedded in a pure straining motion, 248 The increased rate of dissipation in an incompressible suspension, 250 The effective expansion viscosity of a liquid containing gas bubbles, 253

4.12 Changes in the flow due to moving bodies as R increases from 1 to about xoo

255

Contents

Vlll

Chapter 5. Flow at Large Reynolds Number: Effects of Viscosity 5.1

Introduction

5.2

Vorticity dynamics

page 264 266

The intensification of vorticity by extension of vortex-lines, 270

5·3

Kelvin's circulation theorem and vorticity laws for an inviscid

273

fluid The persistence of irrotationality, 276

5 ·4

The source of vorticity in motions generated from rest

277

5 ·5

Steady flows in which vorticity generated at a solid surface is

282

prevented by convection from diffusing far away from it

(a)

Flow along plane and circular walls with suction through the wall, 282 (b) Flow toward a 'stagnation point' at a rigid boundary, 285 (c) Centrifugal flow due to a rotating disk, 290

5 ·6

Steady two-dimensional flow in a converging or diverging

294

channel Purely convergent flow, 297 Purely divergent flow, 298 Solutions showing both outflow and inflow, 301

5 ·7

Boundary layers

302

5 ·8

The boundary layer on a flat plate

308

5 ·9

The effects of acceleration and deceleration of the external

314

stream The similarity solution for an external stream velocity proportional to xm, 316 Calculation of the steady boundary layer on a body moving through fluid, 318 Growth of the boundary layer in initially irrotational flow, 321

5.ro Separation of the boundary layer

325

5. I I The flow due to bodies moving steadily through fluid

331

Flow without separation, 332 Flow with separation, 337

5. I 2 Jets, free shear layers and wakes

343

Narrow jets, 343 Free shear layers, 346 Wakes, 348

5. I 3 Oscillatory boundary layers The damping force on an oscillating body, 355 Steady streaming due to an oscillatory boundary layer, 358 Applications of the theory of steady streaming, 361

353

Contents

IX

5.14 Flow systems with a free surface The boundary layer at a free surface, 364 The drag on a spherical gas bubble rising steadily through liquid, 367 The attenuation of gravity waves, 370

5· 15 Examples of use of the momentum theorem

372

The force on a regular array of bodies in a stream, 372. The effect of a sudden enlargement of a pipe, 373

Chapter 6. lrrotational Flow Theory and its Applications 6.1

The role of the theory of flow of an inviscid fluid

378

6.2

General properties of irrotational flow

380

Integration of the equation of motion, 382 Expressions for the kinetic energy in terms of surface integrals, 383 Kelvin's minimum energy theorem, 384 Positions of a maximum of q and a minimum of p, 384 Local variation of the velocity magnitude, 386

6.3

Steady flow: some applications of Bernoulli's theorem and the momentum theorem

386

Efflux from a circular orifice in an open vessel, 387 Flow over a weir, 391 Jet of liquid impinging on a plane wall, 392 Irrotational ftow which may be made steady by choice of rotating axes, 396

64 .

General features of irrotational flow due to a moving rigid body The The The The The

6.5

398

velocity at large distances from the body, 399 kinetic energy of the fluid, 402 force on a body in translational motion, 404 acceleration reaction, 407 force on a body in accelerating fluid, 409

Use of the complex potential for irrotational flow in two dimensions

409

Flow fields obtained by special choice of the function w(z), 410 Conforinal transformation of the plane of flow, 413 Transformation of a boundary into an infinite straight line, 418 Transformation of a closed boundary into a circle, 420 The circle thearem, 422

6.6

Two-dimensional irrotational flow due to a moving cylinder with circulation

423

A circular cylinder, 424 An elliptic cylinder in translational motion, 427 The force and moment on a cylinder in steady translational motion, 433

6.7

Two-dimensional aerofoils The practical requirements of aerofoils, 435 The generation of circulation round an aerofoil and the basis for Joukowski's hypothesis, 438 Aerofoils obtained by transformation of a circle, 441 Joukowski aerofoils, +44

435

Contents

X

6.8

Axisymmetric irrotational flow due to moving bodies

page 449

Generalities, 449 A moving sphere, 452 Ellipsoids of revolution, 455 Body shapes obtained from source singularities on the axis of symmetry, 458 Semi-infinite bodies, 460

6.9

Approximate results for slender bodies Slender bodies of revolution, 463 Slender bodies in two dimensions, 466 Thin aerofoils in two dimensions, 467

6.10 Impulsive motion of a fluid

471

Impact of a body on a free surface of liquid, 473

6. I I Large gas bubbles in liquid

474

A spherical-cap bubble rising through liquid under gravity, 475 A bubble rising in a vertical tube, 477 A spherical expanding bubble, 479

6.1 2 Cavitation in a liquid Examples of cavity formation in steady flow, 482 Examples of cavity formation in unsteady flow, 485 Collapse of a transient cavity, 486 Steady-state cavities, 49 I

6.13 Free-streamline theory, and steady jets and cavities

493

Jet emerging from an orifice in two dimensions, 495 Two-dimensional flow past a flat plate with a cavity at ambient pressure, 497 Steady-state cavities attached to bodies held in a stream of liquid, 502

Chapter 7. Flow of Effectively Inviscid Fluid with Vorticity 7.1

Introduction The self-induced movement of a line vortex, 509 The instability of a sheet vortex, 5 I I

7.2

Flow in unbounded fluid at rest at infinity The resultant force impulse required to generate the motion, SIS The total kinetic energy of the fluid, 520 Flow with circular vortex-lines, 52I Vortex rings, 522

7·3

Two-dimensional flow in unbounded fluid at rest at infinity

5 27

Integral invariants of the vorticity distribution, 528 Motion of a group of point vortices, 530 Steady motions, 532

