A local study of a double critical point in Taylor-Couette flow

The steady flow, governed by Navier-Stokes equations for an incompressible viscous fluid, between concentric cylinders is considered. The inner cylinder is assumed to be rotating and the outer one at rest. The flow is assumed to be axisymmetric and periodic in the axial direction. At an (m, n) critical point, the eigenfunctions of the operator, linearized around the exact solution, the Couette flow, consist ofm andn axial waves. In a neighbourhood of such a double critical point, using Liapunov-Schmidt method, bifurcation equations are obtained, in ℝ^2. Expressions for the leading coefficients in the truncated system of 2 equations are derived. Using these, the coefficients are computed at a (2, 4) critical point for 2 different radii ratios and the local bifurcation diagrams obtained. Available numerical solutions of the Navier-Stokes system near this double critical point confirm that the reduced bifurcation equations reproduce the qualitative behaviour adequately.