Chapter 1. The Physical Properties of Fluids 1
1.1 Solids，liquids and gases 1
1.2 The continuum hypothesis 4
1.3 Volume forces and surface forces acting on a fluid 7
Representation of surface forces by the stress tensor， 9
The stress tensor in a fluid at rest， 12
1.4 Mechanical equilibrium of a fluid 14
A body ‘floating’ in fluid at rest， 16
Fluid at rest under gravity， 18
1.5 Classical thermodynamics 20
1.6 Transport phenomena 28
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 gases 37
A perfect gas in equilibrium， 38
Departures from the perfect-gas laws， 45
Transport coefficients in a perfect gas， 47
Other manifestations of departure from equilibrium of a perfect gas， 50
1.8 The distinctive properties of liquids 53
Equilibrium properties， 55
Transport coefficients， 57
1.9 Conditions at a boundary between two media 60
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 Field 71
2.1 Specification of the flow field 71
Differentiation following the motion of the fluid， 72
2.2 Conservation of mass 73
Use of a stream function to satisfy the mass-conservation equation， 75
2.3 Analysis of the relative motion near a point 79
Simple shearing motion， 83
2.4 Expression for the velocity distribution with specified rate of expansion and vorticity 84
2.5 Singularities in the rate of expansion.Sources and sinks 88
2.6 The vorticity distribution 92
Line vortices， 93
Sheet vortices， 96
2.7 Velocity distributions with zero rate of expansion and zero vorticity 99
Conditions for 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 space 108
Conditions for ？φ to be determined uniquely， 112
2.9 Three-dimensional flow fields extending to infinity 114
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 infinity 124
Irrotational solenoida flow due to a rigid body in translational motion， 128
Chapter 3. Equations Governing the Motion of a Fluid 131
3.1 Material integrals in a moving fluid 131
Rates of change of material integrals， 133
Conservation laws for a fluid in motion， 135
3.2 The equation of motion 137
Use of the momentum equation in integral form， 138
Equation of motion relative to moving axes， 139
3.3 The expression for the stress tensor 141
Mechanical definition of pressure in a moving fluid， 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 motion 151
3.5 Bernoulli’s theorem for steady flow of a frictionless non-conducting fluid 156
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 flow 164
Isentropic flow， 165
Conditions for the velocity distribution to be approximately solenoidal， 167
3.7 Concluding remarks to chapters 1，2and3 171
Chapter 4.Flow of a Uniform Incompressible Viscous Fluid 174
4.1 Introduction 174
Modification of the pressure to allow for the effect of the body force， 176
4.2 Steady unidirectional flow 179
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 flow 186
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 fluid 195
The layer at a free surface， 197
The layer at a rigid plane boundary， 199
4.5 Flow with circular streamlines 201
4.6 The steady jet from a point source of momentum 205
4.7 Dynamical similarity and the Reynolds number 211
Other dimensionless parameters having dynamical significance， 215
4.8 Flow fields in which inertia forces are negligible 216
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 number 229
A rigid sphere， 230
A spherical drop of a different fluid， 235
A body of arbitrary shape， 238
4.10 Oseen’s improvement of the equation for flow due to moving bodies at small Reynolds number 240
A rigid sphere， 241
A rigid circular cylinder， 244
4.11 The viscosity of a dilute suspension of small particles 246
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 about100 255
Chapter 5. Flow at Large Reynolds Number：Effects of Viscosity 264
5.1 Introduction 264
5.2 Vorticity dynamics 266
The intensification of vorticity by extension of vortex-lines， 270
5.3 Kelvin’s circulation theorem and vorticity laws for an inviscid fluid 273
The persistence of irrotationality， 276
5.4 The source of vorticity in motions generated from rest 277
5.5 Steady flows in which vorticity generated at a solid surface is prevented by convection from diffusing far away from it 282
（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 channel 294
Purely convergent flow， 297
Purely divergent flow， 298
Solutions showing both outflow and inflow， 301
