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Chapter 1. The Physical Properties of Fluids1

1.1 Solids，liquids and gases1

1.2 The continuum hypothesis4

1.3 Volume forces and surface forces acting on a fluid7

Representation of surface forces by the stress tensor，9

The stress tensor in a fluid at rest，12

1.4 Mechanical equilibrium of a fluid14

A body ‘floating’ in fluid at rest，16

Fluid at rest under gravity，18

1.5 Classical thermodynamics20

1.6 Transport phenomena28

The linear relation between flux and the gradient of a scalar intensity，30

The equations for diffusion and heat conduction in isotropic media at rest，32

Molecular transport of momentum in a fluid，36

1.7 The distinctive properties of gases37

A perfect gas in equilibrium，38

Departures from the perfect-gas laws，45

Transport coefficients in a perfect gas，47

Other manifestations of departure from equilibrium of a perfect gas，50

1.8 The distinctive properties of liquids53

Equilibrium properties，55

Transport coefficients，57

1.9 Conditions at a boundary between two media60

Surface tension，60

Equilibrium shape of a boundary between two stationary fluids，63

Transition relations at a material boundary，68

Chapter 2. Kinematics of the Flow Field71

2.1 Specification of the flow field71

Differentiation following the motion of the fluid，72

2.2 Conservation of mass73

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

2.3 Analysis of the relative motion near a point79

Simple shearing motion，83

2.4 Expression for the velocity distribution with specified rate of expansion and vorticity84

2.5 Singularities in the rate of expansion.Sources and sinks88

2.6 The vorticity distribution92

Line vortices，93

Sheet vortices，96

2.7 Velocity distributions with zero rate of expansion and zero vorticity99

Conditions for Vφ to be determined uniquely，102

Irrotational solenoidal flow near a stagnation point，105

The complex potential for irrotational solenoidal flow in two dimensions，106

2.8 Irrotational solenoidal flow in doubly-connected regions of space108

Conditions for ？φ to be determined uniquely，112

2.9 Three-dimensional flow fields extending to infinity114

Asymptotic expressions for ue and uv，114

The behaviour of φ at large distances，117

Conditions for ？φ to be determined uniquely，119

The expression of φ as a power series，120

Irrotational solenoidal flow due to a rigid body in translational motion，122

2.10 Two-dimensional flow fields extending to infinity124

Irrotational solenoida flow due to a rigid body in translational motion，128

Chapter 3. Equations Governing the Motion of a Fluid131

3.1 Material integrals in a moving fluid131

Rates of change of material integrals，133

Conservation laws for a fluid in motion，135

3.2 The equation of motion137

Use of the momentum equation in integral form，138

Equation of motion relative to moving axes，139

3.3 The expression for the stress tensor141

Mechanical definition of pressure in a moving fluid，141

The relation between deviatoric stress and rate-of-strain for a Newtonian fluid，142

The Navier-Stokes equation，147

Conditions on the velocity and stress at a material boundary，148

3.4 Changes in the internal energy of a fluid in motion151

3.5 Bernoulli’s theorem for steady flow of a frictionless non-conducting fluid156

Special forms of Bernoulli’s theorem，161

Constancy of H across a transition region in one-dimensional steady flow，163

3.6 The complete set of equations governing fluid flow164

Isentropic flow，165

Conditions for the velocity distribution to be approximately solenoidal，167

3.7 Concluding remarks to chapters 1，2and3171

Chapter 4.Flow of a Uniform Incompressible Viscous Fluid174

4.1 Introduction174

Modification of the pressure to allow for the effect of the body force，176

4.2 Steady unidirectional flow179

Poiseuille flow，180

Tubes of non-circular cross-section，182

Two-dimensional flow，182

A model of a paint-brush，183

A remark on stability，185

4.3 Unsteady unidirectional flow186

The 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

4.4 The Ekman layer at a boundary in a rotating fluid195

The layer at a free surface，197

The layer at a rigid plane boundary，199

4.5 Flow with circular streamlines201

4.6 The steady jet from a point source of momentum205

4.7 Dynamical similarity and the Reynolds number211

Other dimensionless parameters having dynamical significance，215

4.8 Flow fields in which inertia forces are negligible216

Flow in slowly-varying channels，217

Lubrication theory，219

The Hele Shaw cell，222

Percolation through porous media，223

Two-dimensional flow in a corner，224

Uniqueness and minimum dissipation theorems，227

4.9 Flow due to a moving body at small Reynolds number229

A rigid sphere，230

A spherical drop of a different fluid，235

A body of arbitrary shape，238

4.10 Oseen’s improvement of the equation for flow due to moving bodies at small Reynolds number240

A rigid sphere，241

A rigid circular cylinder，244

4.11 The viscosity of a dilute suspension of small particles246

The flow due to a sphere embedded in a pure straining motion，248

The increased rate of dissipation in an incompressible suspension，250

The effective expansion viscosity of a liquid containing gas bubbles，253

4.12 Changes in the flow due to moving bodies as R increases from 1 to about100255

Chapter 5. Flow at Large Reynolds Number：Effects of Viscosity264

5.1 Introduction264

5.2 Vorticity dynamics266

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

5.3 Kelvin’s circulation theorem and vorticity laws for an inviscid fluid273

The persistence of irrotationality，276

5.4 The source of vorticity in motions generated from rest277

5.5 Steady flows in which vorticity generated at a solid surface is prevented by convection from diffusing far away from it282

