An introduction to fluid dynamics

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