5.7 Boundary layers 302
5.8 The boundary layer on a flat plate 308
5.9 The effects of acceleration and deceleration of the external stream 314
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 layer 325
5.11 The flow due to bodies moving steadily through fluid 331
Flow without separation， 332
Flow with separation， 337
5.12 Jets，free shear layers and wakes 343
Narrow jets， 343
Free shear layers， 346
Wakes， 348
5.13 Oscillatory boundary layers 353
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 surface 364
The boundary layer at a free surface， 364
The drag on a spherical gas bubble rising steadily through liquid， 367
The attenuation of gravity waves， 370
5.15 Examples of use of the momentum theorem 372
The force on a regular array of bodies in a stream， 372
The effect of a sudden enlargement of a pipe， 373
Chapter 6. Irrotational Flow Theory and its Applications 378
6.1 The role of the theory of flow of an inviscid fluid 378
6.2 General properties of irrotational flow 380
Integration of the equation of motion， 382
Expressions for the kinetic energy in terms of surface integrals， 383
Kelvin’s minimum energy theorem， 384
Positions of a maximum of q and a minimum of p， 384
Local variation of the velocity magnitude， 386
6.3 Steady flow：some applications of Bernoulli’s theorem and the momentum theorem 386
Efflux from a circular orifice in an open vessel， 387
Flow over a weir， 391
Jet of liquid impinging on a plane wall， 392
Irrotational flow which may be made steady by choice of rotating axes， 396
6.4 General features of irrotational flow due to a moving rigid body 398
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 dimensions 409
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 circulation 423
A circular cylinder， 424
An elliptic cylinder in translational motion， 427
The force and moment on a cylinder in steady translational motion， 433
6.7 Two-dimensional aerofoils 435
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 bodies 449
Generalities， 449
A moving sphere， 452
Ellipsoids of revolution， 455
Body shapes obtained from source singularities on the axis of symmetry， 458
Semi-infinite bodies， 460
6.9 Approximate results for slender bodies 463
Slender bodies of revolution， 463
Slender bodies in two dimensions， 466
Thin aerofoils in two dimensions， 467
6.10 Impulsive motion of a fluid 471
Impact of a body on a free surface of liquid， 473
6.11 Large gas bubbles in liquid 474
A spherical-cap bubble rising through liquid under gravity， 475
A bubble rising in a vertical tube， 477
A spherical expanding bubble， 479
6.12 Cavitation in a liquid 481
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 cavities 493
Jet emerging from an orifice in two dimensions， 495
Two-dimensional flow past a flat plate with a cavity at ambient pressure， 497
Steady-state cavities attached to bodies held in a stream of liquid， 502
Chapter 7. Flow of Effectively Inviscid Fluid with Vorticity 507
7.1 Introduction 507
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 infinity 517
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 infinity 527
Integral invariants of the vorticity distribution， 528
Motion of a group of point vortices， 530
Steady motions， 532
7.4 Steady two-dimensional flow with vorticity throughout the fluid 536
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 swirl 543
The effect of a change of cross-section of a tube on a stream of rotating fluid， 546
The effect of a change of external velocity on an isolated vortex， 550
7.6 Flow systems rotating as a whole 555
The restoring effect of Coriolis forces， 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 sphere 567
Geostrophic flow， 571
Flow over uneven ground， 573
Planetary waves， 577
7.8 The vortex system of a wing 580
General features of the flow past lifting bodies in three dimensions， 580
Wings of large aspect ratio，and ‘lifting-line’ theory， 583
The trailing vortex system far downstream， 589
Highly swept wings， 591
Appendices 594
1 Measured values of some physical properties of common fluids 594
（a）Dry air at a pressure of one atmosphere， 594
（b）The Standard Atmosphere， 595
（c）Pure water， 595
（d）Diffusivities for momentum and heat at 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 systems 598
Publications referred to in the text 604
Subject Index 609