Potential Flows of Viscous and Viscoelastic Liquids (Cambridge Aerospace Series, 21)

Half-title......Page 3
Series-title......Page 5
Title......Page 7
Copyright......Page 8
Contents......Page 9
Preface......Page 17
List of Abbreviations......Page 19
1 Introduction......Page 21
1.1 Irrotational flow, Laplace\'s equation......Page 22
1.3 Euler\'s equations......Page 23
1.5 Perfect fluids, irrotational flow......Page 24
1.6 Boundary conditions for irrotational flow......Page 25
1.7 Streaming irrotational flow over a stationary sphere......Page 26
2.1 Navier-Stokes equations......Page 28
2.2 Stokes theory of potential flow of viscous fluid......Page 29
2.3 The dissipation method......Page 30
2.4 The distance a wave will travel before it decays by a certain amount......Page 31
3 Boundary conditions for viscous fluids......Page 33
4.1 Helmholtz decomposition......Page 36
4.2 Navier-Stokes equations for the decomposition......Page 37
4.3 Self-equilibration of the irrotational viscous stress......Page 39
4.5 Irrotational flow and boundary conditions......Page 40
4.6.2 Flow between rotating cylinders......Page 41
4.6.3 Stokes flow around a sphere of radius a in a uniform stream U......Page 42
4.6.5 Hadamard-Rybyshinsky solution for streaming flow past a liquid sphere......Page 43
4.6.7 Viscous decay of free-gravity waves......Page 44
4.6.8 Oseen flow......Page 45
4.6.9 Flows near internal stagnation points in viscous incompressible fluids......Page 46
4.6.10 Hiemenz boundary-layer solution for two-dimensional flow toward a \"stagnation point\'\' at a rigid boundary......Page 49
4.6.11 Jeffrey-Hamel flow in diverging and converging channels......Page 51
4.6.13 Lighthill\'s approach......Page 52
4.7 Conclusion......Page 53
5 Harmonic functions that give rise to vorticity......Page 55
6 Radial motions of a spherical gas bubble in a viscous liquid......Page 59
7.1 Analysis......Page 62
7.2 Experiments......Page 66
7.3 Conclusions......Page 70
8.1 Introduction......Page 71
8.1.1 Unexplained and paradoxical features......Page 72
8.1.2 Drainage......Page 73
8.1.3 Brown\'s analysis of drainage......Page 74
8.1.4 Viscous potential flow......Page 75
8.2.1 Ovary ellipsoid......Page 76
8.2.2 Planetary ellipsoid......Page 80
8.2.3 Dimensionless rise velocity......Page 81
8.3 Comparison of theory and experiment......Page 83
8.4 Comparison of theory and correlations......Page 86
8.5 Conclusion......Page 88
9 Rayleigh-Taylor instability of viscous fluids......Page 90
9.3 Analysis......Page 91
9.3.1 Linear theory of Chandrasekhar......Page 93
9.3.2 Viscous potential flow......Page 94
9.5 Comparison of the stability theory with the experiments on drop breakup......Page 96
9.6 Comparison of the measured wavelength of corrugations on the drop surface with the prediction of the stability theory......Page 101
9.7 Fragmentation of Newtonian and viscoelastic drops......Page 104
9.8 Modeling Rayleigh-Taylor instability of a sedimenting suspension of several thousand circular particles in a direct numerical simulation......Page 109
10.1 The flow that is due to the circulation about the cylinder......Page 110
10.2 The streaming flow past the cylinder near a wall......Page 113
10.3 The streaming flow past a cylinder with circulation near a wall......Page 115
11.1 KH instability on an unbounded domain......Page 120
11.2.3 The regularization of Hadamard instability......Page 122
11.3 KH instability in a channel......Page 123
11.3.1 Formulation of the problem......Page 124
11.3.2 Viscous potential flow analysis......Page 125
11.3.2.1 Dispersion relation......Page 126
11.3.2.3 Neutral curves......Page 128
11.3.3 KH instability of inviscid fluid......Page 129
11.3.4 Dimensionless form of the dispersion equation......Page 130
11.3.5 The effect of liquid viscosity and surface tension on growth rates and neutral curves......Page 132
11.3.6 Comparison of theory and experiments in rectangular ducts......Page 134
11.3.7 Critical viscosity and density ratios......Page 138
11.3.8 Further comparisons with previous results......Page 139
11.3.9 Nonlinear effects......Page 141
11.3.10 Combinations of Rayleigh-Taylor and Kelvin-Helmholtz instabilities......Page 143
12.2 Dissipation method according to Lamb......Page 146
12.4 The idea of a pressure correction......Page 147
12.5 Energy equation for irrotational flow of a viscous fluid......Page 148
12.6 Viscous correction of viscous potential flow......Page 150
12.7 Direct derivation of the viscous correction of the normal stress balance for the viscous decay of capillary-gravity waves......Page 152
13.1.1 Pressure correction formulas......Page 154
13.2 Rising spherical gas bubble......Page 155
13.3 Rising oblate ellipsoidal bubble......Page 156
13.4 A liquid drop rising in another liquid......Page 157
13.5.1 Prior work, experiments......Page 159
13.5.2 The energy equation......Page 161
