Pattern Formation in Viscous Flows: The Taylor-Couette Problem and Rayleigh-Bénard Convection

ISBN-13: 9783764360474 / Angielski / Miękka / 1999 / 212 str.

Rita Meyer-Spasche

Opis

It seems doubtful whether we can expect to understand fully the instability of fluid flow without obtaining a mathematical representa tion of the motion of a fluid in some particular case in which instability can actually be ob served, so that a detailed comparison can be made between the results of analysis and those of experiment. - G.l. Taylor (1923) Though the equations of fluid dynamics are quite complicated, there are configurations which allow simple flow patterns as stationary solutions (e.g. flows between parallel plates or between rotating cylinders). These flow patterns can be obtained only in certain parameter regimes. For parameter values not in these regimes they cannot be obtained, mainly for two different reasons: The mathematical existence of the solutions is parameter dependent; or the solutions exist mathematically, but they are not stable. For finding stable steady states, two steps are required: the steady states have to be found and their stability has to be determined."

Spis treści

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.