an introduction to fluid batchelor f.r.s.(cup 2000 631s)

AN INTRODUCTION TO

FLUID DYNAMICSBY

G. K. BATCHELOR, F.R.S.Prof.,1Or of Applid Mathmatit' in 1M Un;v.,tity of Cambridg

.. :.:. CAMBRIDGE::; UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc6n 13.28014 Madrid, Spain Dock House, The Waterfront, Cape Town 800 1, South Africa

http://www.cambridge.org Cambridge University Press 1967, 1973, 2000 This book 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. 1991, 1992, 1994, 1996, 1998 First Cambridge Mathematical Library edition 2000 Reprinted 200 1, 2002 Printed in the United States of America

A catalog record for this book is available from the British Library Library of Congress Catalogue card number: 67-21953ISBN 0 521 66396 2 paperback

v

CONTENTSPrefaceConventions and NotationChapter 1. The Physical Properties of Fluids 1.1 Solids, liquids and gases Volume forces and surface forces acting on a fluidRepresentation of surface forces by the stress tensor, 9 The stress tensor in a fluid at rest, 12

page xiiiXVlll

I 4

1.2 The continuum hypothesisI~3

714

14 Mechanical equilibrium of a fluidA body' floating' in fluid at rest, 16 Fluid at rest under gravity, 18

15 Classical thermodynamics 1.6 Transport phenomenaThe 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

20 28

17 The distinctive properties of gasesA 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, 50

37

1.8 The distinctive properties of liquidsEquilibrium properties, 55 Transport coefficients, 57

53

1.9 Conditions at a boundary between two mediaSurface tension, 60 Equilibrium shape of a boundary between two stationary fluids, 63 Transition relations at a material boundary, 68

60

Chapter 2. Kinematics of the Flow Field 2.1 2.2 23 Specification of the flow fieldDifferentiation following the motion of the fluid, 72

7173 79

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

Analysis of the relative motion near a pointSimple shearing motion, 83

VI

ContentsExpression for the velocity distribution with specified rate of expansion and vorticity Singularities in the rate of expansion. Sources and sinks The vorticity distributionLine vortices, 93 Sheet vortices, 96

2.4

page 8488

2.5 2.6

92 99

2.7

Velocity distributions with zero rate of expansion and zero vorticityConditions for: Vt/J to be determined uniquely, xoa Irrotational solenoidal flow near a stagnation point, xOS The complex potential for irrotational solenoidal flow in two dimensions, x06

2.8 Irrotational solenoidal flow in doubly-connected regions of spaceConditions for Vt/J to be determined uniquely, IIa

108

2.9 Three-dimensional flow fields extending to infinityAsymptotic expressions for u. and lie, xx.... The behaviour of t/J at large distances, xx, Conditions for Vt/J to be determined uniquely, II9 The expression of tP as a power series, 120 Irrotational solenoidal flow due to a rigid body in translational motion, xaa2.10

114

Two-dimensional flow fields extending to infinityIrrotational solenoidal flow due to a rigid body in translational motion, 128

124

3.1

Chapter 3. Equations Governing the Motion of a Fluid Material integrals in a moving fluidRates of change of material integrals, x33 Conservation laws for a fluid in motion, X3S

131

3.2

The equation of motionUse of the momentum equation in integral form, X38 Equation of motion relative to moving axes, x39

137

3.3 The expression for the stress tensor

141

Mechanical definition of pressure in a moving fluid, x.... x The relation between deviatoric stress and rate-of-strain for a Newtonian fluid, x4a The Navier-Stokes equation, x...., Conditions on the velocity and stress at a material boundary, x48

3.4 Changes in the internal energy of a fluid in motion 3.5 Bernoulli's theorem for steady flow of a frictionless nonconducting fluidSpecial forms of Bernoulli's theorem, x6x Constancy of H across a transition region in one-dimensional steady flow, x63

151 156

3.6 The complete set of equations governing fluid flowIsentropic flow, J6S Conditions for the velocity distribution to be approximately solenoidal, x67

164

3.7 Concluding remarks to chapters

I, 2

and 3

171

Contents

Vll

..

Chapter 4. Flow of a Uniform Incompressible Viscous Fluid 4.1 Introduction page 174Modification of the pressure to allow for the effect of the body force, 176

4.2 Steady unidirectional flowPoiseuille flow, 18o Tubes of non-circular cross-section, [82 Two-dimensional flow, 182 A model of a paint-brush, 183 A remark on stability, 18S

179

43 Unsteady unidirectional flowThe smoothing-out of a discontinuity in velocity at a plane, 187 Plane boundary moved suddenly in a fluid at rest, 189 One rigid boundary moved suddenly and one held stationary, 190 Flow due to an oscillating plane boundary, 191 Starting flow in a pipe, 193

186

44 The Ekman layer at a boundary in a rotating fluidThe layer at a free surface, 197 The layer at a rigid plane boundary, 199

195 201 20 5 2112I S

45 Flow with circular streamlines 4. 6 The steady jet from a point source of momentum 47 Dynamical similarity and the Reynolds numberOther dimensionless parameters having dynamical significance,

4. 8 Flow fields in which inertia forces are negligibleFlow in slowly-varying channels, 217 Lubrication theory, 219 The Hele-Shaw cell, 222 Percolation through porous media, 223 Two-dimensional flow in a comer, 224 Uniqueness and minimum dissipation theorems, 227

216

49 Flow due to a moving body at small Reynolds numberA rigid sphere, 230 A spherical drop of a different fluid, 235 A body of arbitrary shape, 238

229

4. 10 Oseen's improvement of the equation for flow due to moving bodies at small Reynolds numberA rigid sphere, 241 A rigid circular cylinder, 244

240

4. 11 The viscosity of a dilute suspension of small particlesThe 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

24 6

4. 12 Changes in the flow due to moving bodies as R increases from 255 1 to about 100

V111

...

