Numerical integration of differential viscoelastic models

Abstract

Various numerical techniques are compared in the integration of the upper-convected Maxwell constitutive model on the basis of given kinematics in the planar stick-slip problem. The techniques include three different finite element methods (Galerkin, streamline-upwind, and streamline-upwind Petrov-Galerkin), and a streamline integration scheme. The results illustrate the failure of Galerkin's method in flows possessing steep solution gradients or singularities. The two upwinding techniques are shown to be superior to Galerkin's method but they remain inaccurate near singularities and in flow regions where the solution gradients are transverse to the streamlines. The streamline integration scheme, which in essence is a method of characteristics applied to the purely hyperbolic constitutive model, leads to stable and accurate stress predictions even close to singularities. In view of these results, we have developed a Picard iterative method based on streamline integration to produce solutions to the full set of governing equations for an Oldroyd-B fluid. Results for the stick-slip flow problem could only be obtained for modestly low values of the Weissenberg number with a rather coarse discretization of the flow domain. The reason for the poor convergence properties of the iterative scheme is the large sensitivity of the computed extra-stress to minute changes in kinematics in flow regions of high stress gradients. Our stress integrator based on streamline integration increases the (iterative) convergence difficulties since it captures those stress gradients with great accuracy. The successful use of streamline integration in viscoelastic calculations requires further work towards decoupled iterative schemes capable of handling large stress sensitivity.