Pattern Formation in Viscous Flows

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 bedetermined.

1 The Taylor Experiment. - 1. 1 Modeling of the Experiment. - 1. 2 Flows between Rotating Cylinders. - 1. 3 Stability of Couette Flow. - 2 Details of a Numerical Method. - 2. 1 Introduction. - 2. 2 The Discretized System. - 2. 3 Computation of Solutions. - 2. 4 Computation of flow Parameters. - 2. 5 Numerical Accuracy. - 3 Stationary Taylor Vortex Flows. - 3. 1 Introduction. - 3. 2 Computations with Fixed Period ? ? 2. - 3. 3 Variation of Flows with Period ? . - 3. 4 Interactions of Secondary Branches. - 3. 5 Re = 2 Recr and the (n, pn) Double Points. - 3. 6 Stability of the Stationary Vortices. - 4 Secondary Bifurcations on Convection Rolls. - 4. 1 Introduction. - 4. 2 The Rayleigh-Bénard Problem. - 4. 3 Stationary Convection Rolls. - 4. 4 The (2, 4) Interaction in a Model Problem. - 4. 5 The (2, 6) Interaction in a Model Problem. - 4. 6 Generalisations and Consequences.