Pattern Formation in Viscous Flows

1 The Taylor Experiment.- 1.1 Modeling of the Experiment.- 1.1.1 Introduction.- 1.1.2 Mathematical Description of the Experiment.- 1.1.3 Narrow Gap Limit and Rayleigh-Bénard Problem.- 1.1.4 End Effects.- 1.2 Flows between Rotating Cylinders.- 1.3 Stability of Couette Flow.- 1.3.1 Equations of Motion for Axisymmetric Perturbations.- 1.3.2 Computation of Marginal Curves.- 1.3.3 Validity of the Principle of Exchange of Stability.- 2 Details of a Numerical Method.- 2.1 Introduction.- 2.1.1 Numerical Model.- 2.1.2 Numerical Methods.- 2.1.3 Validity of the Model.- 2.1.4 Stability.- 2.2 The Discretized System.- 2.2.1 Discretization in the Axial z-Direction.- 2.2.2 Discretization in the Radial r-Direction.- 2.2.3 Boundary Conditions.- 2.2.4 Final Version of the Equations.- 2.3 Computation of Solutions.- 2.3.1 Pseudo-Arclength Continuation and Newton iterations.- 2.3.2 Continuation in the Reynolds number Re.- 2.3.3 Continuation in the Wave Number k.- 2.3.4 Simple Continuation.- 2.3.5 Switching Branches.- 2.4 Computation of flow Parameters.- 2.4.1 Periodicity.- 2.4.2 Computation of um: = u(rm, zm; ?) at rm: = 1 + ?/2, zm = 0.- 2.4.3 Computation of the Torque.- 2.4.4 Computation of Kinetic Energies.- 2.4.5 Computation of the Stream function.- 2.5 Numerical Accuracy.- 2.5.1 Finite Differences.- 2.5.2 Truncation of the Fourier Series.- 2.5.3 Conclusions.- 3 Stationary Taylor Vortex Flows.- 3.1 Introduction.- 3.2 Computations with Fixed Period ? ? 2.- 3.2.1 A Narrow Gap Problem, ? = 0.95.- 3.2.2 A Wide Gap Problem, ? = 0.5.- 3.3 Variation of Flows with Period ?.- 3.3.1 Previous Results on Flows with Wavelengths ? ? 2.- 3.3.2 Continuous Change of Period.- 3.3.3 Flows near Re = 1.5 Recr.- 3.3.4 Flows for Re = 800 ? 3.65 Recr.- 3.4 Interactions of Secondary Branches.- 3.4.1 A Neighborhood of (Re24, ?24) and the Basic (2,4) Fold.- 3.4.2 Connections to the Rayleigh-Bénard Problem.- 3.4.3 The basic (n, 2n)-Fold for Higher Reynolds Numbers.- 3.4.4 The Basic 2-vortex Surface.- 3.5 Re = 2 Recr and the (n, pn) Double Points.- 3.6 Stability of the Stationary Vortices.- 3.6.1 Wavy Vortices.- 3.6.2 Eckhaus and Short-Wavelength Instabilities.- 4 Secondary Bifurcations on Convection Rolls.- 4.1 Introduction.- 4.2 The Rayleigh-Bénard Problem.- 4.2.1 Convection in Fluids.- 4.2.2 Boussinesq Approximation.- 4.2.3 The Rayleigh-Bénard Problem as Limiting Case of the Taylor Problem.- 4.3 Stationary Convection Rolls.- 4.3.1 The Basic Equations.- 4.3.2 Critical Curves of the Primary Solution.- 4.3.3 Pure-Mode Solutions.- 4.4 The (2,4) Interaction in a Model Problem.- 4.4.1 The Model Problem.- 4.4.2 Calculation of Secondary Bifurcation Points on the 2-Roll Solutions.- 4.4.3 Calculation of Secondary Bifurcation Points on the 4-Roll Solutions.- 4.4.4 The Perturbation Approach.- 4.5 The (2,6) Interaction in a Model Problem.- 4.5.1 Calculation of Secondary Bifurcation Points on the 2-Roll Solutions.- 4.5.2 Calculation of Secondary Bifurcation Points on the 6-Roll Solutions.- 4.5.3 Nonlinear Interactions between the Bifurcating Branches.- 4.6 Generalisations and Consequences.- 4.6.1 Other Interactions.- 4.6.2 Linear Superpositions of Pure-Mode Solutions.- 4.6.3 Secondary Bifurcations in the Taylor Problem Revisited.