Preface page ................................................... xv
List of Abbreviations ........................................ xvii
1. Introduction ................................................ 1
1.1. Irrotational flow, Laplace's equation ................... 2
1.2. Continuity equation, incompressible fluids,
isochoric flow .......................................... 3
1.3. Euler's equations ....................................... 3
1.4. Generation of vorticity in fluids governed by
Euler's equations ....................................... 4
1.5. Perfect fluids, irrotational flow ....................... 4
1.6. Boundary conditions for irrotational flow ............... 5
1.7. Streaming irrotational flow over a stationary sphere .... 6
2. Historical notes ............................................. 8
2.1. Navier-Stokes equations ................................. 8
2.2. Stokes theory of potential flow of viscous fluid ........ 9
2.3. The dissipation method ................................. 10
2.4. The distance a wave will travel before it decays by
a certain amount ....................................... 11
3. Boundary conditions for viscous fluids ...................... 13
4. Helmholtz decomposition coupling rotational to
irrotational flow ........................................... 16
4.1. Helmholtz decomposition ................................ 16
4.2. Navier-Stokes equations for the decomposition .......... 17
4.3. Self-equilibration of the irrotational viscous
stress ................................................. 19
4.4. Dissipation function for the decomposed motion ......... 20
4.5. Irrotational flow and boundary conditions .............. 20
4.6. Examples from hydrodynamics ............................ 21
4.6.1. Poiseuille flow ................................. 21
4.6.2. Flow between rotating cylinders ................. 21
4.6.3. Stokes flow around a sphere of radius a in
a uniform stream U .............................. 22
4.6.4. Streaming motion past an ellipsoid .............. 23
4.6.4. Streaming motion past an ellipsoid .............. 23
4.6.5. Hadamard-Rybyshinsky solution for streaming
flow past a liquid sphere ....................... 23
4.6.6. Axisymmetric steady flow around a spherical
gas bubble at finite Reynolds numbers ........... 24
4.6.7. Viscous decay of free-gravity waves ............. 24
4.6.8. Oseen flow ...................................... 25
4.6.9. Flows near internal stagnation points in
viscous incompressible fluids ................... 26
4.6.10.Hiemenz boundary-layer solution for two-
dimensional flow toward a "stagnation point"
at a rigid boundary ............................. 29
4.6.11.Jeffrey-Hamel flow in diverging and
converging channels ............................. 31
4.6.12.An irrotational Stokes flow ..................... 32
4.6.13.Lighthill's approach ............................ 32
4.7. Conclusion ............................................. 33
5. Harmonic functions that give rise to vorticity .............. 35
6. Radial motions of a spherical gas bubble in a viscous
liquid ...................................................... 39
7. Rise velocity of a spherical cap bubble ..................... 42
7.1. Analysis ............................................... 42
7.2. Experiments ............................................ 46
7.3. Conclusions ............................................ 50
8. Ellipsoidal model of the rise of a Taylor bubble in a
round tube .................................................. 51
8.1. Introduction ........................................... 51
8.1.1. Unexplained and paradoxical features ............ 52
8.1.2. Drainage ........................................ 53
8.1.3. Brown's analysis of drainage .................... 54
8.1.4. Viscous potential flow .......................... 55
8.2. Ellipsoidal bubbles .................................... 56
8.2.1. Ovary ellipsoid ................................. 56
8.2.2. Planetary ellipsoid ............................. 60
8.2.3. Dimensionless rise velocity ..................... 61
8.3. Comparison of theory and experiment .................... 63
8.4. Comparison of theory and correlations .................. 66
8.5. Conclusion ............................................. 68
9. Rayleigh-Taylor instability of viscous fluids ............... 70
9.1. Acceleration ........................................... 71
9.2. Simple thought experiments ............................. 71
9.3. Analysis ............................................... 71
9.3.1. Linear theory of Chandrasekhar .................. 73
9.3.2. Viscous potential flow .......................... 74
9.4. Comparison of theory and experiments ................... 76
9.5. Comparison of the stability theory with the
experiments on drop breakup ............................ 76
9.6. Comparison of the measured wavelength of
corrugations on the drop surface with the prediction
of the stability theory ................................ 81
9.7. Fragmentation of Newtonian and viscoelastic drops ...... 84
9.8. Modeling Rayleigh-Taylor instability of a sedimenting
suspension of several thousand circular particles
in a direct numerical simulation ....................... 89
10.The force on a cylinder near a wall in viscous potential
flows ....................................................... 90
10.1.The flow that is due to the circulation about
the cylinder ........................................... 90
10.2.The streaming flow past the cylinder near a wall ....... 93
10.3.The streaming flow past a cylinder with circulation
near a wall ............................................ 95
11.Kelvin-Helmholtz instability ............................... 100
