Application of the natural stress formulation for solving unsteady viscoelastic contraction flows

We present a numerical scheme for a previously unexploited formulation of the equations for unsteady viscoelastic flow. The formulation aligns the polymer stress along particle paths/streamlines, utilising the characteristic curves associated with the hyperbolic part of the constitutive equations. We illustrate the approach for the Oldroyd-B model in the benchmark 4:1 contraction for moderate elasticity numbers. We show that the scheme is able to accurately capture the re-entrant corner singularity for the polymer stresses and the pressure, the latter variable being inaccurately determined by schemes using the traditional formulation in terms of Cartesian polymer stresses. A space-step restriction for stability is derived, which can be numerically limiting in certain recirculation regions. This contrasts with the equivalent space-step restriction for the formulation in Cartesian stresses, which is limiting in flow regions of high velocity gradients, for example, at sharp corners in contraction flows. (C) 2019 Elsevier Inc. All rights reserved. (AU)