Inhaltsangabe
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 stati
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