Contents

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

5. 1 Introduction5.2 Vorticity dynamicsThe intensification of vorticity by extension of vortex-lines, 270

page 264266 273

5.3

Kelvin's circulation theorem and vorticity laws for an inviscid fluidThe persistence of irrotationality, 276

5.4 The source of vorticity in motions generated from rest

277 282

5.5 Steady flows in which vorticity generated at a solid surface isprevented 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, 21}O

5.6 Steady two-dimensional flow in a converging or diverging channelPurely convergent flow, 297 Purely divergent flow, 298 Solutions showing both outflow and inflow, 301

294

5.7 Boundary layers 5.8 The boundary layer on a flat plate

302

308

5.9 The effects of acceleration and deceleration of the external 314 stream The similarity solution for an extemalstream velocity proportional to x"', 316Calculation of the steady boundary layer on a body moving through fluid, 318 Growth of the boundary layer in initially irrotational flow, 321

5.10 Separation of the boundary layer5. II

32 533 1

The flow due to bodies moving steadily through fluidFlow without separation, 33a Flow with separation, 337

5. 12 Jets, free shear layers and wakesNarrow jets, 343 Free shear layers, 346

343

Wake., 348

5.13 Oscillatory boundary layersThe 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 5.14 Flow systems with a free surfaceThe 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

X 1

5.15 Examples of use of the momentum theoremThe force on a regular array of bodies in a stream, 37a The effect of a sudden enlargement of a pipe, 373

372

Chapter 6. Irrotational Flow Theory and its AppUcadons 6.1 The role of the theory of flow of an inviscid fluid 378

6.2 General properties of irrotational flowIntegration 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

380

6.3

Steady flow: some applications of Bernoulli's theorem and the momentum theoremEfflux from a circular orifice in an open vessel, 387 Flow over a weir, 391 Jet of liquid impinging on a plane wall, 392 Irrotational flow which may be made steady by choice of rotating axes, 396

386

6.4 General features of irrotational flow due to a moving rigid body 398The velocity at large distances from the body, 399 The kinetic energy of the fluid, 40:& The force on a body in translational motion, 404The acceleration reaction, 407 The force on a body in accelerating fluid, 409

6.5 Use of the complex potential for irrotational flow in twodimensionsFlow fields obtained by special choice of the function wC.), 410 Conformal transfonnation of the plane of flow, 413 Transfonnation of a boundary into an infinite straight line, 418 Transfonnation of a closed boundary into a circle, 420 The circle theorem, 4za

409

6.6 Two-dimensional irrotational flow due to a moving cylinder with circulationA circular cylinder, 424An elliptic cylinder in translational motion, 427 The force and moment on a cylinder in steady translational motion, 433

423

67 Two-dimensional aerofoilsThe practical requirements of aerofoils, 435 The generation of circulation round an aerofoil and the basis for ]oukowski's hypothesis, 438 Aerofoils obtained by transfonnation of a circle, 441 Joukowski aerofoils, +K

435

x

C;ontents

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 bodiesSlender bodies of revolution, 463 Slender bodies in two dimensions, 466 Thin aerofoils in two dimensions, 467

6.10 Impulsive motion of a fluidImpact of a body on a free surface of liquid, 473

47 1 474

6. I I Large gas bubbles in liquidA spherical-cap bubble rising through liquid under gravity, 475 A bubble rising in a vertical tube, 477 A spherical expanding bubble, 479

6. I 2 Cavitation in a liquidExamples of cavity formation in steady flow, 482 Examples of cavity formation in unsteady flow, 485 Collapse of a transient cavity, 486 Steady-state cavities, 491

6.13 Free-streamline theory, and steady jets and cavitiesJet 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

493

Chapter 7. Flow of Effectively Inviscid Fluid with Vordcity

7. I

IntroductionThe self-induced movement of a line vortex, 509 The instability of a sheet vortex,s I I

507

7.2 Flow in unbounded fluid at rest at infinityThe resultant force impulse required to generate the motion,s 18 The total kinetic energy of the fluid, 520 Flow with circular vortex-lines, sal Vortex rings, saa

517

7.3 Two-dimensional flow in unbounded fluid at rest at infinityIntegral invariants of the vorticity distribution, sa8 Motion of a group of point vortices, 530 Steady motions, S3a

527

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

53 6

Contents7.5 Steady axisymmetric flow with swirlThe 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

XI

page 543

7. 6 Flow systems rotating as a wholeThe restoring effect of Coriolis forces, 555 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

555

7.7 Motion in a thin layer on a rotating sphereGeostrophic flow. 571 Flow over uneven ground. 573 Planetary waves. 577

567

7.8 The vortex system of a wingGeneral features of the flow past lifting bodies in three dimensions, 580 Wings of large aspect ratio. and' lifting-line' theory. 583 The trailing vortex system far downstream. 589 Highly swept wings. 591

580

AppendicesI

Measured values of some physical properties of common fluids 594(0) 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 15C and(e) Surface tension between two fluids. 597

I

atm. 597

2

Expressions for some common vector differential quantities in 598 orthogonal curvilinear co-ordinate systems

