An Introduction to Fluid Dynamics

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ISBN: 9780521663960 | 0521663962
Cover: Paperback
Copyright: 2/28/2000

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First published in 1967, Professor Batchelor’s classic text on fluid dynamics is still one of the foremost texts in the subject. The careful presentation of the underlying theories of fluids is still timely and applicable, even in these days of almost limitless computer power. This re-issue should ensure that a new generation of graduate students see the elegance of Professor Batchelor’s presentation.

Preface

xiii

Conventions and Notation

xviii

The Physical Properties of Fluids

Solids, liquids and gases

(3)

The continuum hypothesis

(3)

Volume forces and surface forces acting on a fluid

(7)

Representation of surface forces by the stress tensor

(3)

The stress tensor in a fluid at rest

(2)

Mechanical equilibrium of a fluid

(6)

A body `floating' in fluid at rest

(2)

Fluid at rest under gravity

(2)

Classical thermodynamics

(8)

Transport phenomena

(9)

The linear relation between flux and the gradient of a scalar intensity

(2)

The equations for diffusion and heat conduction in isotropic media at rest

(4)

Molecular transport of momentum in a fluid

(1)

The distinctive properties of gases

(16)

A perfect gas in equilibrium

(7)

Departures from the perfect-gas laws

(2)

Transport coefficients in a perfect gas

(3)

Other manifestations of departure from equilibrium of a perfect gas

(3)

The distinctive properties of liquids

(7)

Equilibrium properties

(2)

Transport coefficients

(3)

Conditions at a boundary between two media

(11)

Surface tension

(3)

Equilibrium shape of a boundary between two stationary fluids

(5)

Transition relations at a material boundary

(3)

Kinematics of the Flow Field

Specification of the flow field

(2)

Differentiation following the motion of the fluid

(1)

Conservation of mass

(6)

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

(4)

Analysis of the relative motion near a point

(5)

Simple shearing motion

(1)

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

(4)

Singularities in the rate of expansion. Sources and sinks

(4)

The vorticity distribution

(7)

Line vortices

(3)

Sheet vortices

(3)

Velocity distributions with zero rate of expansion and zero vorticity

(9)

Conditions for ∇ &phis; to be determined uniquely

102

(3)

Irrotational solenoidal flow near a stagnation point

105

(1)

The complex potential for irrotational solenoidal flow in two dimensions

106

(2)

Irrotational solenoidal flow in doubly-connected regions of space

108

(6)

Conditions for ∇ &phis; to be determined uniquely

112

(2)

Three-dimensional flow fields extending to infinity

114

(10)

Asymptotic expressions for ue and uv

114

(3)

The behaviour of &phis; at large distances

117

(2)

Conditions for ∇ &phis; to be determined uniquely

119

(1)

The expression of &phis; as a power series

120

(2)

Irrotational solenoidal flow due to a rigid body in translational motion

122

(2)

Two-dimensional flow fields extending to infinity

124

(7)

Irrotational solenoidal flow due to a rigid body in translational motion

128

(3)

Equations Governing the Motion of a Fluid

Material integrals in a moving fluid

131

(6)

Rates of change of material integrals

133

(2)

Conservation laws for a fluid in motion

135

(2)

The equation of motion

137

(4)

Use of the momentum equation in integral form

138

(1)

Equation of motion relative to moving axes

139

(2)

The expression for the stress tensor

141

(10)

Mechanical definition of pressure in a moving fluid

141

(1)

The relation between deviatoric stress and rate-of-strain for a Newtonian fluid

142

(5)

The Navier-Stokes equation

147

(1)

Conditions on the velocity and stress at a material boundary

148

(3)

Changes in the internal energy of a fluid in motion

151

(5)

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

156

(8)

Special forms of Bernoulli's theorem

161

(2)

Constancy of H across a transition region in one-dimensional steady flow

163

(1)

The complete set of equations governing fluid flow

164

(7)

Isentropic flow

165

(2)

Conditions for the velocity distribution to be approximately solenoidal

167

(4)

Concluding remarks to chapters I, 2 and 3

171

(3)

Flow of a Uniform Incompressible Viscous Fluid

Introduction

174

(5)

Modification of the pressure to allow for the effect of the body force

176

(3)

Steady unidirectional flow

179

(7)

Poiseuille flow

180

(2)

Tubes of non-circular cross-section

182

(1)

Two-dimensional flow

182

(1)

A model of a paint-brush

183

(2)

A remark on stability

185

(1)

Unsteady unidirectional flow

186

(9)