11.1.KH instability on an unbounded domain ................. 100
11.2.Maximum growth rate, Hadamard instability, neutral
curves ................................................ 102
11.2.1.Maximum growth rate ............................ 102
11.2.2.Hadamard instability ........................... 102
11.2.3.The regularization of Hadamard instability ..... 102
11.2.4.Neutral curves ................................. 103
11.3.KH instability in a channel ........................... 103
11.3.1.Formulation of the problem ..................... 104
11.3.2.Viscous potential flow analysis ................ 105
11.3.3.KH instability of inviscid fluid ............... 109
11.3.4.Dimensionless form of the dispersion
equation ....................................... 110
11.3.5.The effect of liquid viscosity and surface
tension on growth rates and neutral curves ..... 112
11.3.6.Comparison of theory and experiments in
rectangular ducts .............................. 114
11.3.7.Critical viscosity and density ratios .......... 118
11.3.8.Further comparisons with previous results ...... 119
11.3.9.Nonlinear effects .............................. 121
11.3.10.Combinations of Rayleigh-Taylor and Kelvin-
Helmholtz instabilities ....................... 123
12.Energy equation for irrotational theories of gas-liquid
flow: viscous potential flow, viscous potential flow with
pressure correction, and dissipation method ................ 126
12.1.Viscous potential flow ................................ 126
12.2.Dissipation method according to Lamb .................. 126
12.3.Drag on a spherical gas bubble calculated from the
viscous dissipation of an irrotational flow ........... 127
12.4.The idea of a pressure correction ..................... 127
12.5.Energy equation for irrotational flow of a viscous
fluid ................................................. 128
12.6.Viscous correction of viscous potential flow .......... 130
12.7.Direct derivation of the viscous correction of
the normal stress balance for the viscous decay
of capillary-gravity waves ............................ 132
13.Rising bubbles ............................................. 134
13.1.The dissipation approximation and viscous potential
flow .................................................. 134
13.1.1.Pressure correction formulas ................... 134
13.2.Rising spherical gas bubble ........................... 135
13.3.Rising oblate ellipsoidal bubble ...................... 136
13.4.A liquid drop rising in another liquid ................ 137
13.5.Purely irrotational analysis of a toroidal bubble
in a viscous fluid .................................... 139
13.5.1.Prior work, experiments ........................ 139
13.5.2.The energy equation ............................ 141
13.5.3.The impulse equation ........................... 145
13.5.4.Comparison of irrotational solutions for
inviscid and viscous fluids .................... 145
13.5.5.Stability of the toroidal vortex ............... 148
13.5.6.Boundary-integral study of vortex ring
bubbles in a viscous liquid .................... 152
13.5.7.Irrotational motion of a massless cylinder
under the combined action of Kutta-Joukowski
lift, acceleration of added mass, and viscous
drag ........................................... 153
13.6.The motion of a spherical gas bubble in viscous
potential flow ........................................ 155
13.7.Steady motion of a deforming gas bubble in
a viscous potential flow .............................. 157
13.8.Dynamic simulations of the rise of many bubbles in
a viscous potential flow .............................. 157
14.Purely irrotational theories of the effect of viscosity
on the decay of waves ...................................... 159
14.1.Decay of free-gravity waves ........................... 159
14.1.1.Introduction ................................... 159
14.1.2.Irrotational viscous corrections for
the potential flow solution .................... 160
14.1.3.Relation between the pressure correction and
Lamb's exact solution .......................... 162
14.1.4.Comparison of the decay rate and the wave
velocity given by the exact solution, VPF,
and VCVPF ...................................... 163
14.1.5.Why does the exact solution agree with VCVPF
when к < кс and with VPF when к > кс? .......... 166
14.1.6.Conclusion and discussion ...................... 168
14.1.7.Quasi-potential approximation - vorticity
layers ......................................... 169
14.2.Viscous decay of capillary waves on drops and
bubbles ............................................... 170
14.2.1.Introduction ................................... 171
14.2.2.VPF analysis of a single spherical drop
immersed in another fluid ...................... 172
14.2.3.VCVPF analysis of a single spherical drop
immersed in another fluid ...................... 176
14.2.4.Dissipation approximation (DM) ................. 180
14.2.5.Exact solution of the linearized free-
surface problem ................................ 181
14.2.6.VPF and VCVPF analyses for waves acting on
a plane interface considering surface
tension - comparison with Lamb's solution ...... 183
14.2.7.Results and discussion ......................... 185
14.2.8.Concluding remarks ............................. 192
14.3.Irrotational dissipation of capillary-gravity waves ... 193
14.3.1.Correction of the wave frequency assumed by
Lamb ........................................... 193
14.3.2.Irrotational dissipation of nonlinear
capillary-gravity waves ........................ 195
15.Irrotational Faraday waves on a viscous fluid .............. 197
15.1.Introduction .......................................... 198
15.2.Energy equation ....................................... 199
