Stability of the O(2)-symmetric flow past a sphere in a pipe

The linear stability of the O(2)-symmetric laminar flow past a centrally located sphere in a pipe for a range of sphere sizes is examined. For all values of the blockage ratio studied, the first instability observed as the flow rate increased, occurred at a steady symmetry-breaking bifurcation point. A transformation technique was used to solve the generalized eigenvalue problem which arises when using a mixed finite-element method to determine linear stability. The present results support the conclusion of Natarajan and Acrivos [J. Fluid Mech. 254, 323 (1993)] who have computed the stability of the flow past a sphere in an unbounded domain. The primary instability in three dimensions differs qualitatively from that of the analogous two-dimensional flow past a centrally located cylinder in a channel, which Chen et al. (to appear in J. Fluid Mech.) have shown to occur at a Z2 symmetry breaking Hopf bifurcation point.