7·4

Steady two-dimensional flow with vorticity throughout the fluid Uniform vorticity in a region bounded externally, 538 Fluid in rigid rotation at infinity, 539 Fluid in simple shearing motion at infinity, 541

536

7·5

Contents

xi

Steady axisymmetric flow with swirl

page 543

The effect of a change of cross-section of a tube on a stream of rotating fluid, 546 The effect of a change of external velocity on an isolated vortex, 550

7.6

Flow systems rotating as a whole

555

The restoring effect of Coriolis forces, SSS Steady flow at small Rossby number, 557 Propagation of waves in a rotating fluid, 559 Flow due to a body moving along the axis of rotation, 564

7·7

Motion in a thin layer on a rotating sphere Geostrophic flow, 571 Flow over uneven ground, 573 Planetary waves, 577

7.8

The vortex system of a wing

580

General features of the flow past lifting bodies in three dimensions, s8o Wings of large aspect ratio, and 'lifting-line' theory, 583 The trailing vortex system far downstream, 589 Highly swept wings, 591

Appendices Measured values of some physical properties of common fluids

594

(a) Dry air at a pressure of one atmosphere, 594

(b) The Standard Atmosphere, 595 (c) Pure water, 595

(d)

Diffusivities for momentum and heat at rs oc and I atm, 597

(e) Surface tension between two fluids, 597 2

Expressions for some common vector differential quantities in orthogonal curvilinear co-ordinate systems

Publications referred to in the text Subject Index

Plates I to 24 are between pages 364 and 365

598

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Xlll

PREFACE While teaching fluid dynamics to students preparing for the various Parts of the Mathematical Tripos at Cambridge I have found difficulty over the choice of textbooks to accompany the lectures. There appear to be many books intended for use by a student approaching fluid dynamics with a view to its application in various fields of engineerings but relatively few which cater for a student coming to the subject as an applied mathematician and none which in my view does so satisfactorily. The trouble is that the great strides made in our understanding of many aspects of fluid dynamics during the last 50 years or so have not yet been absorbed into the educational texts for students of applied mathematics. A teacher is therefore obliged to do without textbooks for large parts of his course, or to tailor his lectures to the existing books. This latter alternative tends to emphasize unduly the classical analytical aspects of the subject, and the mathematical theory of irrotational flow in particular, with the probable consequence that the students remain unaware of the vitally important physical aspects of fluid dynamics. Students, and teachers too, are apt to derive their ideas of the content of a subject from the topics treated in the textbooks they can lay hands on, and it is undesirable that so many of the books on fluid dynamics for applied mathematicians should be about problems which are mathematically solvable but not necessarily related to what happens in real fluids* 1 have tried therefore to write a textbook which can be used by students of applied mathematics and which incorporates the physical understanding and information provided by past research. Despite its bulk this book is genuinely an introduction to fluid dynamics; that is to say, it assumes no previous knowledge of the subject and the material in it has been selected to introduce a reader to the important ideas and applications. The book has grown out of a number of courses of lectures, and very little of the material has not been tested in the lecture room. Some of the material is old and well knownf some of it is relatively new; and for all of it i have tried to devise the presentation which appears to be best from a consistent point of view. The book has been prepared as a connected account intended to be read and worked on as a wholes or at least in large portions, rather than to be referred to for particular problems or methods. 1 have had the needs of second-, third- and fourth-year students of applied mathematics in British universities particularly in minds these being the needs with which 1 am most familiar, although 1 hope that engineering students will also find the book useful. The true needs of applied mathe-

xiv

Preface

maticians and engineers are nowadays not far apart. Both require above all an understanding of the fundamentals of fluid dynamics; and this can be achieved without the use of advanced mathematical techniques. Anyone who is familiar with vector analysis and the notation of tensors should have little difficulty with the purely mathematical parts of this work. The book is fairly heavily weighted with theory, but not with mathematics. Attention is paid throughout the book to the correspondence between observation and the various conceptual and analytical models of flow systems. The photographs of flow systems that are included are an essential part of the book? and will help the reader, I hope, to develop a sense of the reality that lies behind the theoretical arguments and analysis. This is particularly important for students who do not have an opportunity of seeing flow phenomena in a laboratory. The various books and lectures by L, Prandtl seem to me to show admirably the way to keep both theory and observation continually in mind, and I have been greatly influenced by them* Prandtl knew in particular the value of a clear photograph of a welldesigned experimental flow system, and many of the photographs taken by him are still the best available illustrations of boundary-layer phenomena. A word is necessary about the selection of topics in this book and the order in which they have been placed. My original intention was to provide between two covers an introduction to all the main branches of fluid dynamics, but I soon found that this comprehensiveness was incompatible with the degree of thoroughness that I also had in mind. 1 decided therefore to attempt only a partial coverage, at any rate so far as this volume is concerned. The first three chapters prepare the ground for a discussion of any branch of fluid dynamics^ and are concerned with the physical properties of fluids, the kinematics of a flow field, and the dynamical equations in general form. The purpose of these three introductory chapters is to show how the various branches of fluid dynamics fit into the subject as a whole and rest on certain idealizations or assumptions about the nature of the fluid or the motion. A teacher is unlikely to wish to include all this preliminary material in a course of lectures, but it can be adapted to suit a specialized course and will 1 hope be useful as background. In the remaining four chapters the fluid is assumed to be incompressible and to have uniform density and viscosity, 1 regard flow of an incompressible viscous fluid as being at the centre of fluid dynamics by virtue of its fundamental nature and its practical importance, Fluids with unusual properties are fashionable in research, but most of the basic dynamical ideas are revealed clearly in a study of rotational flow of a fluid with internal friction; and for applications in geophysics, chemical engineering, hydraulics, mechanical and aeronautical engineering, this