The smoothing-out of a discontinuity in velocity at a plane

187

(2)

Plane boundary moved suddenly in a fluid at rest

189

(1)

One rigid boundary moved suddenly and one held stationary

190

(1)

Flow due to an oscillating plane boundary

191

(2)

Starting flow in a pipe

193

(2)

The Ekman layer at a boundary in a rotating fluid

195

(6)

The layer at a free surface

197

(2)

The layer at a rigid plane boundary

199

(2)

Flow with circular streamlines

201

(4)

The steady jet from a point source of momentum

205

(6)

Dynamical similarity and the Reynolds number

211

(5)

Other dimensionless parameters having dynamical significance

215

(1)

Flow fields in which inertia forces are negligible

216

(13)

Flow in slowly-varying channels

217

(2)

Lubrication theory

219

(3)

The Hele-Shaw cell

222

(1)

Percolation through porous media

223

(1)

Two-dimensional flow in a corner

224

(3)

Uniqueness and minimum dissipation theorems

227

(2)

Flow due to a moving body at small Reynolds number

229

(11)

A rigid sphere

230

(5)

A spherical drop of a different fluid

235

(3)

A body of arbitrary shape

238

(2)

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

240

(6)

A rigid sphere

241

(3)

A rigid circular cylinder

244

(2)

The viscosity of a dilute suspension of small particles

246

(9)

The flow due to a sphere embedded in a pure straining motion

248

(2)

The increased rate of dissipation in an incompressible suspension

250

(3)

The effective expansion viscosity of a liquid containing gas bubbles

253

(2)

Changes in the flow due to moving bodies as R increases from I to about 100

255

(9)

Flow at Large Reynolds Number: Effects of Viscosity

Introduction

264

(2)

Vorticity dynamics

266

(7)

The intensification of vorticity by extension of vortex-lines

270

(3)

Kelvin's circulation theorem and vorticity laws for an inviscid fluid

273

(4)

The persistence of irrotationality

276

(1)

The source of vorticity in motions generated from rest

277

(5)

Steady flows in which vorticity generated at a solid surface is prevented by convection from diffusing far away from it

282

(12)

Flow along plane and circular walls with suction through the wall

282

(3)

Flow toward a `stagnation point' at a rigid boundary

285

(5)

Centrifugal flow due to a rotating disk

290

(4)

Steady two-dimensional flow in a converging or diverging channel

294

(8)

Purely convergent flow

297

(1)

Purely divergent flow

298

(3)

Solutions showing both outflow and inflow

301

(1)

Boundary layers

302

(6)

The boundary layer on a flat plate

308

(6)

The effects of acceleration and deceleration of the external stream

314

(11)

The similarity solution for an external stream velocity proportional to xm

316

(2)

Calculation of the steady boundary layer on a body moving through fluid

318

(3)

Growth of the boundary layer in initially irrotational flow

321

(4)

Separation of the boundary layer

325

(6)

The flow due to bodies moving steadily through fluid

331

(12)

Flow without separation

332

(5)

Flow with separation

337

(6)

Jets, free shear layers and wakes

343

(10)

Narrow jets

343

(3)

Free shear layers

346

(2)

Wakes

348

(5)

Oscillatory boundary layers

353

(11)

The damping force on an oscillating body

355

(3)

Steady streaming due to an oscillatory boundary layer

358

(3)

Applications of the theory of steady streaming

361

(3)

Flow systems with a free surface

364

(8)

The boundary layer at a free surface

364

(3)

The drag on a spherical gas bubble rising steadily through liquid

367

(3)

The attenuation of gravity waves

370

(2)

Examples of use of the momentum theorem

372

(6)

The force on a regular array of bodies in a stream

372

(1)

The effect of a sudden enlargement of a pipe

373

(5)

Irrotational Flow Theory and its Applications

The role of the theory of flow of an inviscid fluid

378

(2)

General properties of irrotational flow

380

(6)

Integration of the equation of motion

382

(1)

Expressions for the kinetic energy in terms of surface integrals

383

(1)

Kelvin's minimum energy theorem

384

(1)

Positions of a maximum of q and a minimum of p

384

(2)

Local variation of the velocity magnitude

386

(1)

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

386

(12)

Efflux from a circular orifice in an open vessel

387

(4)

Flow over a weir

391

(1)

Jet of liquid impinging on a plane wall

392

(4)

Irrotational flow which may be made steady by choice of rotating axes

396

(2)

General features of irrotational flow due to a moving rigid body

398

(11)

The velocity at large distances from the body

399

(3)

The kinetic energy of the fluid

402

(2)