PubUcations referred to in the text Subject Index

609

Plates

I

to 24 are between pages 364 and 365

xiii

PREFACEWhile 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 engineering, 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 ofthe vitally important physical aspects offluid 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. I have tried therefore to write a textbook which can be used by studen~ 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; tJlt lis 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 known, 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 whole, or at least in large portions, rather than to be referred to for particular problems or methods. I have had the needs of second-, third- and fourth-year students of applied mathematics in British universities particularly in mind, these being the needs with which I am most familiar. although I 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 wh~ 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. I 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 I hope be useful as background. In the remaining four chapters the fluid is assumed to be incompressible and to have unifonn density and viscosity. I 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

Pn~u

xv

is still the key branch of fluid dynamics. I 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 I 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, I 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-fluid 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

XVl

J>re/ace

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. I 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 I 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. I 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. I am especially grateful to Philip Chatwin, John Elder, Emin

Preface

xvii

Erdogan, Ken Freeman, Michael McIntyre, 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 officers of Cambridge University Press, with whom it is a pleasure to work. G.K.B. Cambridge April 1967

XVlll

CONVENTIONS AND NOTATIONBold type signifies vector character. x, x' position vectors; Ixl = T S = X - x' relative position vector u velocity at a specified time and position in space;

luI

= q

Dt

D a+u. v ---- = O n

.. 'l " operator glvmg t Ile materIa derlvatlve, or rate 0 f h ange at a c t point moving with the fluid locally; applies only to functions of x and t

System Rectilinear Polar, two dimensions Spherical polar Cylindrical A = V. utal

Co-ordinates

Velocity components

x, y, z orT, (}

Xl> X2' X,

T,

X, U, if; (u2=yl!+Z2)

0, if;

v, w or U1> U2, U. v or 14, Uu U, v, w or Ur, UU, U~ U, v, w or UII:' UCT> U~U,U,

rate ofexpansion (fractional rate ofchange ofvol umeofa materialelement)

= V X u vorticity (twice the local angular velocity of the fluid) e - 1 (OUi + OU/) rate-of-strain tensor i1 - 2 OXt OXjscalar potential of an irrotational velocity distribution (u = Vif;) B vector potential of a solenoidal velocity distribution (u = V x B) 1Jr stream function for a solenoidal velocity distribution; (a) two-dimensional flow: B = (0,0, ljr) oljr oljr 1 oljr 8ljr U = --, v::::: - or 14::::: - - , U - - oy ox T oe Uor (b) axisymmetric flow: ljr 1 oljr 1 oljr cylindrical co-ordinates BA. ::::: -, U = --- tl = - - Y' U II: U AU ' IT U ox . ljr I oljr I oljr polar co-ordmates BiP = -:--(j' Ur = l! (j -(j-' ue = - - ' - ( j T sm T sm a T sm 8r n unit normal to a surface, usually outward if the surface is closed oV, n oA, ox volume, surface and line elements with a specified position in space C~T, noS, 81 material volume, surface and line elements Uti stress tensor; utjnjoA is the i-component of the force exerted across the surface element n SA by the fluid on the side to which n points F =- V'l' conservative body force per unit mass Inertia force (per unit mass) minus the local acceleration Vortex-line line whose tangent is parallel to tal locally Line vortex singular line in vorticity distribution round 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 number of original papers are also referred to, sometimes for historical interest, sometimes because a precise acknowledgement is appropriate, and sometimes, although only rarely, as a guide to further reading on a particular topic. These papers are cited in the text as' Smith (1950)', and the full references for both papers and books are listed at the end of the book.

9

1THE PHYSICAL PROPER TIES OF FLUIDS1.1. Solids, 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 '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 way which leaves the volume unchanged. The implications ofthis 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

[I. 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 two 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 10-8 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 10-7 or lo-scm, 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

1.1]

Solids, liquids and gases

3

the influence of the other.t 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 do' at which the reaction changes sign, one molecule is clearly in a position of stable equilibrium relative to that of the other. do is of order 3-4 x 10-8 em 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 lodo, whereas the average spacing in liquid and solid phases is of

:;"0

u

...

v

-go

e

Repulsion

v ....",

~ ~di;;o.....Ji,--------=::::::====~Attraction

c

d

...

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 do. In gases under ordinary conditions the molecules are thus so far apart from each other that only exceedingly weak cohesive forces act between them, 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 gas', 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.t See, for instance, States of Matter, by E. A. Moelwyn-Hughes (Oliver and Boyd, 1961).

4

The physical properties offluids

[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 mobility, sometimes falling into regular array with other groups and sometimes being split up into smaller groups. The arrangement of the molecules is continually changing, and, 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.Ratio of amplitude of random thermal movement of molecules to do~

Intermolecular forces solid liquid gas strong medium weak

Molecular arrangement ordered partially ordered disordered

Type of statistics needed quantum quantum classical classical

x

of order unity

+

x

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.

1.2. 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

5

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 with spatial distribution of densit),

Local' value of fluid density

Volume of fluid to which instrument responds

Fieure

1.:1.1.

Effect of size of sensitive volume on the density measured by an instrument.

the 'sensitive' 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 '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


[1.2

We are able to regard the fluid as a continuum when, as in the figure, the measured fluid property is constant for sensitive volumes small on the macroscopic scale but large on 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 em and very little variation of the physical and dynamical properties of the fluid occurs over a distance of 10.-3 em (except perhaps in special places such as in a shock wave); thus an instrument with a sensitive volume of 10-9 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' at a point' 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 solids, 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.

