## The Kinematics of Mixing: Stretching, Chaos, and TransportIn spite of its universality, mixing is poorly understood and generally speaking, mixing problems are attacked on a case-by-case basis. This is the first book to present a unified treatment of the mixing of fluids from a kinematical viewpoint. The author's aim is to provide a conceptually clear basis from which to launch analysis and to facilitate an understanding of the numerous mixing problems encountered in nature and technology. After presenting the necessary background in kinematics and fluid dynamics, Professor Ottino considers various examples of dealing with necessary background in dynamical systems and chaos. The book assumes little previous knowledge of fluid dynamics and dynamical systems and can be used as a textbook by final-year undergraduates, graduate students and researchers in applied mathematics, engineering science, geophysics and physics who have an interest in fluid dynamics, continuum mechanics and dynamical systems. It is profusely illustrated in colour, with many line diagrams and half-tones. Systems which illustrate the most important concepts, many exercises and examples are included. |

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### Contents

Introduction | 1 |

Applications and geometrical structure | 9 |

Flow trajectories and deformation | 18 |

Conservation equations change of frame and vorticity | 42 |

Computation of stretching and efficiency | 64 |

Chaos in dynamical systems | 96 |

Chaos in Hamiltonian systems | 130 |

Mixing and chaos in twodimensional timeperiodic flows | 154 |

Mixing and chaos in threedimensional and open flows | 220 |

diffusion and reaction in lamellar structures | 273 |

Cartesian rectors and tensors | 319 |

Lisf of frequently used symbols | 342 |

355 | |

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### Common terms and phrases

analysis axial behavior blob breakup cavity flow chaos chaotic flows chaotic region Chapter components computations Consider constant corresponding cycle defined deformation denoted diffusion dynamical systems efficiency eigenvalues eigenvectors elliptic points equations example experimental filament fixed points fluid mechanics fluid particle frame Franjione function given Hamiltonian systems heteroclinic homoclinic homoclinic orbit homoclinic point horseshoe maps hyperbolic points initial conditions integrable interface intersections journal bearing Khakhar Liapunov exponents linear material lines mixing motion Navier-Stokes equation Newtonian fluid Note obtain orbit orientation Ottino parameter partitioned-pipe mixer period-1 periodic points perturbation phase space Poincare sections possible Problem reaction Reproduced with permission Reynolds number rotation scalar shear flow shown in Figure shows simple shear Smale solution stable and unstable steady streamfunction streamlines stretching and folding striation thickness structure surface tensor theorem time-periodic tracer trajectories transformation twist mapping two-dimensional flows unstable manifolds vector velocity field viscous vorticity