A stable and convergent scheme for viscoelastic flow in contraction channels
We present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels. In our approach we split the viscoelastic terms of the Oldroyd-B constitutive equation using Duhamels formula and discretize the resulting PDEs using a semi-implicit finite difference method based on a LaxWendroff method for hyperbolic terms. In particular, we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well-posed. A projection method is used to impose the incompressibility constraint. We are able to compute the full range of unsteady elastic flows in an abrupt contraction channel  from the viscous limit to the elastic limit  in a stable and convergent manner. We demonstrate the range of our method for unsteady flow of a Maxwell fluid with and without viscosity in planar contraction channels. We also demonstrate stable and convergent results for benchmark high Weissenberg number problems at We=1 and We=10.
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