| The morphogenesis of secondary vortices is investigated for the 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.;For the Upper Convected Maxwell (UCM) model it was found that the secondary flow corresponds to a steady Taylor vortex in the case of small flow elasticities, but to a time-periodic one when elasticity becomes important (Hopf bifurcation). Degenerate Hopf bifurcation theory in the presence of symmetries has been used to show the existence of two different time-periodic solution families. Through a non-linear analysis, using pseudospectral approximations in both space and time, all of the axisymmetric steady and time-periodic bifurcating solutions are shown to be supercritical. Other differential models have also been considered.;In order to determine the stability of the bifurcating branches, the time evolution of finite amplitude axisymmetric perturbations (Taylor cells) to the purely azimuthal, viscoelastic cylindrical Couette flow was numerically simulated. Two time integration numerical methods were developed, both based on a pseudospectral spatial approximation of the variables, efficiently implemented using fast Poisson solvers and optimal filtering routines. Stability results for the UCM fluid were obtained for the supercritical bifurcations, either steady or time-periodic, developed after the onset of instabilities in the primary flow.;Non-axisymmetric disturbances were also considered. The linear stability analysis of the UCM and Oldroyd-B fluids revealed that 3-d inertialess disturbances are more unstable than 2-d ones. The corresponding symmetry analysis of the emerging patterns is also presented.;Through the analysis of the present investigation, it is shown that spectral techniques provide a robust and computationally efficient method for the simulation of complex, non-linear, time-dependent viscoelastic flows. |