13.5.4 Comparison of irrotational solutions for inviscid and viscous fluids......Page 165
13.5.5 Stability of the toroidal vortex......Page 168
13.5.5.2 Small disturbances......Page 169
13.5.6 Boundary-integral study of vortex ring bubbles in a viscous liquid......Page 172
13.5.7 Irrotational motion of a massless cylinder under the combinedaction of Kutta-Joukowski lift, acceleration of added mas......Page 173
13.6 The motion of a spherical gas bubble in viscous potential flow......Page 175
13.8 Dynamic simulations of the rise of many bubbles in a viscous potential flow......Page 177
14.1.1 Introduction......Page 179
14.1.2 Irrotational viscous corrections for the potential flow solution......Page 180
14.1.3 Relation between the pressure correction and Lamb\'s exact solution......Page 182
14.1.4 Comparison of the decay rate and the wave velocity given by the exact solution, VPF, and VCVPF......Page 183
14.1.4.2 VCVPF results......Page 184
14.1.4.3 Comparison of the exact and purely irrotational solutions......Page 185
14.1.5 Why does the exact solution agree with VCVPF when k < kc andwith VPF when k > kc?......Page 186
14.1.6 Conclusion and discussion......Page 188
14.1.7 Quasi-potential approximation - vorticity layers......Page 189
14.2 Viscous decay of capillary waves on drops and bubbles......Page 190
14.2.1 Introduction......Page 191
14.2.2 VPF analysis of a single spherical drop immersed in another fluid......Page 192
14.2.3 VCVPF analysis of a single spherical drop immersed in another fluid......Page 196
14.2.4 Dissipation approximation (DM)......Page 200
14.2.5 Exact solution of the linearized free-surface problem......Page 201
14.2.5.2 Spherical bubble......Page 202
14.2.6VPF and VCVPF analyses for waves acting on a plane interface considering surface tension - comparison with Lamb\'s solution......Page 203
14.2.7 Results and discussion......Page 205
14.2.8 Concluding remarks......Page 212
14.3.1 Correction of the wave frequency assumed by Lamb......Page 213
14.3.2 Irrotational dissipation of nonlinear capillary-gravity waves......Page 215
15 Irrotational Faraday waves on a viscous fluid......Page 217
15.1 Introduction......Page 218
15.2 Energy equation......Page 219
15.3.1 Potential flow......Page 220
15.3.2 Amplitude equations for the elevation of the free surface......Page 221
15.5 Stability analysis......Page 224
15.6 Rayleigh-Taylor instability and Faraday waves......Page 226
15.7 Comparison of purely irrotational solutions with exact solutions......Page 230
15.9 Conclusion......Page 233
16.1.1 Introduction......Page 235
16.1.2 Linearized equations governing capillary instability......Page 237
16.1.4 Viscous potential flow analysis......Page 238
16.1.5 Pressure correction for viscous potential flow......Page 239
16.1.6 Comparison of growth rates......Page 242
16.1.7 Dissipation calculation for capillary instability......Page 250
16.1.8 Discussion of the pressure corrections at the interface of two viscous fluids......Page 252
16.1.9 Capillary instability when one fluid is a dynamically inactive gas......Page 254
16.1.10 Conclusions......Page 257
16.2 Stability of a liquid jet into incompressible gases: Temporal, convective,and absolute instabilities......Page 258
16.2.1 Introduction......Page 259
16.2.2 Problem formulation......Page 260
16.2.3 Dispersion relation......Page 261
16.2.4.1 Inviscid fluids......Page 263
16.2.4.2 Viscous fluids......Page 265
16.2.5 Numerical results of temporal instability......Page 270
16.2.6 Spatial, absolute, and convective instability......Page 271
16.2.7 Algebraic equations at a singular point......Page 275
16.2.8 Subcritical, critical, and supercritical singular points......Page 276
16.2.9 Inviscid jet in inviscid fluid…......Page 281
16.2.10 Exact solution comparison with previous results......Page 282
16.2.11 Summary and discussion......Page 286
16.3.1 Mathematical formulation......Page 287
16.3.2 Normal modes dispersion relation......Page 288
16.3.4 Hadamard instabilities for piecewise discontinuous profiles......Page 289
17 Stress-induced cavitation......Page 292
17.1.1 Mathematical formulation......Page 293
17.1.2 Cavitation threshold......Page 295
17.2 Viscous potential flow analysis of stress-induced cavitation in an aperture flow......Page 298
17.2.1 Analysis of stress-induced cavitation......Page 299
17.2.2 Stream function, potential function, and velocity......Page 301
17.2.3 Cavitation threshold......Page 302
17.2.4 Conclusions......Page 306
17.3 Streaming motion past a sphere......Page 307
17.3.1 Irrotational flow of a viscous fluid......Page 310
17.3.2 An analysis for maximum K......Page 313
17.4.1 Introduction......Page 317
17.4.2 Analysis......Page 319
17.4.3 Conclusions and discussion......Page 324
17.4.4 Appendix......Page 328
18 Viscous effects of the irrotational flow outside boundary layers on rigid solids......Page 330
18.1 Extra drag due to viscous dissipation of the irrotational flow outside the boundary layer......Page 331