The force on a body in translational motion

404

(3)

The acceleration reaction

407

(2)

The force on a body in accelerating fluid

409

(1)

Use of the complex potential for irrotational flow in two dimensions

409

(14)

Flow fields obtained by special choice of the function w(z)

410

(3)

Conformal transformation of the plane of flow

413

(5)

Transformation of a boundary into an infinite straight line

418

(2)

Transformation of a closed boundary into a circle

420

(2)

The circle theorem

422

(1)

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

423

(12)

A circular cylinder

424

(3)

An elliptic cylinder in translational motion

427

(6)

The force and moment on a cylinder in steady translational motion

433

(2)

Two-dimensional aerofoils

435

(14)

The practical requirements of aerofoils

435

(3)

The generation of circulation round an aerofoil and the basis for Joukowski's hypothesis

438

(3)

Aerofoils obtained by transformation of a circle

441

(3)

Joukowski aerofoils

444

(5)

Axisymmetric irrotational flow due to moving bodies

449

(14)

Generalities

449

(3)

A moving sphere

452

(3)

Ellipsoids of revolution

455

(3)

Body shapes obtained from source singularities on the axis of symmetry

458

(2)

Semi-infinite bodies

460

(3)

Approximate results for slender bodies

463

(8)

Slender bodies of revolution

463

(3)

Slender bodies in two dimensions

466

(1)

Thin aerofoils in two dimensions

467

(4)

Impulsive motion of a fluid

471

(3)

Impact of a body on a free surface of liquid

473

(1)

Large gas bubbles in liquid

474

(7)

A spherical-cap bubble rising through liquid under gravity

475

(2)

A bubble rising in a vertical tube

477

(2)

A spherical expanding bubble

479

(2)

Cavitation in a liquid

481

(12)

Examples of cavity formation in steady flow

482

(3)

Examples of cavity formation in unsteady flow

485

(1)

Collapse of a transient cavity

486

(5)

Steady-state cavities

491

(2)

Free-streamline theory, and steady jets and cavities

493

(14)

Jet emerging from an orifice in two dimensions

495

(2)

Two-dimensional flow past a flat plate with a cavity at ambient pressure

497

(5)

Steady-state cavities attached to bodies held in a stream of liquid

502

(5)

Flow of Effectively Inviscid Fluid with Vorticity

Introduction

507

(10)

The self-induced movement of a line vortex

509

(2)

The instability of a sheet vortex

511

(6)

Flow in unbounded fluid at rest at infinity

517

(10)

The resultant force impulse required to generate the motion

518

(2)

The total kinetic energy of the fluid

520

(1)

Flow with circular vortex-lines

521

(1)

Vortex rings

522

(5)

Two-dimensional flow in unbounded fluid at rest at infinity

527

(9)

Integral invariants of the vorticity distribution

528

(2)

Motion of a group of point vortices

530

(2)

Steady motions

532

(4)

Steady two-dimensional flow with vorticity throughout the fluid

536

(7)

Uniform vorticity in a region bounded externally

538

(1)

Fluid in rigid rotation at infinity

539

(2)

Fluid in simple shearing motion at infinity

541

(2)

Steady axisymmetric flow with swirl

543

(12)

The effect of a change of cross-section of a tube on a stream of rotating fluid

546

(4)

The effect of a change of external velocity on an isolated vortex

550

(5)

Flow systems rotating as a whole

555

(12)

The restoring effect of Coriolis forces

555

(2)

Steady flow at small Rossby number

557

(2)

Propagation of waves in a rotating fluid

559

(5)

Flow due to a body moving along the axis of rotation

564

(3)

Motion in a thin layer on a rotating sphere

567

(13)

Geostrophic flow

571

(2)

Flow over uneven ground

573

(4)

Planetary waves

577

(3)

The vortex system of a wing

580

(14)

General features of the flow past lifting bodies in three dimensions

580

(3)

Wings of large aspect ratio, and `lifting-line' theory

583

(6)

The trailing vortex system far downstream

589

(2)

Highly swept wings

591

(3)

Appendices

1 Measured values of some physical properties of common fluids

594

(4)

(a) Dry air at a pressure of one atmosphere

594

(1)

(b) The Standard Atmosphere

595

(1)

595

(2)

(d) Diffusivities for momentum and heat at 15 °C and I atm

597

(1)

(e) Surface tension between two fluids

597

(1)

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

598

(6)

Publications referred to in the text

604

(5)

Subject Index

609

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An Introduction to Fluid Dynamics

Summary

Table of Contents

Supplemental Materials