Viscoelastic Taylor–Couette flow: bifurcation analysis in the presence of symmetries

Abstract

The morphogenesis of secondary vortices is investigated for the axisymmetric flow of a viscoelastic fluid, confined between two independently rotating, infinitely long cylinders, in the region near the onset of instability of the purely azimuthal Couette flow. The Oldroyd-B constitutive equation is used to model viscoelasticity. Three characteristic regions in the parameter space, corresponding to three distinct solution families have been investigated where the onset of instability is due primarily to inertia, both inertia and elasticity, and exclusively elasticity, respectively. The secondary flow corresponds to a steady Taylor vortex in the first case, but to a time-periodic one when elasticity becomes important (Hopf bifurcation). Degenerate Hopf bifurcation theory in the presence of symmetries ( O (2) × S 1 ) has been used to show the existence of two different time-periodic solution families, each following either one of two possible patterns, the rotating wave or the standing wave. Through a computer-aided nonlinear analysis, all of the steady and time-periodic bifurcating solutions are shown to be supercritical, implying that one and only one is stable. These results are consistent with the conclusions of time-dependent numerical simulations which have demonstrated an exchange of stabilities from the rotating to the standing wave pattern emerging after the bifurcation, as the elasticity of the fluid increases.