1.3]

Volume forces and surface forces acting on a fluid

7

1.3. Volume forces and surface forces acting on a fluidIt 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 ofthe 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 8V surrounding the point whose position vector is x by

F(x, t)p8V;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

[1.3

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 layert 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 be written as the vector E(n,x,t)8A, (1.3.2) where n is the unit normal to the element. The convention to be adopted here is that E is the stress exerted by the fluid on the side of the surface element to which n points, on the fluid on the side which n points away from; so a normal component of E with the same sense as n represents a tension. The force per unit area, E, 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 - E(n, x, t) 8A, and since this is also the force represented by E( - n, x, t) 8A we see that E 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. I) 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 t Unless the element is chosen to have such small linear dimensions that the short-rangeforces exerted by external matter are still significant at the centre of the element; but the clement would then contain only a few molecules at most, and representation of the fluid as a continuum would not be possible.

1.3]


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 material 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 ofsurface 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 E 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 ofE on the direction of the normal to the surface element across which it acts. c Consider all the forces acting instantaneously on the fluid within an element of volume 8V in the shape of a tetrahedron as shown in figure 1.3.1. The three orthogonal faces have areas 8A 1 , 8A 2 , 8A s, and b unit (outward) normals - a, - b, -c, and the fourth inclined face has area 8A and unit normal n. Surface forces will act on the fluid in the tetrahedron across each of the four faces, and their sum isE(n)8A+E( -a)8A 1Figure 1.3.1. A volume element in the shape o,f a tetrahedron with three orthogonal faces.

+E( - b)8A 2 +E( -c)8A s ;

the dependence of E on x and t is not displayed here, because these variables have the same values (approximately, in the case of x) for all four contributions. In view of the orthogonality of three of the faces, three relations like

8A 1 = a.n8Aare available, and the i-component of the sum of the surface forces can therefore be written as

Now the total body force on the fluid within the tetrahedron is proportional to the volume 8V, which is of smaller order than 8A in the linear t The suffix notation for vector components has been used here, with the usual conventionof vector and tensor analysis that terms containing a repeated suffix are to be regarded as summed over all three possible values of the suffix. Both the suffix form and the representation in bold-face type without suffixes will be used for vectors in this book, the choice being made usually with an eye to neatness of the formulae.

10


[1.3

dimensions of the tetrahedron. The mass of the fluid in the tetrahedron is also of order 8V, and so too is the product of the mass and the acceleration of the fluid in the tetrahedron, provided that both the local density and acceleration are finite. Thus if the linear dimensions of the tetrahedron are made to approach zero without change of its shape, the first two terms of the equation mass x acceleration=

resultant of body forces + resultant of surface forces

approach zero as 8V, whereas the third term apparently approaches zero only as 8A. In these circumstances the equation can be satisfied only if the coefficient of 8A in (1.3.3) vanishes identically (with the implication that information about the resultant surface force on the element requires a higher degree of approximation which takes account of the difference between the values of:t at different positions on the surface of the element), . glvmg

.

Thus the component of stress in a given direction represented by the suffix i across a plane surface element with an arbitrary orientation specified by the unit normal n is related to the same component of stress across any three orthogonal plane surface elements at the same position in the fluid in the same way as if it were a vector with orthogonal components ~~(a), ~~(b),:E~(c).

The vectors nand E do not depend in any way on the choice of the axes of reference, and the expression within curly brackets in (1.3.4) must represent th.e (i,j)-component of a quantity which is similarly independent of the axes. In other words the expression within curly brackets is one component of a second-order tensor, t (T# say, and:E~(n) = (T~ini'

(135)

(Tii is the i-component of the force per unit area exerted across a plane surface element normal to the j-direction, at position x in the fluid and at time t, and the tensor of which it is the general component is called the stress tensor. Specification of the local stress in the fluid is now provided by (T~i' which is independent of n, in place of E(n). A similar argument can be used to demonstrate that the nine components of the stress tensor are not all independent. This time we consider the moments of the various forces acting on the fluid within a volume V of arbitrary shape. The i-component of the total moment, about a point 0 tA general familiarity with the elementary properties of tensors will be assumed in this book. Only Cartesian tensors (that is, tensors for which the suffixes denote components with respect to rectangular co-ordinate axes) will be used. Two special tensors which will appear often are the Kronecker delta tensor Bi/ , such that Bil = I when i = i and 8u = 0 when i 9= i, and the alternating tensor 6i/lc, with value zero unless i. i. k are all different, in which case the value is + I or - I according as i, i. k are or are not in cyclic order.

1.3]


II

within this volume, exerted by the surface forces at the boundary of the volume is ei,ikri uklnl dA ,

f

where r is the position vector of the surface element n8A relative to O. This integral over a closed surface can be transformed by the divergence theorem to the volume integral

J

ei,Jk

8(rl Ukl) dV~l

'

(1.3.6)

If now the volume V is reduced to zero in such a way that the configuration made up of the boundary of the volume and the fixed point 0 retains the same shape, the first term on the right-hand side of (1.3.6) becomes small as V whereas the second term approaches zero more quickly as Vi. The total moment about 0 exerted on the fluid element by the body forces is clearly of order Vi when V is small, t and so too is the rate of change of the angular momentum of the fluid instantaneously in V. Thus Jei,ik ukidV is apparently of larger order in V than all the other terms in the moment equation, and as a consequence it must be identically zero. This is possible for all choices of the position of 0 and the shape of V, when u'i is continUous in x, only if