Preface

xv

is still the key branch of fluid dynamics. 1 regret that many important topics such as gas dynamics, surface waves, motion due to buoyancy forces, turbulence, heat and mass transfer, and magneto-fluid dynamics, are apparently ignored? but the subject is simply too large for proper treatment in one volume. If the reception given to the present book suggests that a second volume would be welcome, I may try later to make the coverage more nearly complete. As to the order of material in chapters 4 to 7, the description of motion of a viscous fluid and of flow at large Reynolds number precedes the discussion of irrotational flow (although the many purely kinematical properties of an irrotational velocity distribution have a natural place in chapter 2) and of motion of an inviscid fluid with vorticity. My reason for adopting this unconventional arrangement is not that I believe the' classical' theory of irrotational flow is less important than is commonly supposed. It is- simply that results concerning the flow of inviscid fluid can be applied realistically only if the circumstances in which the approximation of zero viscosity is valid are first made clear. The mathematical theory of irrotational flow is a powerful weapon for the solution of problems, but in itself it gives no information about whether the whole or a part of a given flow field at large Reynolds number will be approximately irrotational For that vital information some understanding of the effects of viscosity of a real fluid and of boundary-layer theory is essential; and, whereas the understanding was lacking when Lamb wrote his classic treatise Hydrodynamics, it is available today, i believe that the first book, at least in English, to show how so many common flow systems could be understood in terms of boundary layers and separation and vorticity movement was Modern Developments in Fluid Dynamics, edited by Sydney Goldstein. That pioneering book published in 1938 was aimed primarily at research workers, and 1 have tried to take the further step of making the understanding of the flow of real fluids accessible to students at an early stage of their study of fluid dynamics, Desirable though it is for study of the flow of viscous fluids to precede consideration of an inviscid fluid and irrotational flow, 1 appreciate that a lecturer may have his hand forced by the available lecturing time. In the case of mathematics students who are to attend only one course on fluid dynamics? of length under about 30 lectures^ it would be foolish to embark on a study of viscous fluid flow and boundary layers in preparation for a description of inviscid-fiuid flow and its applications, since too little time would be left for this topic; the lecturer would need to compromise with scientific logic, and could perhaps take his audience from chapters 2 and 3

xvi

Preface

to chapter 6, with some of the early sections of chapters 5 and 7 included. It is a difficulty inherent in the teaching of fluid dynamics to mathematics undergraduates that a partial introduction to the subject is unsatisfactory, tending to leave them with analytical procedures and results but no information about when they are applicable. Furthermore, students do take some time to grasp the principles of fluid dynamics^ and I suggest that 40 to 50 lectures are needed for an adequate introduction of the subject to non-specialist students. However, a book is not subject to the same limitations as a course of lectures. 1 hope lecturers will agree that it is desirable for students to be able to see all the material set out in logical order, and to be able to improve their own understanding of the subject by reading, even if in a course of lectures many important topics such as boundarylayer separation must be ignored. Exercises are an important part of the process of understanding and mastering so analytical a subject as fluid dynamics, and the reading of this text should be accompanied by the working of illustrative exercises. I should have liked to be able to provide many suitable questions and exercises, but a search among those already published in various places did not produce many in keeping with the approach adopted in this book. Moreover, the published exercises are concentrated on a small number of topics. The lengthy task of devising and compiling suitable exercises over the whole field of 'modern' fluid dynamics has yet to be undertaken. Consequently only a few exercises will be found at the end of sections. To some extent exercises ought to be chosen to suit the particular background and level of the class for which they are intended, and it may be that a lecturer can turn into exercises for his class many portions of the text not included explicitly in his course of lectures, as I have done in my own teaching. It is equally important that a course of lectures on the subject matter of this book should be accompanied by demonstrations of fluid flow. Here the assistance of colleagues in a department of engineering may be needed, The many films on fluid dynamics that are now available are particularly valuable for classes of applied mathematicians who do not undertake any laboratory work. By one means or another, a teacher should show the relation between his analysis and the behaviour of real fluids; fluid dynamics is much less interesting if it is treated largely as an exercise in mathematics, 1 am indebted to a large number of people for their assistance in the preparation of this book. Many colleagues kindly provided valuable comments on portions of the manuscript, and enabled me to see things more clearly. 1 am especially grateful to Philip Chatwin, John Elder, Emin

Preface

xvii

Erdogan, Ken Freeman, Michael Mclntyre, Keith Moffatt, John Thomas and Ian Wood who helped with the heavy task of checking everything in the proof. My thanks go also to those who supplied me with diagrams or photographs or who permitted reproduction from an earlier publication; to Miss Pamela Baker and Miss Anne Powell, who did the endless typing with patience and skill; and to the oiEEcers of Cambridge University Press, with whom it is a pleasure to work. G. K, Be Cambridge April ig6f

XV1U

CONVENTIONS AND NOTATION Bold type signifies vector character, x, x ' position vectors; |x| = r s = x — x ' relative position vector u velocity at a specified time and position in space; | u | = q D d ••- = — -f u . V operator giving the material derivative, or rate of change at a point moving with the fluid locally; applies only to functions of x and t System Rectilinear Polar, two dimensions Spherical polar Cylindrical A = V.u

Velocity components

Co-ordinates Xf yt

2> o r

X|,

X2J

Xg

rt0

r,0,

xy cr, (a2=y2~{-z2)

M, V, W

OF Uly U2f

%

w, v or uTf UQ u, v, w or urt Ufff tifp u, vy w or uXi uai u^

rate of expansion (fractional rate of change of volume of a material element)

w = V x u vorticity (twice the local angular velocity of the fluid) 1 [duj duA e a = - 17;—!— I rate-of-strain tensor 2 \CXi

