Galerkin finite element analysis of complex viscoelastic flows

Abstract

Possible causes for the difficulties with computing steady two-dimensional viscoelastic flows are explored through a sequence of Galerkin finite element calculations for the flow of an upper-convected Maxwell fluid through abrupt and smooth axisymmetric contractions and between eccentric cylinders with the inner one rotating. Calculations in all three flow problems are terminated by a limiting value of the Deborah number, where the family of numerical solutions beginning at zero De turns back to lower values of De. The critical value of De decreases drastically with mesh refinement near the reentrant corner of the abrupt contraction, indicating that the strength of the stress singularity may not be representable by a continuous approximation. Calculations for the smooth contraction and the eccentric cylinder geometries show the development of oscillations in the stress and velocity fields preceding the limit point. Results including fluid inertia indicate that the oscillations may be connected with the hyperbolic character of the coupled equation set and thus suggest new methods of solution.