Finite-amplitude time-periodic states in viscoelastic taylor-couette flow described by the UCM model

Abstract

Time-dependent simulations using the EEME finite-element method are reported for calculation of the linear stability and nonlinear dynamics of the transition to time-periodic, viscoelastic flow in axisymmetric TaylorCouette flow between parallel cylinders. The linear stability analysis is based on time integration of the linearized finite-element equations and reproduces the oscillatory linear instability recently analyzed by Larson et al. past a critical Deborah number De c. Linear analysis presented here shows that the viscoelastic instability corresponds to a secondary flow structure composed of multiple toroidal flow cells nested radially, and that these cells travel across the gap between the cylinders. Nonlinear simulations demonstrate the existence of supercritical, i.e. for De - De c > 0, time-periodic flow states. For only small changes in De > De c, the flatness of the neutral stability curve with variation in the flow cell height leads to nonlinear interactions between flows that are closely spaced in De. These interactions will make the observation of simple oscillations near onset extremely difficult.