15.3.VPF and VCVPF ......................................... 200
15.3.1.Potential flow ................................. 200
15.3.2.Amplitude equations for the elevation of
the free surface ............................... 201
15.4.Dissipation method .................................... 204
15.5.Stability analysis .................................... 204
15.6.Rayleigh-Taylor instability and Faraday waves ......... 206
15.7.Comparison of purely irrotational solutions with
exact solutions ....................................... 210
15.8.Bifurcation of Faraday waves in a nearly square
container ............................................. 213
15.9.Conclusion ............................................ 213
16.Stability of a liquid jet into incompressible gases and
liquids .................................................... 215
16.1.Capillary instability of a liquid cylinder in
another fluid ......................................... 215
16.1.1.Introduction ................................... 215
16.1.2.Linearized equations governing capillary
instability .................................... 217
16.1.3.Fully viscous flow analysis .................... 218
16.1.4.Viscous potential flow analysis ................ 218
16.1.5.Pressure correction for viscous potential
flow ........................................... 219
16.1.6.Comparison of growth rates ..................... 222
16.1.7.Dissipation calculation for capillary
instability .................................... 230
16.1.8.Discussion of the pressure corrections at
the interface of two viscous fluids ............ 232
16.1.9.Capillary instability when one fluid is
a dynamically inactive gas ..................... 234
16.1.10.Conclusions ................................... 237
16.2.Stability of a liquid jet into incompressible gases:
Temporal, convective, and absolute instabilities ...... 238
16.2.1.Introduction ................................... 239
16.2.2.Problem formulation ............................ 240
16.2.3.Dispersion relation ............................ 241
16.2.4.Temporal instability ........................... 243
16.2.5.Numerical results of temporal instability ...... 250
16.2.6.Spatial, absolute, and convective
instability .................................... 251
16.2.7.Algebraic equations at a singular point ........ 255
16.2.8.Subcritical, critical, and supercritical
singular points ................................ 256
16.2.9.Inviscid jet in inviscid fluid (Re → ∞,
т = 0) ......................................... 261
16.2.10.Exact solution; comparison with previous
results ....................................... 262
16.2.11.Summary and discussion ........................ 266
16.3.Viscous potential flow of the Kelvin-Helmholtz
instability of a cylindrical jet of one fluid into
the same fluid ........................................ 267
16.3.1.Mathematical formulation ....................... 267
16.3.2.Normal modes; dispersion relation .............. 268
16.3.3.Growth rates and frequencies ................... 269
16.3.4.Hadamard instabilities for piecewise
discontinuous profiles ......................... 269
17.Stress-induced cavitation .................................. 272
17.1.Theory of stress-induced cavitation ................... 273
17.1.1.Mathematical formulation ....................... 273
17.1.2.Cavitation threshold ........................... 275
17.2.Viscous potential flow analysis of stress-induced
cavitation in an aperture flow ........................ 278
17.2.1.Analysis of stress-induced cavitation .......... 279
17.2.2.Stream function, potential function, and
velocity ....................................... 281
17.2.3.Cavitation threshold ........................... 282
17.2.4.Conclusions .................................... 286
17.2.5.Navier-Stokes simulation ....................... 287
17.3.Streaming motion past a sphere ........................ 287
17.3.1.Irrotational flow of a viscous fluid ........... 290
17.3.2.An analysis for maximum К ...................... 293
17.4.Symmetric model of capillary collapse and rupture ..... 297
17.4.1.Introduction ................................... 297
17.4.2.Analysis ....................................... 299
17.4.3.Conclusions and discussion ..................... 304
17.4.4.Appendix ....................................... 308
18.Viscous effects of the irrotational flow outside
boundary layers on rigid solids ............................ 310
18.1.Extra drag due to viscous dissipation of the
irrotational flow outside the boundary layer .......... 311
18.1.1.Pressure corrections for the drag on
a circular gas bubble .......................... 312
18.1.2.A rotating cylinder in a uniform stream ........ 315
18.1.3.The additional drag on an airfoil by
the dissipation method ......................... 324
18.1.4.Discussion and conclusion ...................... 327
18.2.Glauert's solution of the boundary layer on
a rapidly rotating cylinder in a uniform stream
revisited ............................................. 329
18.2.1.Introduction ................................... 330
18.2.2.Unapproximated governing equations ............. 334
18.2.3.Boundary-layer approximation and Glauert's
equations ...................................... 334
18.2.4.Decomposition of the velocity and pressure
field .......................................... 335
18.2.5.Solution of the boundary-layer flow ............ 336
18.2.6.Higher-order boundary-layer theory ............. 347
18.2.7.Discussion and conclusion ...................... 350
18.3.Numerical study of the steady-state uniform flow
past a rotating cylinder .............................. 352
18.3.1.Introduction ................................... 353
18.3.2.Numerical features ............................. 355