OXjj

scalar potential of an irrotational velocity distribution (u = V0) B vector potential of a solenoidal velocity distribution (u = v7 x B) f stream function for a solenoidal velocity distribution; (a) two-dimensional flow: B = (o, o, f) 1 df df df df u = —-, v = —-—- or uT = ~rTr u0 = - oy ox (b) axisymmetric flow: ¥ dip 1 df f cylindrical co-ordinates BA = —, 1 df 1 df polar co-ordinates B& v = —.—-, r sin 0 n unit normal to a surface, usually outward if the surface is closed SVf TidAy Sx volume, surface and line elements with a specified position in space ST9 mSS$ SI material volume, surface and line elements crit stress tensor; (r^TtjSA is the /-component of the force exerted across the surface element nSA by the fluid on the side to which 11 points F = — VY conservative body force per unit mass Inertia force (per unit mass) m i n u s the local acceleration Vortex-line line whose tangent is parallel to fai locally Line vortex singular line in vorticity distribution r o u n d which the circulation is non-zero Books which may provide collateral reading are cited in detail in the text, usually in footnotes. A comparatively small n u m b e r of original papers are also referred to, sometimes for historical interest, sometimes because a precise acknowledgement is appropriate^ and sometimeSj although only rarelyf as a guide to further reading on a particular topic. T h e s e papers are cited in t h e text as s S m i t h (1950) \ and the full references for both papers and books are listed at the end of the book.

1 THE PHYSICAL OF

PROPERTIES

FLUIDS

L I . SolidSj liquids and gases The defining property of fluids, embracing both liquids and gases, lies in the ease with which they may be deformed. A piece of solid material has a definite shape, and that shape changes only when there is a change in the external conditions. A portion of fluid, on the other hand, does not have a preferred shape, and different elements of a homogeneous fluid may be rearranged freely without affecting the macroscopic properties of the portion of fluid. The fact that relative motion of different elements of a portion of fluid can, and in general does, occur when forces act on the fluid gives rise to the science of fluid dynamics. The distinction between solids and fluids is not a sharp one, since there are many materials which in some respects behave like a solid and in other respects like a fluid. A c simple' solid might be regarded as a material of which the shape, and the relative positions of the constituent elements^ change by a small amount only, when there is a small change in the forces acting on it. Correspondingly, a€ simple? fluid (there is no one term in general use) might be defined as a material such that the relative positions of the elements of the material change by an amount which is not small when suitably chosen forces? however small in magnitude, are applied to the material. But, even supposing that these two definitions could be made quite precise, it is known that some materials do genuinely have a dual character. A thixotropic substance such as jelly or paint behaves as an elastic solid after it has been allowed to stand for a time, but if it is subjected to severe distortion by shaking or brushing it loses its elasticity and behaves as a liquid. Pitch behaves as a solid normally, but if a force is imposed on it for a very long time the deformation increases indefinitely, as it would for a liquid. Even more troublesome to the analyst are those materials like concentrated polymer solutions which may simultaneously exhibit solid-like and fluidlike behaviour. Fortunately, most common fluids, and air and water in particular, are quite accurately simple in the above sense, and this justifies a concentration of attention on simple fluids in an introductory text. In this book we shall suppose that the fluid under discussion cannot withstand any tendency by applied forces to deform it in a wTay which leaves the volume unchanged. The implications of this definition will emerge later, after we have examined the nature of forces that tend to deform an element of fluid. In the meantime [i]

2

The physical properties of

fluids

[x.i

it should be noted that a simple fluid may offer resistance to attempts to deform it; what the definition implies is that the resistance cannot prevent the deformation from occurring, or, equivalently, that the resisting force vanishes with the rate of deformation, Since we shall be concerned exclusively with the kind of idealized material described here as a simple fluid, there is no need to use the term further. We shall therefore refer only to fluids in subsequent pages. The distinction between liquids and gases is much less fundamental, so far as dynamical studies are concerned. For reasons related to the nature of intermolecular forces, most substances can exist in either of twro stable phases which exhibit the property of fluidity, or easy deformability. The density of a substance in the liquid phase is normally much larger than that in the gaseous phase, but this is not in itself a significant basis for distinction since it leads mainly to a difference in the magnitudes of forces required to produce given magnitudes of acceleration rather than to a difference in the types of motion. The most important difference between the mechanical properties of liquids and gases lies in their bulk elasticity, that is, in their compressibility. Gases can be compressed much more readily than liquids, and as a consequence any motion involving appreciable variations in pressure will be accompanied by much larger changes in specific volume in the case of a gas than in the case of a liquid. Appreciable variations in pressure in a fluid must be reckoned with in meteorology, as a .result of the action of gravity on the whole atmosphere, and in very rapid motions, of the kind which occur in ballistics and aeronautics, resulting from the motion of solid bodies at high speed through the fluid. It will be seen later that there are common circumstances in which motions of a fluid are accompanied by only slight variations in pressure, and here gases and liquids behave similarly since in both cases the changes in specific volume are slight. The gross properties of solids, liquids and gases are directly related to their molecular structure and to the nature of the forces between the molecules. We may see this superficially from a consideration of the general form of the force between two typical molecules in isolation as a function of their separation, At small values of the distance d between the centres of the molecules, of order io~ 8 cm for molecules of simple type, the mutual reaction is a strong force of quantum origin, being either attractive or repulsive according to the possibility of 'exchange' of electron shells. When exchange is possible, the force is attractive and constitutes a chemical bond; when exchange is not possible, the force is repulsive, and falls off very rapidly as the separation increases. At larger distances between the centres, say of order IG~~7 or io~ 6 cm, the mutual reaction between the two molecules (assumed to be un-ionized, as is normally the case at ordinary temperatures) is a weakly attractive force. This cohesive force, is believed to fall off first as d~~7 and ultimately as d~8 when d is large, and may be regarded, crudely speaking, as being due to the electrical polarization of each molecule under