（a）Flow along plane and circular walls with suction through the wall，282

（b）Flow toward a ‘stagnation point’ at a rigid boundary，285

（c）Centrifugal flow due to a rotating disk，290

5.6 Steady two-dimensional flow in a converging or diverging channel294

Purely convergent flow，297

Purely divergent flow，298

Solutions showing both outflow and inflow，301

5.7 Boundary layers302

5.8 The boundary layer on a flat plate308

5.9 The effects of acceleration and deceleration of the external stream314

The similarity solution for an external stream velocity proportional to xm，316

Calculation of the steady boundary layer on a body moving through fluid，318

Growth of the boundary layer in initially irrotational flow，321

5.10 Separation of the boundary layer325

5.11 The flow due to bodies moving steadily through fluid331

Flow without separation，332

Flow with separation，337

5.12 Jets，free shear layers and wakes343

Narrow jets，343

Free shear layers，346

Wakes，348

5.13 Oscillatory boundary layers353

The damping force on an oscillating body，355

Steady streaming due to an oscillatory boundary layer，358

Applications of the theory of steady streaming，361

5.14 Flow systems with a free surface364

The boundary layer at a free surface，364

The drag on a spherical gas bubble rising steadily through liquid，367

The attenuation of gravity waves，370

5.15 Examples of use of the momentum theorem372

The force on a regular array of bodies in a stream，372

The effect of a sudden enlargement of a pipe，373

Chapter 6. Irrotational Flow Theory and its Applications378

6.1 The role of the theory of flow of an inviscid fluid378

6.2 General properties of irrotational flow380

Integration of the equation of motion，382

Expressions for the kinetic energy in terms of surface integrals，383

Kelvin’s minimum energy theorem，384

Positions of a maximum of q and a minimum of p，384

Local variation of the velocity magnitude，386

6.3 Steady flow：some applications of Bernoulli’s theorem and the momentum theorem386

Efflux from a circular orifice in an open vessel，387

Flow over a weir，391

Jet of liquid impinging on a plane wall，392

Irrotational flow which may be made steady by choice of rotating axes，396

6.4 General features of irrotational flow due to a moving rigid body398

The velocity at large distances from the body，399

The kinetic energy of the fluid，402

The force on a body in translational motion，404

The acceleration reaction，407

The force on a body in accelerating fluid，409

6.5 Use of the complex potential for irrotational flow in two dimensions409

Flow fields obtained by special choice of the function w（z），410

Conformal transformation of the plane of flow，413

Transformation of a boundary into an infinite straight line，418

Transformation of a closed boundary into a circle，420

The circle theorem，422

6.6 Two-dimensional irrotational flow due to a moving cylinder with circulation423

A circular cylinder，424

An elliptic cylinder in translational motion，427

The force and moment on a cylinder in steady translational motion，433

6.7 Two-dimensional aerofoils435

The practical requirements of aerofoils，435

The generation of circulation round an aerofoil and the basis for Joukowski’s hypothesis，438

Aerofoils obtained by transformation of a circle，441

Joukowski aerofoils，444

6.8 Axisymmetric irrotational flow due to moving bodies449

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 bodies463

Slender bodies of revolution，463

Slender bodies in two dimensions，466

Thin aerofoils in two dimensions，467

6.10 Impulsive motion of a fluid471

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

6.11 Large gas bubbles in liquid474

A spherical-cap bubble rising through liquid under gravity，475

A bubble rising in a vertical tube，477

A spherical expanding bubble，479

6.12 Cavitation in a liquid481

Examples of cavity formation in steady flow，482

Examples of cavity formation in unsteady flow，485

Collapse of a transient cavity，486

Steady-state cavities，491

6.13 Free-streamline theory，and steady jets and cavities493

Jet emerging from an orifice in two dimensions，495

Two-dimensional flow past a flat plate with a cavity at ambient pressure，497

Steady-state cavities attached to bodies held in a stream of liquid，502

Chapter 7. Flow of Effectively Inviscid Fluid with Vorticity507

7.1 Introduction507

The self-induced movement of a line vortex，509

The instability of a sheet vortex，511

7.2 Flow in unbounded fluid at rest at infinity517

The resultant force impulse required to generate the motion，518

The total kinetic energy of the fluid，520

Flow with circular vortex-lines，521

Vortex rings，522

7.3 Two-dimensional flow in unbounded fluid at rest at infinity527

Integral invariants of the vorticity distribution，528

Motion of a group of point vortices，530

Steady motions，532

7.4 Steady two-dimensional flow with vorticity throughout the fluid536

Uniform vorticity in a region bounded externally，538

Fluid in rigid rotation at infinity，539

Fluid in simple shearing motion at infinity，541

7.5 Steady axisymmetric flow with swirl543

The effect of a change of cross-section of a tube on a stream of rotating fluid，546

The effect of a change of external velocity on an isolated vortex，550

7.6 Flow systems rotating as a whole555

The 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

7.7 Motion in a thin layer on a rotating sphere567

Geostrophic flow，571

Flow over uneven ground，573

Planetary waves，577

7.8 The vortex system of a wing580

General 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

Appendices594

1 Measured values of some physical properties of common fluids594

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

（b）The Standard Atmosphere，595

（c）Pure water，595

（d）Diffusivities for momentum and heat at 15 ℃ and I atm，597

（e）Surface tension between two fluids，597

2 Expressions for some common vector differential quantities in orthogonal curvilinear co-ordinate systems598

Publications referred to in the text604

Subject Index609