everywhere in the fluid; for if e'ik Uki were non-zero in some region of the fluid, we should be able to choose a small volume V for which the integral is non-zero, giving a contradiction.! The relation (1.3.7) shows that the stress tensor is symmetrical, that is, u'i = u", and has only six independent components. The three diagonal components of U ii are normal stresses in the sense that each of them gives the normal component of surface force acting across a plane surface element parallel to one of the co-ordinate planes. The six nondiagonal components of ui,j are tangential stresses, sometimes also called shearing stresses, since in both fluids and solids they are set up by a shearing motion or displacement in which parallel layers of matter slide relative to each other. Figure 1.3.2 shows the first approximation to the various surface forces acting in the (Xl' x2)-plane on a small rectangular element with sides 8xI and 8x2 and unit depth in the xa-direction; the components of the stress do not have exactly the same values on opposite sides of the rectangle, and the differences, of order 8xI or 8x2, will need to be taken into account when the equation of motion of an element of fluid is formulated. It is always possible to choose the directions of the orthogonal axes of reference so that the non-diagonal elements of a symmetrical second-ordert In the absence of any body couple' of order V. like the couple exerted on a polarized

t

dielectric medium by an imposed electric field. This deduction about the integrand of an integral which is zero for all choices of the range of integration will be needed often, for volume. surface and line integrals.

12

The physt'cal properties of fluids

[1.3

tensor are all zero. Referred to such principal axes of the stress tensor O'ij at a given point x, the diagonal elements of the stress tensor become principal stresses, 0'~1' U~2' U~3 say; and it is a well-known property of second-order tensors that changes of directions of orthogonal axes of reference leave the sum of the diagonal elements unchanged, so that Relative to these new axes the components of the force per unit area acting across an element of area with normal (ni, n~, n~) are

Figure 1.3.a. The surface forces acting on a rectangular element of fluid of unit depth.

A normal stress uil acting across an element normal to the first of the new axes corresponds to a state of tension (or compression if O'h is negative) in the direction of that axis, and similarly for 0'~2 and O'~. Thus the general state of the fluid near any given point may be regarded as a superposition of tensions in three orthogonal directions.The stress tensor in a fluid at rest We have defined a fluid as being unable to withstand any tendency by applied forces to deform it without change of volume. This definition has consequences for the form of the stress tensor in a fluid at rest. To see this, consider the surface forces exerted on the fluid within a sphere by the surrounding fluid, the radius of the sphere being small so that 0'# is approximately uniform over the surface. We choose axes coinciding (locally) with principal axes of uij, and take the further step of writing the stress tensor, which now has zero non-diagonal elements, as the sum of the two tensors

olUii

0

0)

and

(0'~1 -

!O'i"00

U~2 -lUi"0

0

,0

0)

.

(1.3.9)

o

lUii

O'sa-lUi"

The first of these tensors has spherical symmetry, or isotropy, and the corresponding contribution to the force per unit area exerted on the surface

1.3]


13

of the sphere at a point where the normal is n is !u"n. This uniform compression (for the sign of iu" is usually negative) of the fluid in the sphere tends to change its volume and can certainly be withstood by the fluid in the sphere while at rest. The second of the tensors in (1.3.9) is the departure of the stress tensor from an isotropic form. The diagonal elements of this tensor have zero sum, in view of (1.3.8), and thus represent normal stresses of which at least one is a tension and at least one a compression. The corresponding contribution to the force per unit area exerted on the surface of the sphere at a point where the normal vector is (n~, n~, n~) has components (relative to the new axes)

In other words, the sphere is embedded in fluid which is in a state of uniform tension in the direction of one axis, together with uniform compression in the (orthogonal) direction of another axis, and uniform tension or compression in the third orthogonal direction (the algebraic sum of the three

- -0- -. tttftf(h)

I~ ! IJ1

-...

(a)

Figure 1.3.3. Two contributions to the stress at the surface of a spherical element of fluid; (a) an isotropic compression, and (b) uniform tension in the direction of one principal axis of the stress tensor together with uniform compression in the direction of another principal axi.

tensions and compressions being zero), as indicated in figure 1.3.3. This second contribution thus tends to deform the spherical element of fluid into an ellipsoid, without any necessary change of volume; nor can this deforming surface force be balanced by any volume force on the fluid, because the latter is of a different order of magnitude in the small volume of the spherical element. The spherical element of fluid cannot withstand such a tendency to deform it by applied forces (that is, by forces due to agencies external to the element), so that a state of rest is not compatible with the existence of non-zero values of any of the force components (1.3.10). Hence, in a fluid at rest, the principal stresses Uil' U~2' U~3 are all the same and equal to lUH' at all points in the fluid; that is, the stress tensor in a fluid at rest is everywhere isotropic, all orthogonal axes of reference are principal axes for the stress tensor, and only normal stresses act.

14


[1.4

Fluids at rest are normally in a state of compression, and it is therefore convenient to write the stress tensor in a fluid at rest as

(1.3. 11 )where p ( = -100",,) may be termed the static-fluid preslUret and is in general a function of x. It follows that in a fluid at rest the contact force per unit area exerted across a plane surface element in the fluid with unit normal n is - pn, and is a normal force of the same magnitude for all directions of the normal n at a given point. This well-known property of the static-fluid pressure, of , acting equally in all directions', is often established as a consequence of an assumption that in a fluid at rest the tangential stresses are zero; the argument is simply a consideration of the balance of forces on an element of fluid of simple geometrical shape, such as the tetrahedron with three orthogonal faces! or a portion of a cylinder with one plane section normal to the generators and one inclined to them. An assumption that tangential stresses are zero in a fluid at rest is reasonable, for in the absence of any bulk motion it seems unlikely that the random molecular configuration and motion could have any statistical directional preferences, in which event the reaction due to molecular forces and flux of momentum across a surface element would be purely normal. However, it seems preferable to derive the properties of the stress tensor in fluid at rest from the more primitive assertion that fluids cannot withstand any attempt to change their shape.