I.I]

Solids, liquids and gases

3

the influence of the other.f The mutual reaction as a function of d for two molecules not forming a chemical bond thus has the form shown in figure 1.1.1. At separation dm at which the reaction changes sign? one molecule is clearly in a position of stable equilibrium relative to that of the other, d0 is of order 3-4 x io~ 8 cm for most simple molecules. From a knowledge of the mass of a molecule and the density of the corresponding substance* it is possible to calculate the average distance between the centres of adjoining molecules. For substances composed of simple molecules, the calculation shows that the average spacing of the molecules in a gaseous phase at normal temperature and pressure is of the order of iod0> whereas the average spacing in liquid and solid phases is of

3

u

"o

e

Repulsion

"73 C

q

o Attraction

Figure 1.1.1. Sketch of the force exerted by one (un-ionized) simple molecule on another as a function of the distance d between their centres.

order dQ. In gases under ordinary conditions the molecules are thus so far apart from each other that only exceedingly weak cohesive forces act between thems except on the rare occasions when two molecules happen to come close together; and in the kinetic theory of gases it is customary to postulate a * perfect gas1, for which the potential energy of a molecule in the force fields of its neighbours is negligible by comparison with its kinetic energy; that is? a gas in which each molecule moves independently of its neighbours except when making an occasional 'collision*, in liquid and solid phases? on the other hand, a molecule is evidently well within the strong force fields of several neighbours at all times. The molecules are here packed together almost as closely as the repulsive forces will allow. In the case of a solid the arrangement of the molecules is virtually permanent and may have a simple periodic structure^ as in a crystal.; the molecules oscillate about their stable positions (the kinetic energy of this oscillation being part of the thermal energy of the solid), but the molecular lattice remains intact until the temperature of the solid is raised to the melting point. f See, for instance, States of Matter, by E. A. Moelwyn-Hughes (Oliver and Boyd, 1961).

4

The physical properties of

fluids

[1,2

The density of most substances falls by several per cent on melting (the increase in density in the transition from ice to water being exceptional), and it is paradoxical that such a small change in the molecular spacing is accompanied by such a dramatic change in the mobility of the material Knowledge of the liquid state is still incomplete, but it appears that the arrangement of the molecules is partially ordered, with groups of molecules as a whole having mobility5 sometimes falling into regular array with other groups and sometimes being split up into smaller groups. The arrangement of the molecules is continually changing, and5 as a consequence, any force applied to the liquid (other than a bulk compression) produces a deformation which increases in magnitude for so long as the force is maintained. The manner in which some of the molecular properties of a liquid stand between those of a solid and a gas is shown in the following table. In the matter of the simplest macroscopic quantity, viz. density, liquids stand much closer to solids; and in the matter of fluidity, liquids stand wholly with gases.

Intermolecular forces

Ratio of amplitude of random thermal movement of molecules to d0

solid liquid

strong medium

1

Molecular arrangement ordered partially ordered disordered

Type of statistics needed quantum quantum + classical classical

The molecular mechanism by which a liquid resists an attempt to deform it is not the same as that in a gas, although, as we shall see, the differential equation determining the rate of change of deformation has the same form in the two cases. L2o The continuum hypothesis The molecules of a gas are separated by vacuous regions with linear dimensions much larger than those of the molecules themselves. Even in a liquid, in which the molecules are nearly as closely packed as the strong short-range repulsive forces will allow, the mass of the material is concentrated in the nuclei of the atoms composing a molecule and is very far from being smeared uniformly over the volume occupied by the liquid. Other properties of a fluid, such as composition or velocity, likewise have a violently non-uniform distribution when the fluid is viewed on such a small scale as to reveal the individual molecules, However, fluid mechanics is normally concerned with the behaviour of matter in the large? on a macroscopic scale large compared with the distance between molecules, and it will not often happen that the molecular structure of a fluid need be taken into account explicitly. We shall suppose, throughout this book? that the macroscopic behaviour of fluids is the same as if they were perfectly continuous in

1.2]

The continuum hypothesis

structure; and physical quantities such as the mass and momentum associated with the matter contained within a given small volume will be regarded as being spread uniformly over that volume instead of, as in strict reality, being concentrated in a small fraction of it, The validity of the simpler aspects of this continuum hypothesis under the conditions of everyday experience is evident. Indeed the structure and properties of air and water are so obviously continuous and smoothlyvarying, when observed with any of the usual measuring devices, that no different hypothesis would seem natural. When a measuring instrument is inserted in a fluid, it responds in some way to a property of the fluid within some small neighbouring volume^ and provides a measure which is effectively an average of that property over

Variation due to molecular fluctuations Variation associated A to spattai d'f^nbuuoii of dens^y

'Local' value of fi*id density

Volume of fluid to which instrument responds F i g u r e 1.2.1. Effect of size of sensitive v o l u m e 011 the density m e a s u r e d by an i n s t r u m e n t .

the csensitive* volume (and sometimes also over a similar small sensitive time). The instrument is normally chosen so that the sensitive volume is small enough for the measurement to be a c local? one; that is, so that further reduction of the sensitive volume (within limits) does not change the reading of the instrument. The reason why the particle structure of the fluid is usually irrelevant to such a measurement is that the sensitive volume that is --small enough for the measurement to be * local' relative to the macroscopic scale is nevertheless quite large enough to contain an enormous number of molecules^ and amply large enough for the fluctuations arising from the different properties of molecules to have no effect on the observed average. Of course, if the sensitive volume is made so small as to contain only a few molecules, the number and kind of molecules in the sensitive volume at the instant of observation will fluctuate from one observation to another and the measurement will vary in an irregular way with the size of the sensitive volume. Figure 1.2.1 illustrates the way in which a measurement of density of the fluid would depend on the sensitive volume of the instrument.

