Viscoelastic flow around a confined cylinder: 2-D linear stability analysis leading to asymmetric flow

Abstract

We study the two-dimensional creeping flow of a viscoelastic fluid around a cylinder confined between two plates parallel to its axis. First, we solve the governing equations under steady state with our novel stabilized finite-element formulation to obtain converged solutions even at very high Weissenberg numbers. Then, we examine the stability of this solution by perturbing all flow variables and solving the corresponding eigenvalue problem. At critical conditions, a stable asymmetric flow arises, in which more fluid passes from either the upper or the lower gap between the cylinder and the channel wall. Both shear-thinning and elasticity play a crucial role on the onset and subsequent evolution of the instability. Energy analysis shows that the terms of the constitutive equation corresponding to apparent strain-rate thinning and material extensibility are responsible for the flow destabilization. The instability is present at a wider range of flow conditions when the material is more elastic and when the solvent contribution is smaller. The instability is also promoted by increasing the confinement. Beyond the critical conditions, asymmetric flow profiles vanish when the flow is so intense that thinning effects are not important anymore. The critical Weissenberg number for instability inception and cessation depends on material properties and geometry exponentially and linearly, respectively. Furthermore, the instability arises even in a seemingly non-shear-thinning fluid, i.e. one with constant shear viscosity in simple shear, when the solvent contribution is minimal, because of the apparent thinning effect that is created by the convection of the viscoelastic stresses. Finally, models with extension-rate thinning trigger the instability at limited flow conditions, when the shear viscosity decreases with the shear rate, and the normal stresses at the wake of the cylinder are still important. These results agree with previous experiments and simulations, and give new insights on the physical mechanism that triggers this flow instability.