Foreword ........................................................ x
Preface ...................................................... xiii
Video resources ............................................... xvi
1 Introduction ................................................. 1
1.1 Phase space, phase portrait ............................. 1
1.2 Stability of a fixed point .............................. 2
1.3 Bifurcations ............................................ 6
1.4 Examples from hydrodynamics ............................ 12
1.5 Non-normality of the linearized operator ............... 30
1.6 Exercises .............................................. 36
2 Instabilities of fluids at rest ............................. 43
2.1 Introduction ........................................... 43
2.2 The Jeans gravitational instability .................... 44
2.3 The Rayleigh-Taylor interface instability .............. 53
2.4 The Rayleigh-Plateau capillary instability ............. 64
2.5 The Rayleigh-Bénard thermal instability ................ 68
2.6 The Bénard-Marangoni thermocapillary instability ....... 76
2.7 Discussion ............................................. 79
2.8 Exercises .............................................. 80
3 Stability of open flows: basic ideas ........................ 88
3.1 Introduction ........................................... 88
3.2 A criterion for linear stability ....................... 96
3.3 Convective and absolute instabilities .................. 98
3.4 Exercises ............................................. 102
4 Inviscid instability of parallel flows ..................... 104
4.1 Introduction .......................................... 104
4.2 General results ....................................... 107
4.3 Instability of a mixing layer ......................... 116
4.4 The Couette-Taylor centrifugal instability ............ 126
4.5 Exercises ............................................. 134
5 Viscous instability of parallel flows ...................... 139
5.1 Introduction .......................................... 139
5.2 General results ....................................... 145
5.3 Plane Poiseuille flow ................................. 154
5.4 Poiseuille flow in a pipe ............................. 162
5.5 Boundary layer on a flat surface ...................... 162
5.6 Exercises ............................................. 169
6 Instabilities at low Reynolds number ....................... 171
6.1 Introduction .......................................... 171
6.2 Films falling down an inclined plane .................. 174
6.3 Sheared liquid films .................................. 193
6.4 Exercises ............................................. 199
7 Avalanches, ripples, and dunes ............................. 201
7.1 Introduction .......................................... 201
7.2 Avalanches ............................................ 202
7.3 Sediment transport by a flow .......................... 208
7.4 Ripples and dunes: a preliminary dimensional
analysis .............................................. 218
7.5 Subaqueous ripples under a continuous flow ............ 220
7.6 Subaqueous ripples in oscillating flow ................ 230
7.7 Subaqueous dunes ...................................... 238
7.8 Exercises ............................................. 244
8 Nonlinear dynamics of systems with few degrees of freedom .. 246
8.1 Introduction .......................................... 246
8.2 Nonlinear oscillators ................................. 249
8.3 Systems with few degrees of freedom ................... 260
8.4 Illustration: instability of a sheared interface ...... 264
8.5 Exercises ............................................. 268
9 Nonlinear dispersive waves ................................. 274
9.1 Introduction .......................................... 274
9.2 Instability of gravity waves .......................... 275
9.3 Instability due to resonant interactions .............. 279
9.4 Instability to modulations ............................ 287
9.5 Resonances revisited .................................. 294
9.6 Exercises ............................................. 295
10 Nonlinear dynamics of dissipative systems .................. 299
10.1 Introduction .......................................... 299
10.2 Weakly nonlinear dynamics ............................. 300
10.3 Saturation of the primary instability ................. 305
10.4 The Eckhaus secondary instability ..................... 305
10.5 Instability of a traveling wave ....................... 311
10.6 Coupling to a field at large scales ................... 317
10.7 Exercises ............................................. 323
11 Dynamical systems and bifurcations ......................... 326
11.1 Introduction .......................................... 326
11.2 Phase space and attractors ............................ 327
11.3 Linear stability ...................................... 334
11.4 Invariant manifolds and normal forms .................. 338
11.5 Structural stability and genericity ................... 345
11.6 Bifurcations .......................................... 351
11.7 Exercises ............................................. 365
Appendix A: The Saint-Venant equations ........................ 369
A.l Outflow from a slice of fluid ......................... 369
A.2 Mass conservation ..................................... 370
A.3 Momentum conservation ................................. 371
A.4 Modeling the wall friction ............................ 372
A.5 Consistent depth-averaged equations ................... 373
References .................................................... 375
Index ......................................................... 387
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