Hydrodynamic Instabilities

Cover......Page 1
Title......Page 5
Copyright......Page 6
Contents......Page 9
Foreword......Page 12
Preface......Page 15
Video resources......Page 18
1.1 Phase space, phase portrait......Page 21
1.2.1 Fixed points......Page 22
1.2.2 Linear stability of a fixed point......Page 23
1.3.1 Definition......Page 26
1.3.2 Saddle--node bifurcation......Page 27
1.3.3 Pitchfork bifurcation......Page 29
1.4.1 Stability of a soap film......Page 32
1.4.2 Stability of a bubble......Page 36
1.4.3 Stability of a colloidal suspension......Page 40
1.4.4 Convection in a ring......Page 43
1.4.5 Double diffusion of heat and matter......Page 47
1.5.1 Algebraic transient growth......Page 50
1.5.2 Optimal excitation of an unstable mode......Page 53
1.6.1 The forced harmonic oscillator......Page 56
1.6.2 Particle in a double-well potential......Page 57
1.6.3 Avalanches in a sand pile......Page 58
1.6.5 A first-order phase transition......Page 59
1.6.6 A model of soap-film instability......Page 60
1.6.8 Optimal excitation of an unstable mode......Page 61
1.6.9 Subcritical bifurcation via a transient growth......Page 62
2.1 Introduction......Page 63
2.2.1 Acoustic waves......Page 64
2.2.2 The effect of gravity at large scales......Page 67
2.2.3 Discussion......Page 71
2.3.1 Dimensional analysis......Page 73
2.3.2 Perturbation equations......Page 76
2.3.3 Linearization, normal modes, and the dispersion relation......Page 79
2.3.4 Discussion......Page 80
2.3.5 The effects of horizontal walls and viscosity......Page 81
2.4.1 Description......Page 84
2.4.2 Dimensional analysis......Page 86
2.5.1 Description......Page 88
2.5.2 The instability mechanism (Pr 1)......Page 91
2.5.3 Study of stability within the Boussinesq approximation......Page 93
2.6.1 Description......Page 96
2.6.2 Dimensional analysis......Page 97
2.7.2 General characteristics of a threshold instability......Page 99
2.8.1 Rayleigh--Taylor instability between walls......Page 100
2.8.2 Instability of a suspended thin film......Page 101
2.8.3 Rayleigh--Plateau instability on a wire......Page 102
2.8.4 Stability of a planar front between two fluids in a porous medium......Page 103
2.8.5 The Darrieus--Landau instability of a flame front......Page 105
3.1.1 Linear dynamics of a wave packet......Page 108
3.1.2 Stability in the Lyapunov sense, asymptotic stability......Page 112
3.1.3 Linear stability and instability......Page 113
3.2.1 Spatio-temporal evolution of a general perturbation......Page 116
3.3.1 The criterion for absolute instability......Page 118
3.3.3 Illustrations......Page 120
3.3.4 The Gaster relation......Page 121
3.4.1 Dispersion of a wave packet......Page 122
3.4.2 Spatial branches of a convective instability......Page 123
4.1 Introduction......Page 124
4.2.1 Linearized equations for small perturbations......Page 127
4.2.2 The Squire theorem......Page 129
4.2.3 The Rayleigh equation of two-dimensional perturbations......Page 130
4.2.4 The Rayleigh inflection point theorem......Page 133
4.2.5 Jump conditions between two layers of uniform vorticity......Page 135
4.3 Instability of a mixing layer......Page 136
4.3.1 Kelvin--Helmholtz instability of a vortex sheet......Page 137
4.3.2 The case of nonzero vorticity thickness......Page 142
4.3.3 Viscous effects......Page 145
4.4.1 Introduction......Page 146
4.4.2 The steady flow and its instability......Page 147
4.4.3 The instability criterion for inviscid flow......Page 150
4.4.4 The effect of viscosity: the Taylor number......Page 151
4.5.1 The Kelvin--Helmholtz instability with gravity and capillarity......Page 154
4.5.3 Internal waves in a density-stratified shear flow......Page 155
4.5.4 Instability of inviscid Couette--Taylor flow......Page 156
4.5.5 Instability of a viscous film......Page 157
5.1 Introduction......Page 159
5.1.1 Instability of Poiseuille flow in a tube......Page 160
5.1.2 Instability of a boundary layer......Page 163
5.2.1 The linearized perturbation equations......Page 165
5.2.2 The Squire theorem......Page 166
5.2.3 The Orr--Sommerfeld equation......Page 168
5.2.4 The viscous instability mechanism......Page 171
5.3.1 Marginal stability, eigenmodes......Page 174
5.3.2 Experimental study for small perturbations......Page 176
5.3.3 Transient growth......Page 178
5.5.1 Experimental demonstration......Page 182
5.5.2 Local analysis......Page 183
5.5.3 Eigenmodes, marginal stability, and nonparallel effects......Page 184
5.5.4 Transient growth......Page 188
5.6.3 Solution of the Orr--Sommerfeld equation for Couette flow......Page 189
5.6.4 Instability due to linear resonance......Page 190
6.1 Introduction......Page 191
6.2 Films falling down an inclined plane......Page 194
6.2.1 Base flow and characteristic scales......Page 195
6.2.2 Formulation of the stability problem......Page 196
6.2.3 A long-wave interfacial instability......Page 198
6.2.4 The instability mechanism......Page 203
6.2.5 Experimental study......Page 206
6.3.1 Introduction......Page 213
6.3.2 The long-wave instability mechanism......Page 214
6.3.3 Waves of shorter wavelength......Page 217
6.3.4 Instability of a falling film revisited......Page 218
