Hydrodynamic Instabilities

The instability of fluids is a key topic in classical fluid mechanics because it has huge repercussions for applied disciplines such as chemical engineering, hydraulics, aeronautics and meteorology. This modern introduction is written for any student, researcher or practitioner working in the area, for whom an understanding of hydrodynamic instabilities is essential. Based on a decade's experience of teaching postgraduate students in fluid dynamics, this book brings the subject to life by emphasizing the physical mechanisms involved. The theory of dynamics provides the basic structure of the exposition and, wherever possible, Charru discusses the phenomena in terms of characteristic scales and dimensional analysis. Asymptotic methods also play an important role. The book includes numerous experimental studies, with references to videos and multimedia material, as well as over 150 exercises which introduce the reader to new problems.

Foreword	p. x
Preface	p. xiii
Video resources	p. xvi
Introduction	p. 1
Phase space, phase portrait	p. 1
Stability of a fixed point	p. 2
Bifurcations	p. 6
Examples from hydrodynamics	p. 12
Non-normality of the linearized operator	p. 30
Exercises	p. 36
Instabilities of fluids at rest	p. 43
Introduction	p. 43
The Jeans gravitational instability	p. 44
The Rayleigh-Taylor interface instability	p. 53
The Rayleigh-Plateau capillary instability	p. 64
The Rayleigh-Benard thermal instability	p. 68
The Benard-Marangoni thermocapillary instability	p. 75
Discussion	p. 79
Exercises	p. 80
Stability of open flows: basic ideas	p. 88
Introduction	p. 88
A criterion for linear stability	p. 96
Convective and absolute instabilities	p. 98
Exercises	p. 102
Inviscid instability of parallel flows	p. 104
Introduction	p. 104
General results	p. 107
Instability of a mixing layer	p. 116
The Couette-Taylor centrifugal instability	p. 126
Exercises	p. 134
Viscous instability of parallel flows	p. 139
Introduction	p. 139
General results	p. 145
Plane Poiseuille flow	p. 154
Poiseuille flow in a pipe	p. 162
Boundary layer on a flat surface	p. 162
Exercises	p. 169
Instabilities at low Reynolds number	p. 171
Introduction	p. 171
Films falling down an inclined plane	p. 174
Sheared liquid films	p. 193
Exercises	p. 199
Avalanches, ripples, and dunes	p. 201
Introduction	p. 201
Avalanches	p. 202
Sediment transport by a flow	p. 208
Ripples and dunes: a preliminary dimensional analysis	p. 218
Subaqueous ripples under a continuous flow	p. 220
Subaqueous ripples in oscillating flow	p. 230
Subaqueous dunes	p. 238
Exercises	p. 244
Nonlinear dynamics of systems with few degrees of freedom	p. 246
Introduction	p. 246
Nonlinear oscillators	p. 249
Systems with few degrees of freedom	p. 260
Illustration: instability of a sheared interface	p. 264
Exercises	p. 268
Nonlinear dispersive waves	p. 274
Introduction	p. 274
Instability of gravity waves	p. 275
Instability due to resonant interactions	p. 279
Instability to modulations	p. 287
Resonances revisited	p. 294
Exercises	p. 295
Nonlinear dynamics of dissipative systems	p. 299
Introduction	p. 299
Weakly nonlinear dynamics	p. 300
Saturation of the primary instability	p. 305
The Eckhaus secondary instability	p. 305
Instability of a traveling wave	p. 311
Coupling to a field at large scales	p. 317
Exercises	p. 323
Dynamical systems and bifurcations	p. 326
Introduction	p. 326
Phase space and attractors	p. 327
Linear stability	p. 334
Invariant manifolds and normal forms	p. 338
Structural stability and genericity	p. 345
Bifurcations	p. 351
Exercises	p. 365
The Saint-Venant equations	p. 369
Outflow from a slice of fluid	p. 369
Mass conservation	p. 370
Momentum conservation	p. 371
Modeling the wall friction	p. 372
Consistent depth-averaged equations	p. 373
References	p. 375
Index	p. 387
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Hydrodynamic Instabilities

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