The Couette-Taylor Problem / Edition 1

by Pascal Chossat, Gerard Iooss

Overview

1. 1 A paradigm About one hundred years ago, Maurice Couette, a French physicist, designed 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 .

Product Details

ISBN-13:	9781461287308
Publisher:	Springer New York
Publication date:	09/26/2011
Series:	Applied Mathematical Sciences , #102
Edition description:	Softcover reprint of the original 1st ed. 1994
Pages:	234
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

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.