Viscoelastic stagnation flows: regularity, instability, bifurcations

Flows of viscoelastic fluids containing stagnation points are an interesting subject in both applied and research aspects. Significant strains arising in the vicinity of such points give ground to a number of specific and complex features. Some of them, purely elastic instabilities, flow asymmetries, bifurcations, reversals, have made an impact on the present work. These non-linear phenomena pose challenges in simulation. One of them is oftentimes observed divergence of numerical solutions for benchmark stagnation flows (near the stationary points) simulated by the upper convective Maxwell (UCM) rheological state equation. It has made some authors find such singularities "invincible" and following from the nature of the UCM model itself. The present work argues this position by presenting asymptotic and numerical solutions for two benchmark types of stagnation UCM flows. In many cases both flows turn out regular in their stagnation points including the case of high Weissenberg numbers. Good agreement is demonstrated between the analytical and numerical results. An explanation is proposed of flow asymmetries onset in terms of the vorticity accumulation at the stagnation point.