Numerical analysis of the branching of couette flow between concentric cylinders for various wave numbers

Abstract

The character of stability loss of the circular Couette flow, when the Reynolds number R passes through the critical value R₀, is investigated within a broad range of variation of the wave numbers. The Lyapunov-Schmidt method is used [1, 2]; the boundary-value problems for ordinary differential equations arising in the case of its realization are solved numerically on a computer. It is shown that the branching character substantially depends on the wave number α. For all a, excluding a certain interval (α₁, α₂), the usual postcritical branching takes place: at a small supercriticality the circular flow loses stability and is “softly” excited into a secondary stationary flow — stable Taylor vortices. For wave numbers from the interval (α₁,α₂) a hard excitation of Taylor vortices takes place: at a small subcriticality R=R₀−ε² the secondary mode is unstable and merges with the Couette flow for ε→0; however, for a small supercriticality in the neighborhood of a circular flow there exist no stationary modes which are different.