1.4. Mechanical equilibrium of a fluid A rigid body is in equilibrium when the resultant force and the resultant couple exerted on it by external agencies are both zero. The conditions for equilibrium of a fluid are less simple, because the different elements of fluid can move relative to each other and must separately be in equilibrium. The forces acting on any given portion of fluid are, as stated in the previous section, volume forces due to external agencies and surface forces exerted across the boundary by the surrounding matter. These volume and surface forces must balance if the fluid is to remain at rest. In the notation of the previous section, the total body force acting on the fluid lying within a volume Vis JpFdV,in which both p and F may be functions of position in the fluid. The total

t

t

The term hydrostatic pressure is often used, but the implied association with water has only historical justification and may be misleading. The terms C hydrodynamics' and aerodynamics' are likewise unnecessarily restrictive, and are being superseded by the more general term fluid dynamics'. Put 1:i(n) = "i1:(n), 1: i(a) = oi1:(a), etc., in (1.3.4) and then take the scalar product of both sides of the equation with a, b, and c in tum.

1.4]

Mechanical equilibrium of a fluid

IS

contact force exerted by the surrounding matter at the surface A bounding the volume V (when the fluid is at rest) is

- fpndA,in which p is also in general a function of the position vector x and n is the unit outward normal to the surface A. This latter integral may be transformed to an integral over the volume V by the analogue of the divergence theorem for a scalar quantity, giving - fVp dV. Hence a necessary condition for equilibrium of the fluid is that

f(pF - Vp)dV =

0,

(1.4. 1)

for all choices of the volume V lying entirely in the fluid, which is possible only if the integrand itself (assumed to be continuous in x) is zero everywhere in the fluid. The necessary condition for equilibrium is then that pF = Vp

(1.4.2)

everywhere in the fluid. If (1.4.1) holds for all choices of V, the resultant force on each element of the fluid is zero. Moreover, our use of a SYmmetrical stress tensor ensures that the couple on each volume element of fluid is zero, so that when ( 1.4.2) is satisfied the resultant couple on the fluid within a volume V of arbitrary shape and size is zero (in the absence of any body couple acting on the fluid), as may be verified directly. Equation (1.4.2) is therefore the necessary and sufficient condition for the fluid to be in equilibrium. In the case of a solid, for which the tangential stresses are not necessarily zero, the corresponding condition is an equation like (1.4.2) in which the (i-component of the) righthand side has the more general form - O(T'l//oxi' The restriction imposed by equation (1.4.2) lies in the fact that only for certain distributions of p and F, viz. those for which pF (the body force per unit volume) can be expressed as the gradient of a scalar quantity, does there exist a pressure distribution satisfying (1.4.2). When the distribution of pF does have the form required for equilibrium, p is constant over any surface which is everywhere normal to the body force. The nature of the restriction on p and F takes a more specific form in the common case in which the body force per unit mass (F) represents a conservative field and can be written as - V'l', where 'l' is the potential energy per unit mass associated with this field. In this case the condition for -pV'l' = Vp, (1.4.3) equilibrium is or, on taking the curl of both sides,

(Vp) x (V'l') = o.Thus the level-surfaces of p and 'l' must coincide, and, when this condition is satisfied, these are also the level surfaces of p and we may write

dp/d'Y = -pry).

16


[1.4

The particular case in which V'Y has the same direction everywhere, so that 'Y, p and p are constant on each one of a family of parallel planes, occurs in discussions of the earth's atmosphere. The density of an element of fluid may be affected by the pressure to which it is subjected, and also by other factors, so that further discussion of the implications of (1.4.3) requires information about p. However, in a case in which the fluid has uniform density p, the solution of ( 1.4.3) is simply

p = Po-p'Y,where Po is a constant.

(145)

A body 'floating' in fluid at restThe common notion of floating relates to a rigid body partially immersed at a horizontal free surface of liquid at rest under gravity, but the term may be used more generally. A body may be said to float when it is wholly immersed in fluid (some of the fluid may be liquid and some gaseous, giving partial immersion in everyday terminology) and both it and the fluid are at rest under the action of volume forces. The primary result for a floating body is Archimedes' theorem, which is usually stated and proved for the case of a body supported by the buoyancy force due to the action of gravity on a uniform liquid. This is the most important field of application of the theorem, but the additional generality of the form of the theorem to be established here has some value. Suppose that a body of volume V and bounding surface A is immersed in fluid and that the body and the fluid are at rest. The resultant force on the body due entirely to the presence of the fluid is

- fPDdA,where D is the outward normal to the body surface. The pressure p in the fluid is determined by the equilibrium relation (1.4.2), and, taking our cue from the conventional form of Archimedes' theorem, we wish to use (1.4.2) to express this resultant surface force in terms of the total volume force on fluid which in some sense is able to take the place of the body. We need to know how fluid can replace the body without disturbing the equilibrium and without changing conditions in the surrounding fluid. A definite answer may be given in a case in which F = - V'Y and'Y is a prescribed function of position in space. The level surfaces of ' may be continued through the region occupied by the body, and the uniform value that the density p must have on each level surface of'Y for fluid in this region to be in equilibrium is equal to the value of p on the same level surface outside the region. In other words, we have a specification for the distribution of density of fluid which can take the place of the body. The total volume force on this replacement fluid is

- f pV'J!' dV,

1.4]


