O(2)-symmetry breaking bifurcation: with application to the flow past a sphere in a pipe

The steady, axisymmetric laminar flow of a Newtonian fluid past a centrally-located sphere in a pipe first loses stability with increasing flow rate at a steady, O(2)-symmetry breaking bifurcation point. Using group theoretic results, a number of authors have suggested techniques for locating singularities in branches of solutions which are invariant with respect to the symmetries of an arbitrary group. We present these arguments for the O(2) symmetry encountered here and discuss their implementation for O(2)-symmetric problems. In particular, we describe how a bifurcation point may first be detected and then accurately located using an “extended system”. Also it is shown how to decide numerically if the bifurcating branch is subcritical or super-critical. The numerical solutions were obtained using the finite-element code ENTWIFE. This has enabled us to compute the symmetry-breaking bifurcation point for a range of sphere to pipe diameter ratios. We also introduce a wire along the centreline of the pipe downstream of the sphere, and show its effect on the critical Reynolds number to be small.