The consistent streamline-upwind/Petrov-Galerkin method for viscoelastic flow revisited

In an earlier paper, Marchal and Crochet introduced two mixed finite-element methods for calculating viscoelastic flow. The first one, based on a consistent streamline-upwind/Petrov-Galerkin integration of the constitutive equations, was disregarded because it produced wiggles in the numerical calculation of the stick-slip flow of a Maxwell fluid. The second method, based on a non-consistent streamline-upwind integration, was found to be stable at high values of the Weissenberg number; however, it introduces artificial diffusivity of the order of the element size, h, which decreases with mesh refinement. In the present paper, we re-examine the first method. We show that it is both stable and accurate for solving flows of a Maxwell fluid in smooth geometries. The test problems are the flow around a sphere in a tube and the flow through a corrugated tube. The results coincide with those of other accurate methods for solving the same problems. Finally, we show that the results obtained with the second method converge towards the most accurate results when the element size decreases. In particular, we show that the velocity field is little affected by numerical diffusion in the stress constitutive equations.