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 a t a steady O(2)-symmetry breaking bifurcation point. Using group theoretic results, a number of authors have suggested techniques for locating singula rities in branches of solutions that are invariant with respect to the symm etries of an arbitrary group. These arguments are presented for the O(2)-sy mmetry encountered here and their implementation for O(2)symmetric problems is discussed. In particular, how a bifurcation point may first be detected and then accurately located using an 'extended system' is described. Also shown is how to decide numerically if the bifurcating branch is subcritical or supercritical. The numerical solutions were obtained using the finite e lement code ENTWIFE. This has enabled the computation of the symmetry break ing bifurcation point for a range of sphere-to-pipe diameter ratios. A wire along the centerline of the pipe downstream of the sphere is also introduc ed, and its effect on the critical Reynolds number is shown to be small. Co pyright (C) 2000 John Wiley & Sons, Ltd.