INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2000; 32: 175–200 O (2)-symmetry breaking bifurcation: with application to the flow past a sphere in a pipe K. A. Cliffe a,1 , A. Spence b,2 and S. J. Tavener c, * a AEA Technology, Harwell Laboratory, Didcot, Oxfordshire OX11 0RA, UK b School of Mathematics, Uniersity of Bath, Claerton Down, Bath BA27AY, UK c Department of Mathematics, The Pennsylania State Uniersity, Uniersity Park, PA 16802, USA SUMMARY 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 that are invariant with respect to the symmetries of an arbitrary group. These arguments are presented for the O(2)-symmetry 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 element code ENTWIFE. This has enabled the computation of the symmetry breaking 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 introduced, and its effect on the critical Reynolds number is shown to be small. Copyright © 2000 John Wiley & Sons, Ltd. KEY WORDS: flow past a sphere; numerical bifurcation; O(2)-symmetry breaking 1. INTRODUCTION The external flow past a sphere has attracted considerable attention for at least the past 60 years, since the pioneering experimental work of Mo ¨ ller [1]. More recent experimental investigations by Goldburg and Florsheim [2], Margarvey and Bishop [3], Nakamura [4] and Wu and Faeth [5], and computational studies by Natarajan and Acrivos [6], Tomboulides et al. [7] and Tavener [8], have all concluded that the initially steady axisymmetric flow past the sphere loses stability to a steady, asymmetric flow above a critical flow rate. The experimental * Correspondence to: Department of Mathematics, The Pennsylvania State University, 416 McAllister Building, University Park, PA 16802, USA. Tel.: +1 814 8653873; fax: +1 814 8653735; e-mail: tavener@math.psu.edu 1 E-mail: andrew.cliffe@aeat.co.uk 2 E-mail: a.spence@maths.bath.ac.uk CCC 0271–2091/2000/020175 – 26$17.50 Copyright © 2000 John Wiley & Sons, Ltd. Receied 23 July 1998 Reised 13 January 1999