17

where the integral is taken over the region which was occupied by the body, and this force is balanced by the contact force at the boundary A, which is unchanged by the replacement of the body by fluid. Thus the 'buoyancy' force on an immersed body due to the action of a volume force on the surrounding fluid (at rest) is

JpV'Y dV,

= -

JpF dV,

where the density p at a point within the region occupied by the body is determined by continuation of the distribution in the surrounding fluid in the manner described above. A body immersed in fluid loses 'weight' equal to the 'weight' of the fluid 'displaced', where 'weight' and 'displaced' can both be given rather more general meanings than those intended by Archimedes. The practical implications of these principles are examined in textbooks on hydrostaticst and need not be recounted here. However, the reader may be interested to consider briefly the application of the principles to one problem different from those involving only gravity and uniform liquids. Suppose, for instance, that a vessel containing fluid of non-uniform density is rotating steadily about the vertical z-axis and that the fluid has taken up the same steady rotation. Relative to axes rotating with the vessel, with angular velocity.Q say, the fluid is at rest and is acted on by a body force per unit mass with vertical component - g due to gravity and with radial component .Q2(X2+y2)1 in a horizontal plane due to the effective centrifugal force. Thus we have F = - V'Y, 'Y = gz_!.Q2(X2+y2), and the level surfaces of'Y are equal paraboloids of revolution, with vertical axes, translated vertically from each other (figure 1.4.1). For equilibrium it is necessary that p be constant on each of these paraboloids; and then p is also constant on each paraboloid. If now a solid body, say a sphere of uniform density, is immersed in the fluid in this vessel and is at rest relative to it, the fluid exerts a certain buoyancy force on the body. There arises the question: can this buoyancy be balanced by the same volume forces (gravity and centrifugal force) acting on the body itself? In other words, if the body is placed at a certain position in the fluid, will it remain there? We need to find a position for the centre of the sphere such that the sphere displaces its own mass of fluid, which selects (approximately) a certain value of'Y (figure 1.4.1), and such that the same centrifugal force acts on the displaced fluid as on the solid sphere. It is evident that such a position cannot be found off the axis of rotation, because the tilting of the surfaces of equal density implies a greater centrifugal force on the displaced fluid, given that it has the same mass as the sphere, than on the sphere. Hence a uniform sphere would 'fall' downt See, for instance, Statics, by H. Lamb (Cambridie University Pre,1933).

18


[1.4

the paraboloid of revolution on which it must lie to displace its own mass of fluid and would come to rest at the axis. The same is true of a sphere at a free surface of rotating liquid, since this is simply a particular distribution of density with respect to 'Y. On the other hand, if the sphere is sufficiently non-uniform in density, say by being weighted on one side, it is clearly possible for the total centrifugal force on the sphere to be greater than that on displaced fluid of the same total mass, in which case the sphere moves outward on a paraboloid of revolution until it meets the wall of the vessel.,~"

"~I"';'~/8",.&.,' ' ' : P if

,

--"',;"..... . ~':"~:='l~~~--- ----:-Lf' ,'.... " ,... , ------ ----" '.......... .....- --. "', '... ....... - ... -- '" " ...,_..... -.... ............. ..., ........

... "Jet

4'C'~

.::

.1.r Uif #

~I\

:II

A position of sphere in ~/ which it displaces its ,,~" /~ own mass of fluid ,,' ",'"

...

' ....

"

~~

--~'

,,,,'''',,,,,~' ",'" , ~,,

,,~

"

... '.... "

......

...

..... _--

Equilibrium position of a uniform sphere

........

-

---- - - - - - - -

...' ------ .-.......----'

...... ~-

~' ,,','~ -,' ""

~, ~

.--"

,I'

(xl+y,> f

Figure 1.4.1. Non-uniform fluid at rest under the action of gravity and centrifugal force.

Fluid at rest under grOlVityThe case in which gravity is the only volume force acting on the fluid is both important and simple. Two extreme situations may be distinguished. In the first one, the mass of fluid concerned is large and isolated so that the gravitational attraction of other parts of the fluid provides the volume force on any element of the fluid, as in the case of a gaseous star. At the other extreme, the mass of fluid concerned is much smaller than that of neighbouring matter and the gravitational field is approximately uniform over the region occupied by the fluid. In the case of a self-gravitating fluid, we have F= - V'Y, where the gravitational potential 'Y is related to the distribution of density by the equation V2'Y = 41TGp, (1.4. 6) G being the constant of gravitation. On combining (1.4.6) with equation (1.4.3) for the pressure in a fluid at rest, we obtain

V. (;')

= -41TGp.

(1.4.7)

It is also necessary, as found earlier, that the level-surfaces of 'Y, p and p coincide. On expressing the differential operator in ( 1.4.7) in terms of curvilinear co-ordinates (not necessarily orthogonal) such that the level-surfaces

1.4]


19

of p coincide with one set of parametric surfaces, we see that the kinds of solution are severely restricted. Rigorous enumeration of the solutions is difficult, but the only possibilities seem to be solutions in which p and p are functions only of (i) one co-ordinate of a rectilinear system, or (ii) the radial co-ordinate ofa cylindrical polar system, or (iii) the radial co-ordinate r of a spherical polar system, corresponding to symmetrical' stars' in one, two or three dimensions. In the last case, describing a spherically symmetrical distribution of density and pressure, (1.4.7) becomes

~ (~Z)

= -41TGrap,

(1.4. 8)

and further progress cannot be made without information about the distribution of density. In real stars the density is in general not a function of p alone, but solutions of (1.4.8) corresponding to an assumed simple relationship between p and p are sometimes useful for comparison with more complicated models. If we assume for instance that

p ex: p1+1/n (n ~ 0),it is possible to integrate ( 1 .4.8) numerically for any value of n. Two analytical and representative solutions are also available. When n = 0, corresponding to a fluid of uniform density, Po say, we have

p = f1TGpg( aa - ra),where r = a may be interpreted as the outer boundary of the star. When n = 5, it may be verified that .I. 27aset p = Cpa = (21TG)f(a 2+r2)S; the pressure and density here are non-zero for all r and there is no definite outer boundary, but the total mass of the star is finite. In the case of a uniform body force due to gravity, we have F

= g( = const.),Vp

'P'

= -g. x,

(1.4.9)(1.4.10)

and the equation for the pressure in a fluid at rest is

= pg.

