A Spatial Center Manifold Approach to a Hydrodynamical Problem with O(2) Symmetry

Abstract

We consider bifurcations from the 3D Poiseuille flow between parallel plates. In the “classical” statement the problem possesses SO(2) × O(2) symmetry group and from the beginning seems to be completely analogous the Couette-Taylor problem, but in fact it is quite different as in the most physically interesting range of parameters the most dangerous are pure 2D disturbances. In contrast to the classical studies, we make no assumptions on the behavior in the spanwise direction, except the uniform closeness of the bifurcating solution to the basic flow. However, we impose time periodicity as well as spatial periodicity with period 2π/α in streamwise direction. This allows to apply the “spatial dynamics” approach taking the spanwise variable as an evolutionary one. For a certain range of parameters α, we are able to reduce the bifurcation problem to a spatial center manifold on which the flow is described by a steady Ginzburg-Landau equation. All relevant coefficients can be taken from the analysis of the purely 2D Poiseuille problem. For small β the study of the reduced problem demonstrates that both, spirals and ribbons, bifurcate subcritical, in contrast to the Couette-Taylor problem. These are solutions which are additionally 2π/β periodic in the spanwise direction.