6.4.4 Stability using the depth-averaged equations......Page 219
7.1 Introduction......Page 221
7.2 Avalanches......Page 222
7.2.1 Particle flow on a rough inclined plane......Page 223
7.2.2 Linear stability......Page 225
7.2.3 Experiments......Page 226
7.3.1 Dimensional analysis......Page 228
7.3.2 The speed of mobile particles......Page 230
7.3.3 The number density of mobile particles......Page 232
7.3.4 The particle flux......Page 233
7.3.5 Particle relaxation effects......Page 234
7.4.1 Aeolian ripples and dunes......Page 238
7.4.2 Subaqueous ripples and dunes......Page 239
7.5 Subaqueous ripples under a continuous flow......Page 240
7.5.1 Phase advance of the shear stress......Page 241
7.5.2 Instability......Page 243
7.5.3 Discussion......Page 247
7.6.1 Introduction......Page 250
7.6.2 Observations......Page 251
7.6.3 Steady streaming over a wavy bottom......Page 254
7.6.4 Instability......Page 257
7.7.1 Introduction......Page 258
7.7.2 A simple model......Page 259
7.7.3 Stability on a flat rigid bed......Page 260
7.7.4 Stability on an erodible bed......Page 261
7.8.3 Dunes: constant friction coefficient......Page 264
7.8.4 Dunes: nonconstant friction coefficient......Page 265
8.1 Introduction......Page 266
8.2.1 A strongly dissipative oscillator in a double-well potential......Page 269
8.2.2 The van der Pol oscillator: amplitude saturation......Page 271
8.2.3 The Duffing oscillator: the frequency correction......Page 275
8.2.4 Forced oscillators......Page 278
8.3 Systems with few degrees of freedom......Page 280
8.3.2 The amplitude equations......Page 281
8.3.3 Reduction of the dynamics near threshold......Page 282
8.4 Illustration: instability of a sheared interface......Page 284
8.5.2 The van der Pol oscillator: restabilization......Page 288
8.5.4 The van der Pol oscillator subject to a constant forcing......Page 289
8.5.5 The parametrically forced oscillator......Page 290
8.5.6 Weakly nonlinear dynamics of the KS--KdV equation......Page 292
9.1 Introduction......Page 294
9.2.1 Stokes waves......Page 295
9.2.2 The Benjamin--Feir instability......Page 298
9.3.1 The model problem......Page 299
9.3.2 A nonlinear Klein--Gordon wave......Page 301
9.3.3 Instability of a monochromatic nonlinear wave......Page 305
9.4 Instability to modulations......Page 307
9.4.1 Linear dynamics of a wave packet: envelope equation......Page 308
9.4.2 Nonlinear dynamics: the nonlinear Schrödinger equation......Page 309
9.4.3 Stability of a quasi-monochromatic wave......Page 310
9.4.5 Derivation of the NLS equation for the Klein--Gordon wave......Page 312
9.5 Resonances revisited......Page 314
9.6.1 A nonlinear wave including two harmonics (1)......Page 315
9.6.2 A nonlinear wave including two harmonics (2)......Page 316
9.6.3 A nonlinear Korteweg--de Vries wave......Page 318
10.1 Introduction......Page 319
10.2.1 Linear evolution of a wave packet......Page 320
10.2.2 Weakly nonlinear effects: the Ginzburg--Landau equation......Page 322
10.2.3 Example of the derivation of the Ginzburg--Landau equation......Page 323
10.4.1 The instability criterion......Page 325
10.4.2 Interpretation in terms of phase dynamics......Page 327
10.4.3 Some experimental illustrations......Page 329
10.5.1 Evolution of a wave packet......Page 331
10.5.3 The Benjamin--Feir--Eckhaus instability......Page 333
10.5.4 Tollmien--Schlichting waves and the transition to turbulence......Page 335
10.6.1 Galilean invariance and conservation laws......Page 337
10.6.2 Coupled evolution equations......Page 339
10.6.3 Wave stability......Page 340
10.6.4 An experimental illustration......Page 341
10.7.1 Derivation of the GL equation for the Swift--Hohenberg model......Page 343
10.7.2 Translational invariance and Galilean invariance......Page 345
11.1 Introduction......Page 346
11.2.1 Flow generated by a vector field and orbits in phase space......Page 347
11.2.2 Dissipative and conservative systems, and attractors......Page 349
11.2.3 Poincaré sections......Page 351
11.3.1 Solution of the linearized system......Page 354
11.3.3 Types of fixed point......Page 355
11.3.4 ``Resemblance'' of nonlinear and linearized fields......Page 356
11.4.1 Stable and unstable manifolds of a hyperbolic fixed point......Page 358
11.4.2 The center manifold......Page 360
11.4.3 The normal form of a vector field......Page 362
11.5.1 The basic problem......Page 365
11.5.2 Definitions......Page 367
11.5.3 Structural stability conditions......Page 368
11.6.2 Definition of a bifurcation......Page 371
11.6.3 Codimension of a bifurcation......Page 372
11.6.4 The saddle--node bifurcation......Page 374
11.6.5 The Hopf bifurcation......Page 379
11.6.6 An example of a bifurcation of codimension two......Page 380
11.7.3 Integration of linear differential systems......Page 385
11.7.7 Resonances of eigenvalues......Page 386
11.7.12 Bifurcation diagram (2)......Page 387
11.7.14 Hopf bifurcation of the Lorenz system......Page 388
A.1 Outflow from a slice of fluid......Page 389
A.2 Mass conservation......Page 390
A.3 Momentum conservation......Page 391
A.4 Modeling the wall friction......Page 392
A.5 Consistent depth-averaged equations......Page 393
References......Page 395
index......Page 407