The Couette-Taylor Problem

1. 1 A paradigm About one hundred years ago, Maurice Couette, a French physicist, de signed an apparatus consisting of two coaxial cylinders, the space between the cylinders being filled with a viscous fluid and the outer cylinder being rotated at angular velocity O2. The purpose of this experiment was, follow ing an idea of the Austrian physicist Max Margules, to deduce the viscosity of the fluid from measurements of the torque exerted by the fluid on the inner cylinder (the fluid is assumed to adhere to the walls of the cylinders). At least when O is not too large, the fluid flow is nearly laminar and 2 the method of Couette is valuable because the torque is then proportional to 110 , where II is the kinematic viscosity of the fluid. If, however, O is 2 2 increased to a very large value, the flow becomes eventually turbulent. A few years later, Arnulph Mallock designed a similar apparatus but allowed the inner cylinder to rotate with angular velocity 01, while O2 = o. The surprise was that the laminar flow, now known as the Couette flow, was not observable when 0 exceeded a certain "low" critical value Ole, even 1 though, as we shall see in Chapter II, it is a solution of the model equations for any values of 0 and O .

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This book presents a systematic and unified approach to the nonlinear stability problem and transitions in the Couette-Taylor problem, by the means of analytic and constructive methods. The most "elementary" one-parameter theory is first presented with great detail. More complex situations are then analyzed (mode interactions, imperfections, non-spatially periodic patterns). The whole analysis is based on the mathematically rigorous theory of center manifold and normal forms, and symmetries are fully taken into account. These methods are very general and can be applied to other hydrodynamical instabilities, or more generally to physical problems modelled by partial differential equations. Non-mathematician readers can skip the mathematically "hard" parts of the book and still catch the ideas and results. This book is primarily intended for graduate students and researchers in fluid mechanics, and more generally for applied mathematicians and physicists who are interested in the analysis of instabilities in systems governed by partial differential equations.

Contenu
I Introduction.- I.1 A paradigm.- I.2 Experimental results.- I.3 Modeling for theoretical analysis.- I.4 Arrangements of topics in the text.- II Statement of the Problem and Basic Tools.- II.1 Nondimensionalization, parameters.- II.2 Functional frame and basic properties.- II.3 Linear stability analysis.- II.4 Center Manifold Theorem.- III Taylor Vortices, Spirals and Ribbons.- III.1 Taylor vortex flow.- III.2 Spirals and ribbons.- III.3 Higher codimension bifurcations.- IV Mode Interactions.- IV.1 Interaction between an axisymmetric and a nonaxisymmetric mode.- IV.2 Interaction between two nonaxisymmetric modes.- V Imperfections on Primary Bifurcations.- V.1 General setting when the geometry of boundaries is perturbed.- V.2 Eccentric cylinders.- V.3 Little additional flux.- V.4 Periodic modulation of the shape of cylinders in the axial direction.- V.5 Time-periodic perturbation.- VI Bifurcation from Group Orbits of Solutions.- VI.1 Center manifold for group orbits.- VI.2 Bifurcation from the Taylor vortex flow.- VI.3 Bifurcation from the spirals.- VI.4 Bifurcation from ribbons.- VI.5 Bifurcation from wavy vortices, modulated wavy vortices.- VI.6 Codimension-two bifurcations from Taylor vortex flow.- VII Large-scale EfTects.- VII. 1 Steady solutions in an infinite cylinder.- VII.2 Time-periodic solutions in an infinite cylinder.- VII.3 Ginzburg-Landau equation.- VIII Small Gap Approximation.- VIII.1 Introduction.- VIII.2 Choice of scales and limiting system.- VIII.3 Linear stability analysis.- VIII.4 Ginzburg-Landau equations.