The three functions 'F, P and p are constant on each horizontal plane normal to g, and hence depend only on g. x. If we choose the z-axis of a rectilinear co-ordinate system to be vertical (positive upwards) so that g.x = -gz, (1.4.10) becomes

dp/dz = -gp(z).

(1.4.11)

Again this is as much as we can deduce from the condition of mechanical equilibrium alone.

20


[I.S

When the fluid is of uniform density, we obtain from (1.4.11) the linear relation between pressure and height well-known in the study ofhydrostatics :

p = Po-pg%.

(1.4. 12 )

In the case of the earth's atmosphere, p decreases with decrease of the pressure owing to the compressibility of the air, although thermal effects are usually present and no single functional relation between p and p is adequate. As a crude approximation one may put

pip = const.,

=

gH say,

corresponding to Boyle's law for a perfect gas of uniform temperature and constitution ( 1.7). The pressure in an atmosphere for which this relation holds is found from (1.4. I I) to be

p = poe-etH,where Po is the pressure at ground level, % = o. Thus both p and p diminish by a factor e-1 over a height interval H, and the constant H may be termed the 'scale-height' of the atmosphere. For air at oC, H = gokIn. When the temperature is not uniform, plpg may still be regarded as a local scaleheight. Observed average values of the pressure, density and temperature at different heights in the atmosphere will be found in appendix I (b).Exercises

n about a horizontal axis. Show that the surfaces of equal pressure are circularcylinders whose common axis is at a height g/Q2 above the axis of rotation. Obtain an expression for the pressure at the centre of a self-gravitating spherical star of which the density at distance r from the centre is2.

I.

A closed vessel full of water is rotating with constant angular velocity

p = pcCI-fJr).Show that if the mean density be twice the surface density, the pressure at the centre is greater, by a factor ~8J., than if the star had uniform. density with the same total mass.

1.S. Classical thermodynamics In our subsequent discussion of the dynamics of fluids we shall need to make use of some of the concepts of classical thermodynamics and of the relations between various thermodynamic quantities, such as temperature and internal energy. Classical thermodynamics is concerned, at any rate as the bulk of the subject stands, with equilibrium states of uniform matter, that is, with states in which all local mechanical, physical and thermal quantities are virtually independent of both position and time. Thermodynamical results may be applied directly to fluids at rest when their properties are uniform. Comparatively little is known of the thermodynamics

I.sj

Classical thermodynamics

21

of non-equilibrium states. However, observation shows that results for equilibrium states are approximately valid for the non-equilibrium nonuniform states common in practical fluid dynamics; large though the departures from equilibrium in a moving fluid may appear to be, they are apparently small in their effect on thermodynamical relationships. The purpose of this section is to recapitulate briefly the laws and results of equilibrium thermodynamics and to set down for future reference the relations that will be needed later. For a proper account of the fundamentals ofthe subject the reader should refer to one ofthe many text-books available. t The concepts of thermodynamics are helpful to the student of fluid mechanics for the additional reason that in both subjects the objective is a set of results which apply to matter as generally as possible, without regard for the different molecular properties and mechanisms at work. Additional results may of course be obtained by taking into account any known molecular properties of a fluid, as proves to be possible for certain gases with the aid of kinetic theory (see 1.7). It is taken as a fact of experience that the state of a given mass of fluid in equilibrium (the word being used here and later to imply spatial as well as temporal uniformity) under the simplest possible conditions is specified uniquely by two parameters, which for convenience may be chosen as the specific volume f) (= 1/p, where p is the density) and the pressure p as defined above. All other quantities describing the state of the fluid are thus functions of these two parameters ofstate. One of the most important of these quantities is the temperature. A mass of fluid in equilibrium has the same temperature as a test mass of fluid also in equilibrium if the two masses remain in equilibrium when placed in thermal contact (that is, when separated only by a wall allowing transmission of heat); and the second law of thermodynamics provides an absolute measure of the temperature of a fluid, as we shall note later. The relation between the temperature T and the two parameters of state, which we may write as

f(P, f), T) =

0,

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.! Work and heat are regarded as equivalent formst See, for instance, Classical Thermodynamics, by A. B. Pippard (Cambridge University

t

Press, 1957). 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


[I. 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 6E=Q+W. The internal energy E is a function of the parameters of state, and the change M, 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 = 0 and the change of state of the fluid is said to be adiabatic. 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,t and equal to p say, so that the work done on unit mass of the fluid as a consequence of compression leading to a small decrease in volume t is - P8fJ. Thus for a reversible transition from one state to another, neighbouring, state we have 8E = 8Q-p8fJ. (1.5.3) A finite reversible change of this kind can be described by summing (1.5.3) oveI: 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 fJ 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 asc = 8Q/8T,

(1.5.4)

t t

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 co

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