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 .

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.1.1 Basic formulation.- II.1.2 Nondimensionalization.- II.1.3 Couette flow and the perturbation.- II.1.4 Symmetries.- II.1.5 Small gap case.- II.1.5.1 Case when the average rotation rate is very large versus the difference ?1 — ?2.- II.1.5.2 Case when the rotation rate of the inner cylinder is very large.- II.2 Functional frame and basic properties.- II.2.1 Projection on divergence-free vector fields.- II.2.2 Alternative choice for the functional frame.- II.2.3 Main results for the nonlinear evolution problem.- II.3 Linear stability analysis.- II.4 Center Manifold Theorem.- III Taylor Vortices, Spirals and Ribbons.- III.1 Taylor vortex flow.- III.1.1 Steady-state bifurcation with O(2)-symmetry.- III.1.2 Identification of the coefficients in the amplitude equation.- III.1.3 Geometrical pattern of the Taylor cells.- III.2 Spirals and ribbons.- III.2.1 The Hopf bifurcation with O(2)-symmetry.- III.2.2 Application to the Couette-Taylor problem.- III.2.3 Geometrical structure of the flows.- III.2.3.1 Spirals.- III.2.3.2 Ribbons.- III.3 Higher codimension bifurcations.- III.3.1 Weakly subcritical Taylor vortices.- III.3.2 Competition between spirals and ribbons.- IV Mode Interactions.- IV.1 Interaction between an axisymmetric and a nonaxisymmetric mode.- IV.1.1 The amplitude equations (6 dimensions).- IV.1.2 Restriction of the equations to flow-invariant subspaces.- IV.1.3 Bifurcated solutions.- IV.1.3.1 Primary branches.- IV.1.3.2 Wavy vortices.- IV.1.3.3 Twisted vortices.- IV.1.4 Stability of the bifurcated solutions.- IV.1.4.1 Taylor vortices.- IV.1.4.2 Spirals.- IV.1.4.3 Ribbons.- IV.1.4.4 Wavy vortices.- IV.1.4.5 Twisted vortices.- IV.1.5 A numerical example.- IV.1.6 Bifurcation with higher codimension.- IV.2 Interaction between two nonaxisymmetric modes.- IV.2.1 The amplitude equations (8 dimensions).- IV.2.2 Restriction of the equations to flow-invariant subspaces.- IV.2.3 Bifurcated solutions.- IV.2.3.1 Primary branches.- IV.2.3.2 Interpenetrating spirals (first kind).- IV.2.3.3 Interpenetrating spirals (second kind).- IV.2.3.4 Superposed ribbons (first kind).- IV.2.3.5 Superposed ribbons (second kind).- IV.2.4 Stability of the bifurcated solutions.- IV.2.4.1 Stability of the m-spirals.- IV.2.4.2 Stability of the (m + 1)-spirals.- IV.2.4.3 Stability of the m-ribbons.- IV.2.4.4 Stability of the (m + 1)-ribbons.- IV.2.4.5 Stability of the interpenetrating spirals SI(0,3) and SI(1,2).- IV.2.4.6 Stability of the interpenetrating spirals SI(1,3) and SI(0,2).- IV.2.4.7 Stability of the superposed ribbons RS(0).- IV.2.4.8 Stability of the superposed ribbons RS(?).- IV.2.5 Further bifurcations.- IV.2.6 Two numerical examples.- V Imperfections on Primary Bifurcations.- V.1 General setting when the geometry of boundaries is perturbed.- V.1.1 Reduction to an equation in H(Qh).- V.1.2 Amplitude equations.- V.2 Eccentric cylinders.- V.2.1 Effect on Taylor vortices.- V.2.2 Computation of the coefficient b.- V.2.3 Effect on spirals and ribbons.- V.3 Little additional flux.- V.3.1 Perturbed Taylor vortices lead to traveling waves.- V.3.2 Identification of coefficients d and e.- V.3.3 Effects on spirals and ribbons.- V.4 Periodic modulation of the shape of cylinders in the axial direction.- V.4.1 Effect on Taylor vortices.- V.4.2 Effects on spirals and ribbons.- V.5 Time-periodic perturbation.- V.5.1 Perturbed Taylor vortices.- V.5.2 Perturbation of spirals and ribbons.- VI Bifurcation from Group Orbits of Solutions.- VI.1 Center manifold for group orbits.- VI.1.1 Group orbits of first bifurcating solutions.- VI.1.1.1 Taylor vortex flow.- VI.1.1.2 Spirals.- VI.1.1.3 Ribbons.- VI.1.2 The center manifold reduction for a group-orbit of steady solutions.- VI.2 Bifurcation from the Taylor vortex flow.- VI.2.1 The stationary case.- VI.2.1.1 No symmetry breaking.- VI.2.1.2 Breaking reflectional symmetry creates a traveling wave.- VI.2.1.3 Doubling the axial wave length.- VI.2.1.4 Doubling the axial wave length and breaking reflectional symmetry.- VI.2.2 Hopf bifurcation from Taylor vortices.- VI.3 Bifurcation from the spirals.- VI.4 Bifurcation from ribbons.- VI.4.1 The stationary case.- VI.4.1.1 No symmetry breaking.- VI.4.1.2 Breaking the twist symmetry.- VI.4.1.3 Breaking the reflectional symmetry (stationary bifurcation creating a traveling wave).- VI.4.1.4 Breaking the reflectional and twist symmetries.- VI.4.2 Hopf bifurcation from ribbons.- VI.4.2.1 Breaking the twist symmetry.- VI.4.2.2 Breaking the twist and reflectional symmetries.- VI.5 Bifurcation from wavy vortices, modulated wavy vortices.- VI.5.1 Hopf bifurcation of wavy vortices into modulated wavy vortices.- VI.5.2 Steady bifurcation of the wavy vortices into a quasi-periodic flow with a slow drift.- VI.6 Codimension-two bifurcations from Taylor vortex flow.- VII Large-scale EfTects.- VII. 1 Steady solutions in an infinite cylinder.- VII.1.1 A center manifold for steady Navier-Stokes equations.- VII.1.2 Resolution of the four-dimensional amplitude equations.- VII.1.2.1 The normal form.- VII.1.2.2 Integrability of the reduced system..- VII.1.2.3 Periodic solutions of the amplitude equations.- VII.1.2.4 Other solutions of the amplitude equations.- VII.1.2.5 Quasi-periodic solutions.- VII.1.2.6 Eckhaus points E and E?.- VII.1.2.7 Homoclinic solutions.- VII.2 Time-periodic solutions in an infinite cylinder.- VII.2.1 Center manifold for time-periodic Navier-Stokes equations.- VII.2.2 Spectrum of $$\kappa \mu ,\omega $$ near criticality.- VII.2.3 Resolution of the four-dimensional amplitude equations. New solutions.- VII.3 Ginzburg-Landau equation.- VIII Small Gap Approximation.- VIII.1 Introduction.- VIII.2 Choice of scales and limiting system.- VIII.2.1 Choice of scales.- VIII.2.2 Limiting system.- VIII.3 Linear stability analysis.- VIII.4 Ginzburg-Landau equations.- VIII.4.1 Case (i).- VIII.4.2 Case (ii).- VIII.4.3 Case (iii).