The first instability in spherical Taylor-Couette flow

Abstract

In this paper continuation methods are applied to the axisymmetric Navier-Stokes equations in order to investigate how the stability of spherical Couette flow depends on the gap size σ. We find that the flow loses its stability due to symmetry-breaking bifurcations and exhibits a transition with hysteresis into a flow with one pair of Taylor vortices if the gap size is sufficiently small, i.e. if σ ≤σ_B. In wider gaps, i.e. for σ_B < σ ≤ σ_F, both flows, the spherical Couette flow and the flow with one pair of Taylor vortices, are stable. We predict that the latter exists in much wider gaps than previous experiments and calculations showed. Taylor vortices do not exist if σ > σ_F. The numbers σ_B and σ_F are computed by calculating the instability region of the spherical Couette flow and the region of existence of the flow with one pair of Taylor vortices.