《Potential flows of viscous and viscoelastic fluids》
《Potential flows of viscous and viscoelastic fluids》粘性流体和粘弹性流体的势流
作者：
Daniel Joseph
University of Minnesota
Toshio Funada
Numazu College of Technology
JingWang
University of Minnesota
出版社：Cambridge
出版时间：2008年
目录
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
4.6.5 Hadamard–Rybyshinsky solution for streaming flow past a liquid
sphere 23
vii
viii Contents
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
Contents ix
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
x Contents
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 k < kc and
with VPF when k > kc ? 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
Contents xi
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 →∞, m = 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
xii Contents
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 K 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
Contents xiii
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 B 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 −1/( a 2 ) 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
xiv Contents
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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