Testing viscoelastic numerical schemes using the Oldroyd-B fluid in Newtonian kinematics

We focus here on using a Newtonian velocity field to evaluate numerical schemes for two different formulations of viscoelastic flow. The two distinct formulations we consider, correspond to either using a fixed basis for the elastic stress or one that uses the flow directions or streamlines. The former is the traditional Cartesian stress formulation, whilst the later may be referred to as the natural stress formulation of the equations. We choose the Oldroyd-B fluid and three benchmarks in computational rheology: the 4:1 contraction flow, the stick-slip and cross-slot problems. In the context of the contraction flow, fixing the kinematics as Newtonian, actually gives a larger stress singularity at the re-entrant corner, the matched asymptotics of which are presented here. Numerical results for temporal and spatial convergence of the two formulations are compared first in this decoupled velocity and elastic stress situation, to assess the performance of the two approaches. This may be regarded as an intermediate test case before proceeding to the far more difficult fully coupled velocity and stress situation. We also present comparison results between numerics and asymptotics for the stick-slip problem. Finally, the natural stress formulation is used to investigate the cross-slot problem, again in a Newtonian velocity field.