18.3.3.Results and discussion ......................... 359
18.3.4.Concluding remarks ............................. 372
19.Irrotational flows that satisfy the compressible Navier-
Stokes equations ........................................... 374
19.1.Acoustics ............................................. 375
19.2.Spherically symmetric waves ........................... 377
19.3.Liquid jet in a high-Mach-number airstream ............ 378
19.3.1.Introduction ................................... 378
19.3.2.Basic partial differential equations ........... 379
19.3.3.Cylindrical liquid jet in a compressible gas ... 380
19.3.4.Basic isentropic relations ..................... 380
19.3.5.Linear stability of the cylindrical liquid
jet in a compressible gas; dispersion
equation ....................................... 381
19.3.6.Stability problem in dimensionless form ........ 383
19.3.7.Inviscid potential flow ........................ 386
19.3.8.Growth-rate parameters as functions of M for
different viscosities .......................... 386
19.3.9.Azimuthal periodicity of the most dangerous
disturbance .................................... 387
19.3.10.Variation of the growth-rate parameters
with the Weber number .......................... 388
19.3.11.Convective/absolute instability ............... 389
19.3.12.Conclusions ................................... 393
20.Irrotational flows of viscoelastic fluids .................. 395
20.1.Oldroyd В model ....................................... 395
20.2.Asymptotic form of the constitutive equations ......... 396
20.2.1.Retarded motion expansion for the UCM model .... 396
20.2.2.The expanded UCM model in potential flow ....... 397
20.2.3.Potential flow past a sphere calculated with
the expanded UCM model ......................... 397
20.3.Second-order fluids ................................... 398
20.4.Purely irrotational flows ............................. 400
20.5.Purely irrotational flows of a second-order fluid ..... 400
20.6.Reversal of the sign of the normal stress at a point
of stagnation ......................................... 401
20.7.Fluid forces near stagnation points on solid bodies ... 402
20.7.1.Turning couples on long bodies ................. 402
20.7.2.Particle-particle interactions ................. 402
20.7.3.Sphere-wall interactions ....................... 403
20.7.4.Flow-induced microstructure .................... 404
20.8.Potential flow over a sphere for a second-order
fluid ................................................. 406
20.9.Potential flow over an ellipse ........................ 408
20.9.1.Normal stress at the surface of the ellipse .... 409
20.9.2.The effects of the Reynolds number ............. 410
20.9.3.The effects of-α1(pα2) ......................... 412
20.9.4.The effects of the aspect ratio ................ 412
20.10.The moment on the ellipse ............................ 413
20.11.The reversal of the sign of the normal stress at
stagnation points .................................... 414
20.12.Flow past a flat plate ............................... 416
20.13.Flow past a circular cylinder with circulation ....... 416
20.14.Potential flow of a second-order fluid over a
triaxial ellipsoid ................................... 417
20.15.Motion of a sphere normal to a wall in a second-
order fluid .......................................... 418
20.15.1.Low Reynolds numbers ......................... 419
20.15.2.Viscoelastic Potential Flow ................... 422
20.15.3.Conclusions ................................... 425
21.Purely irrotational theories of stability of
viscoelastic fluids ........................................ 426
21.1.Rayleigh-Taylor instability of viscoelastic drops
at high Weber numbers ................................. 426
21.1.1.Introduction ................................... 426
21.1.2.Experiments .................................... 427
21.1.3.Theory ......................................... 428
21.1.4.Comparison of theory and experiment ............ 437
21.2.Purely irrotational theories of the effects of
viscosity and viscoelasticity on capillary
instability of a liquid cylinder ...................... 443
21.2.1.Introduction ................................... 443
21.2.2.Linear stability equations and the exact
solution ....................................... 444
21.2.3.Viscoelastic potential flow .................... 446
21.2.4.Dissipation and the formulation for the
additional pressure contribution ............... 447
21.2.5.The additional pressure contribution for
capillary instability .......................... 448
21.2.6.Comparison of the growth rate .................. 449
21.2.7.Comparison of the stream functions ............. 451
21.2.8.Discussion ..................................... 456
21.3.Steady motion of a deforming gas bubble in
a viscous potential flow .............................. 460
22.Numerical methods for irrotational flows of viscous
fluid ...................................................... 461
22.1.Perturbation methods .................................. 461
22.2.Boundary-integral methods for inviscid potential
flow .................................................. 462
22.3.Boundary-integral methods for viscous potential
flow .................................................. 464
Appendix A. Equations of motion and strain rates for
rotational and irrotational flow in Cartesian,
cylindrical, and spherical coordinates ............ 465
Appendix B. List of frequently used symbols and concepts ...... 471
References .................................................... 473
Index ......................................................... 487
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