## Introduction to Hydrodynamic StabilityInstability of flows and their transition to turbulence are widespread phenomena in engineering and the natural environment, and are important in applied mathematics, astrophysics, biology, geophysics, meteorology, oceanography and physics as well as engineering. This is a textbook to introduce these phenomena at a level suitable for a graduate course, by modelling them mathematically, and describing numerical simulations and laboratory experiments. The visualization of instabilities is emphasized, with many figures, and in references to more still and moving pictures. The relation of chaos to transition is discussed at length. Many worked examples and exercises for students illustrate the ideas of the text. Readers are assumed to be fluent in linear algebra, advanced calculus, elementary theory of ordinary differntial equations, complex variable and the elements of fluid mechanics. The book is aimed at graduate students but will also be very useful for specialists in other fields. |

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### Contents

General Introduction | 1 |

12 The Methods of Hydrodynamic Stability | 6 |

13 Further Reading and Looking | 8 |

Introduction to the Theory of Steady Flows Their Bifurcations and Instability | 10 |

22 Instability | 19 |

23 Stability and the Linearized Problem | 28 |

KelvinHelmholtz Instability | 45 |

33 Governing Equations for Perturbations | 47 |

72 Instability of Couette Flow | 125 |

73 Görtler Instability | 130 |

Stability of Parallel Flows | 138 |

82 General Properties of Rayleighs Stability Problem | 144 |

83 Stability Characteristics of Some Flows of an Inviscid Fluid | 149 |

84 Nonlinear Perturbations of a Parallel Flow of an Inviscid Fluid | 154 |

Viscous Fluid | 156 |

86 Some General Properties of the OrrSommerfeld Problem | 160 |

34 The Linearized Problem | 48 |

35 Surface Gravity Waves | 50 |

37 RayleighTaylor Instability | 51 |

38 Instability Due to Shear | 52 |

Capillary Instability of a Jet | 62 |

Development of Instabilities in Time and Space | 68 |

52 Weakly Nonlinear Theory | 74 |

53 The Equation of the Perturbation Energy | 82 |

RayleighBénard Convection | 93 |

62 The Linearized Problem | 95 |

63 The Stability Characteristics | 97 |

64 Nonlinear Convection | 100 |

Centrifugal Instability | 123 |

87 Stability Characteristics of Some Flows of a Viscous Fluid | 167 |

88 Numerical Methods of Solving the OrrSommerfeld Problem | 171 |

89 Experimental Results and Nonlinear Instability | 172 |

810 Stability of Axisymmetric Parallel Flows | 178 |

Routes to Chaos and Turbulence | 208 |

92 Routes to Chaos and Turbulence | 211 |

Case Studies in Transition to Turbulence | 215 |

102 Transition of Flow of a Uniform Stream Past a Bluff Body | 219 |

103 Transition of Flows in a Diverging Channel | 225 |

237 | |

249 | |

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### Common terms and phrases

amplitude approximation asymptotically axisymmetric basic flow basic velocity boundary conditions channel complex constant Couette flow critical layer critical value cylinder Deduce define density domain of flow Drazin & Reid dynamics eigenfunctions eigenvalue problem energy example exponentially Figure flow is stable function given gives Hopf bifurcation hydrodynamic hydrodynamic stability incompressible inviscid fluid instability of flows inviscid fluid Jeffery-Hamel flows Kelvin-Helmholtz instability Landau equation linear stability linear theory linearized problem marginal curve marginally stable mathematical method Navier-Stokes equations normal modes null solution ordinary-differential Orr-Sommerfeld equation Orr-Sommerfeld problem physical pitchfork bifurcation plane Poiseuille flow plate Poiseuille flow Rayleigh number Rayleigh stability Rayleigh-Benard convection Rayleigh-Taylor instability Reynolds stress shear layer small perturbations solve spatial stability characteristics steady flow steady solutions streamfunction symmetry Taking normal modes three-dimensional Tollmien-Schlichting waves transition to turbulence unstable mode velocity profile vortex sheet vorticity equation wavenumber waves zero