6

The physical properties of

fluids

[1.2

We are able to regard the fluid as a continuum when, as in the figures the measured fluid property is constant for sensitive volumes small on the macroscopic scale but large 011 the microscopic scale. One or two numbers will indicate the great difference between the length scale representative of the fluid as a whole and that representative of the particle structure For most laboratory experiments with fluids^ the linear dimensions of the region occupied by the fluid is at least as large as 1 cm and very little variation of the physical and dynamical properties of the fluid occurs over a distance of io"~3 cm (except perhaps in special places such as in a shock wave); thus an instrument with a sensitive volume of io~ 9 cm3 would give a measurement of a local property. Small though this volume is, it contains about 3 x 1010 molecules of air at normal temperature and pressure (and an even larger number of molecules of water) which is large enough, by a very wide margin, for an average over the molecules to be independent of their number. Only under extreme conditions of low gas density, as in the case of flight of a missile or satellite at great heights above the earth's surface, or of very rapid variation of density with position, as in a shock wave, is there difficulty in choosing a sensitive volume which gives a local measurement and which contains a large number of molecules. Our hypothesis implies that it is possible to attach a definite meaning to the notion of value s at a point J of the various fluid properties such as density, velocity and temperature, and that in general the values of these quantities are continuous functions of position in the fluid and of time. On this basis we shall be able to establish equations governing the motion of the fluid which are independent, so far as their form is concerned, of the nature of the particle structure—so that gases and liquids are treated together—and indeed, independent of whether any particle structure exists. A similar hypothesis is made in the mechanics of solids5 and the two subjects together are often designated as continuum mechanics. Natural though the continuum hypothesis may be, it proves to be difficult to deduce the properties of the hypothetical continuous medium that moves in the same way as a real fluid with a given particle structure. The methods of the kinetic theory of gases have been used to establish the equations determining the 'local' velocity (defined as above) of a gas, and, with the help of simplifying assumptions about the collisions between molecules^ it may be shown that the equations have the same form as for a certain continuous fluid although the values of the molecular transport coefficients (see §1.6) are not obtained accurately. The mathematical basis for the continuum treatment of gases in motion is beyond our scope, and it is incomplete for liquids, so that we must be content to make a hypothesis. There is ample observational evidence that the common real fluids? both gases and liquids, move as if they were continuous, under normal conditions and indeed for considerable departures from normal conditions, but some of the properties of the equivalent continuous media need to be determined empirically.

I,J]

Volume forces and surface forces acting on a

fluid

7

1 3 . Volume forces and surface forces acting on a fluid It is possible to distinguish two kinds of forces which act on matter in bulk, in the first group are long-range forces like gravity which decrease slowly with increase of distance between interacting elements and which are still appreciable for distances characteristic of natural fluid flows. Such forces are capable of penetrating into the interior of the fluid, and act on all elements of the fluid. Gravity is the obvious and most important example, but two other kinds of long-range force of interest in fluid mechanics are electromagnetic forces, which may act when the fluid carries an electric charge or when an electric current passes through it, and the fictitious forces, such as centrifugal force, which appear to act on mass elements when their motion is referred to an accelerating set of axes. A consequence of the slow variation of one of these long-range forces with position of the element of fluid on which it is acting is that the force acts equally on all the matter within a small element of volume and the total force is proportional to the size of the volume element. Long-range forces may thus also be called volume or body forces. When writing equations of motion in general form, we shall designate the total of all body forces acting at time t on the fluid within an element of volume SV surrounding the point whose position vector is x by F(x9t)p8V;

(1.3.1)

the factor p has been inserted because the two common types of body force per unit volume—gravity and the fictitious forces arising from the use of accelerating axes—are in fact proportional to the mass of the element on which they act. in the case of the earth's gravitational field the force per unit mass is — F = g, the vector g being constant in time and directed vertically downwards, In the second group are short-range forces, which have a direct molecular origin, decrease extremely rapidly with increase of distance between interacting elements, and are appreciable only when that distance is of the order of the separation of molecules of the fluid. They are negligible unless there is direct mechanical contact between the interacting elements, as in the case of the reaction between two rigid bodies, because without that contact none of the molecules of one of the elements is sufficiently close to a molecule of the other element. The short-range forces exerted between two masses of gas in direct contact at a common boundary are due predominantly to transport of momentum across the common boundary by migrating molecules. In the case of a liquid the situation is more complex because there are contributions to the short-range or contact forces from transport of momentum across the common boundary by molecules in oscillatory motion about some quasi-stationary position and from the forces between molecules on the two