18.1.1 Pressure corrections for the drag on a circular gas bubble......Page 332
18.1.2.1 Dissipation calculation......Page 335
18.1.2.2 Boundary-layer analysis......Page 336
18.1.2.3 Pressure correction and the additional drag......Page 341
18.1.3 The additional drag on an airfoil by the dissipation method......Page 344
18.1.4 Discussion and conclusion......Page 347
18.2 Glauert\'s solution of the boundary layer on a rapidly rotating cylinder in auniform stream revisited......Page 349
18.2.1 Introduction......Page 350
18.2.3 Boundary-layer approximation and Glauert\'s equations......Page 354
18.2.4 Decomposition of the velocity and pressure field......Page 355
18.2.5 Solution of the boundary-layer flow......Page 356
18.2.6 Higher-order boundary-layer theory......Page 367
18.2.7 Discussion and conclusion......Page 370
18.3 Numerical study of the steady-state uniform flow past a rotating cylinder......Page 372
18.3.1 Introduction......Page 373
18.3.2 Numerical features......Page 375
18.3.3 Results and discussion......Page 379
18.3.3.1 Validation of the numerical approach......Page 380
18.3.3.2 Vortical region thickness from the numerical flow field......Page 382
18.3.3.3 Drag and lift coefficients and pressure distribution from the numerical solution......Page 385
18.3.3.4 Viscous dissipation......Page 391
18.3.4 Concluding remarks......Page 392
19 Irrotational flows that satisfy the compressible Navier-Stokes equations......Page 394
19.1 Acoustics......Page 395
19.2 Spherically symmetric waves......Page 397
19.3.1 Introduction......Page 398
19.3.2 Basic partial differential equations......Page 399
19.3.4 Basic isentropic relations......Page 400
19.3.5 Linear stability of the cylindrical liquid jet in a compressible gas dispersion equation......Page 401
19.3.6 Stability problem in dimensionless form......Page 403
19.3.8 Growth-rate parameters as functions of M for different viscosities......Page 406
19.3.9 Azimuthal periodicity of the most dangerous disturbance......Page 407
19.3.10 Variation of the growth-rate parameters with the Weber number......Page 408
19.3.11 Convective/absolute instability......Page 409
19.3.12 Conclusions......Page 413
20.1 Oldroyd B model......Page 415
20.2.1 Retarded motion expansion for the UCM model......Page 416
20.2.3 Potential flow past a sphere calculated with the expanded UCM model......Page 417
20.3 Second-order fluids......Page 418
20.5 Purely irrotational flows of a second-order fluid......Page 420
20.6 Reversal of the sign of the normal stress at a point of stagnation......Page 421
20.7.2 Particle-particle interactions......Page 422
20.7.3 Sphere-wall interactions......Page 423
20.7.4 Flow-induced microstructure......Page 424
20.8 Potential flow over a sphere for a second-order fluid......Page 426
20.9 Potential flow over an ellipse......Page 428
20.9.1 Normal stress at the surface of the ellipse......Page 429
20.9.2 The effects of the Reynolds number......Page 430
20.9.4 The effects of the aspect ratio......Page 432
20.10 The moment on the ellipse......Page 433
20.11 The reversal of the sign of the normal stress at stagnation points......Page 434
20.13 Flow past a circular cylinder with circulation......Page 436
20.14 Potential flow of a second-order fluid over a triaxial ellipsoid......Page 437
20.15 Motion of a sphere normal to a wall in a second-order fluid......Page 438
20.15.1 Low Reynolds numbers......Page 439
20.15.2 Viscoelastic Potential Flow......Page 442
20.15.3 Conclusions......Page 445
21.1.1 Introduction......Page 446
21.1.2.1 Displacement-time graphs and accelerations......Page 447
21.1.3 Theory......Page 448
21.1.3.1 Stability analysis......Page 449
21.1.3.2 Viscoelastic potential flow analysis of stability......Page 455
21.1.3.3 Comparison of the exact and the potential flow analysis......Page 456
21.1.4 Comparison of theory and experiment......Page 457
21.2.1 Introduction......Page 463
21.2.2 Linear stability equations and the exact solution......Page 464
21.2.3 Viscoelastic potential flow......Page 466
21.2.4 Dissipation and the formulation for the additional pressure contribution......Page 467
21.2.5 The additional pressure contribution for capillary instability......Page 468
21.2.6 Comparison of the growth rate......Page 469
21.2.7 Comparison of the stream functions......Page 471
21.2.8 Discussion......Page 476
21.3 Steady motion of a deforming gas bubble in a viscous potential flow......Page 480
22.1 Perturbation methods......Page 481
22.2 Boundary-integral methods for inviscid potential flow......Page 482
22.3 Boundary-integral methods for viscous potential flow......Page 484
Appendix A: Equations of motion and strain rates for rotational and irrotational flow in Cartesian, cylindrical, and spherical......Page 485
Appendix B: List of frequently used symbols and concepts......Page 491
References......Page 493
Index......Page 507