8

The physical properties of

fluids

[IA

sides of the common boundary; both these contributions have large magnitude, but they act approximately in opposite directions and their resultant normally has a much smaller magnitude than either. However, as already remarked, the laws of continuum, mechanics do not depend on the nature of the molecular origin of these contact forces and we need not enquire into the details of the origin in liquids, at this stage. If an element of mass of fluid is acted on by short-range forces arising from reactions with matter (either solid or fluid) outside this element, these short-range forces can act only on a thin layerf adjacent to the boundary of the fluid element, of thickness equal to the ' penetration ? depth of the forces. The total of the short-range forces acting on the element is thus determined by the surface area of the element, and the volume of the element is not directly relevant. The different parts of a closed surface bounding an element of fluid have different orientations, so that it is not useful to specify the short-range forces by their total effect on a finite volume element of fluid; instead we consider a plane surface element in the fluid and specify the local short-range force as the total force exerted on the fluid on one side of the element by the fluid on the other side. Provided the penetration depth of the short-range forces is small compared with the linear dimensions of the plane surface element? this total force exerted across the element will be proportional to its area 8A and its value at time t for an element at position x can he ^ r ntten as the vector „ A aA - „ _ -\ w where n is the unit normal to the element. The convention to be adopted here is that S is the stress exerted by the fluid on the side of the surface element to which ii points, on the fluid on the side which n points away from; so a normal component of S with the same sense as n represents a tension. The force per unit area, S, is called the local stress. The way in which it depends on n is determined below. The force exerted across the surface element on the fluid on the side to which n points is of course — Z(n, x, t) 8A, and since this is also the force represented by S( — n, x, t) 8A we see that Z must be an odd function of n. In chapter 3 we shall formulate equations describing the motion of a fluid which is subject to long-range or body forces represented by (1.3.1) and short-range or surface forces represented by (1.3,2). Forces of these two kinds act also on solids? and their existence is perhaps more directly evident to the senses for a solid than for a fluid medium. In the case of a solid body which is rigid, only the short-range forces acting at the surface of the body (say, as a result of mechanical contact with another rigid body) are relevant, and it is a simple matter to determine the body's motion when the total body force and the total surface force acting on it are known. When the solid body f Unless the element is chosen to have such small linear dimensions that the short-range forces exerted by external matter are still significant at the centre of the element; but the element would then contain only a few molecules at most, and representation of the fluid as a continuum would not be possible.

.,:, 31

Volume forces and surface forces acting on a

fluid

9

is deformable, and likewise in the case of a fluid, the different material elements are capable of different movements, and the distribution of the body and surface forces throughout the matter must be considered; moreover. both the body and surface forces may be affected by the relative motion of materia! elements. The way in which body forces depend on the local properties of the fluid is evident, at any rate in the cases of gravity and the fictitious forces due to accelerating axes, but the dependence of surface forces on the local properties and motion of the fluid will require examination. Representation of surface forces by the stress tensor Some information about the stress S may be deduced from its definition as a force per unit area and the law of motion for an element of mass of the fluid. First we determine the dependence of S on the direction of the normal to the surface element across which it acts. Consider all the forces acting instanfe taneously on. the fluid within an element ! of volume SV in the shape of a tetrahedron yk as shown in figure 1.3.1, The three ortho/? \ gonal faces have areas 8AV 8A2J SA& and / J 8A, \ v unit (outward) normals — a, — b ? — c, and /s -:fL ^ the fourth inclined face has area 8A and '* . A^ „--" unit normal n . Surface forces will act on ,• •• ^~,?, T) = o,

(1.5,1)

thereby exhibiting formally the arbitrariness of the choice of the two parameters of state, is called an equation of state. For every quantity like temperature which describes the fluid, but excluding the two parameters of state of course, there is an equation of state, Another important quantity describing the state of the fluid is the internal energy per unit mass, E say. J Work and heat are regarded as equivalent forms f See, for instance. Classical Thermodynamics, by A. B. Pippard (Cambridge University Press? 1957). I The usual practice in the literature of thermodynamics is to use a capital letter for the total amount of some extensive quantity like internal energy in the system under consideration, and a small letter for the amount per unit mass. Introduction of the latter quantity alone is sufficient in fluid dynamics, and the use of a capital letter for it is conventional.

22

The physical properties of

fluids

[1,5

of energy, and the change in the internal energy of a mass of fluid at rest consequent on a change of state is defined, by the first law of thermodynamics, as being such as to satisfy conservation of energy when account is taken of both heat given to the fluid and work done on the fluid. Thus if the state of a given uniform mass of fluid is changed by a gain of heat of amount Q per unit mass and by the performance of work on the fluid of amount W per unit mass, the consequential increase in the internal energy per unit mass is F &E = Q+W. (1.5.2) The internal energy E is a function of the parameters of state, and the change A2?, which may be either infinitesimal or finite^ depends only on the initial and final states; but Q and W are measures of external effects and may separately (but not in sum) depend also on the particular way in which the transition between the two states is made. If the mass of fluid is thermally isolated from its surroundings so that no exchange of heat can occur, Q = o and the change of state of the fluid is said to be adiabaiic. There are many ways of performing work on the system, although compression of the fluid by inward movement of the bounding walls is of special relevance in fluid mechanics. An analytical expression for the work done by compression is available in the important case in which the change occurs reversibly. This word implies that the change is carried out so slowly that the fluid passes through a succession of equilibrium states, the direction of the change being without effect. At each stage of a reversible change the pressure in the fluid is uniform,f and equal top say? so that the work done on unit mass of the fluid as a consequence of compression leading to a small decrease in volume J is —pSv. Thus for a reversible, transition from one state to another, neighbouring, state we have SE = SQ-pSv.

(1.5,3)

A finite reversible change of this kind can be described by summing (1.5.3) over the succession of infinitesimal steps making up the finite change; the particular path by which the initial and final equilibrium states are joined is relevant here, because p is not in general a function of v alone. A practical quantity of some importance is the specific heat of the fluid? that is, the amount of heat given to unit mass of the fluid per unit rise in temperature in a small reversible change. A complete discussion of specific heat is best preceded by the second law of thermodynamics, but we may first see a direct consequence of the first law. The specific heat may be written as c = 8Q/8T, x

(1.5.4)

If a fluid at rest is acted on by a body force, the pressure varies throughout the fluid,, as we have seen, but the pressure variation may be made negligibly small by consideration of a portion of fluid of small volume. The thermodynamical arguments refer to local properties of the fluid when a body force acts. I Note that our definition of a simple fluid in §1.1 implies that no work is done on a fluid during a reversible change if only the shape and not the volume changes.

i.5]

Classical thermodynamics

23

and is not determined uniquely until we specify further the conditions under which the reversible change occers. An equilibrium state of the fluid may be represented as a point on a (/>, #)-plane, or indicator diagram, and a small reversible change (8p,8v) starting from a point A (see figure 1.5.1/ may proceed in any direction, if the only work done on the fluid is that done by compression, the heat 8Q which must be supplied to unit mass is determined by (1,5.3) as

and the change in temperature is

"•-(g).»+(sV The specific heat thus depends on the ratio SpjSv, that iss on the choice of direction of the change from A. f

*! 1

Adsabsatic ^> ** change

\

X x

1 \J^ X*

'S.X X^'^

Isothermal , change

A X>.

Figure 1.5,1, Indicator diagram for the equilibrium states of a fluid.

Two well-defined particular choices are changes parallel to the axes of the indicator diagram, giving the^rX^XX specif!" heats

^•5-5) Now ST ¥aries sinusoidaiiy as the point representing the final stace moves round a circle of small radius centred on A, being zero on the isotherm through A and a maximum in a direction HI normal to the isotherm. Likewise 8Q varies sinusoidaiiy, being zero on the adiabatic line through A and a maximum in a direction 11 normal to it. Thus if (mm mp) and (nm np) be the components of the two unit vectors, C = *«(*£)"»"•

n c

=

p(*8)max.

24

The physical properties of

fluids

[1,5

and since —mjmp and —nv/np are the gradients of the isothermal and adiabatic lines, we have for the ratio of the principal specific heats, denoted by % 1

~" €v " V ™ p ~ W a d i a b . / W r

0l

W / T / W/adiab.'

^'5'°

The weighted ratio —vSp/8v of the increments in J> and v in a small reversible change is the bulk modulus of elasticity of the fluid; also useful for fluid dynamical purposes is its reciprocal, the coefficient of compressibility — Sv/(v Sp)} or SpJ(pSp). Like the specific heat, the bulk modulus takes a different value for each direction of the change in the indicator diagram. Adiabatic and isothermal changes correspond to two particular directions, with special physical significance, and, somewhat surprisingly, the first law requires the ratio of the two corresponding bulk moduli to equal the ratio of the principal specific heats, It is clearly possible to draw a line defining the direction of a small reversible change involving no gain or loss of heat through each point of the indicator diagram, and to regard the family of these adiabatic lines as lines of equal value of some new function of state. The properties of this function are the subject of the second law of thermodynamics. The second law can be stated in a number of apparently different but equivalent ways, none of which is easy to grasp. Our use of the law will be indirect, and will not require any of the usual statements, it will in fact be sufficient for our purposes to know that the second law of thermodynamics implies the existence of another extensive property of the fluid in equilibrium (even for systems with more than two independent parameters of state), termed the entropy, such that, in a reversible transition from an equilibrium state to another, neighbouring, equilibrium state, the increase in entropy is proportional to the heat given to the fluid, and that the constant of proportionality, itself a function of state, depends only on the temperature and can be chosen as the reciprocal of the temperature. Thus, with entropy per unit mass of a fluid denoted by Sf we have ~. 0 ^ J- oS = oy, (1.5.7; where 8Q is the infinitesimal amount of heat given reversibly to unit mass of the fluid. This is the means by which the thermodynamic or absolute scale (unrelated to the properties of any particular material) of temperature is defined. An adiabatic reversible transition takes place at constant entropy, and so is said to be iseniropic. Moreover, it is a consequence of the second law that, in an adiabatic irreversible change, the entropy cannot diminish (when T is chosen to be positive); any change in the entropy must be an increase. Since both (1.5.3) ariC^ (1 -5 • 7) ^pply to reversible changes, it follows that, for a small reversible change in which work is done on the fluid by comPresslon>

T8S = 3E+p8v.

(1.5,8)

T 5]

Classical thermodynamics

25

Now the initial and final values of S and 2?, as of all other functions of state, are fully determined by the initial and final states, and consequently the relation (1.5.8), which contains only functions of state, must be valid for any infinitesimal transition in which work is done by compression, whether reversible or not. If the transition is irreversible, the equality (1,5.7) is not valid, and neither is that between 8W and — p Sv. Another function of state which, like internal energy and entropy, proves to be convenient for use in fluid mechanics, particularly when effects of compressibility of the fluid are important, is the enthalpy, or heat function. The enthalpy of unit mass of fluid, 1 say, is defined as I = E+pv, (1.5,9) and has the dimensions of energy per unit mass. A small change in the parameters of state corresponds to small changes in the functions /, E and S which are related by 8I = 8E+p8v + v 8p, = T8S + v8p

(1,5.10)

in view of (1.5.8). The relation (1.5.10), like (1.5.8), involves only functions of state, and is consequently independent of the manner in which the fluid might be brought from one to the other of the two neighbouring states, For a reversible small change at constant pressure, it appears from (1=5/7) that Yet another important function of state with the dimensions of energy is the Helmholtz free energy, of which the amount per unit mass is defined as F = E-TS. The small change in F consequent on small changes in the parameters of state is given by ^S8T3 SF = __p Sv showing that the gain in free energy per unit mass in a small isothermal change, whether reversible or not, is equal to —p 8v; and when this small isothermal change is reversible, the gain in free energy is equal to the work done on the system. Four useful identities, known as Maxwell's thermodynamic relations, follow from the above definitions of the various functions of state. To obtain the first of these relations, we note from (1,5.8) that, if v and 5 are now regarded as the two independent parameters of state on which all functions of state depend, the two partial, derivatives of E are /8E\

(&E\

.

where the subscript serves as a reminder of the variable held constant, The double derivative d2E/dv 8S may now be obtained in two different ways? yielding the relation 8

i\ -

8SL '

r)-

26

T~-?