Finite element methdos for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity

Abstract

Adding a Newtonian solvent to most differential viscoelastic constitutive equations mathematically regularizes the coupled set formed by the momentum, continuity and constitutive equations. The momentum and continuity equations form an elliptic saddle point problem for velocity and pressure, and the constitutive equation is hyperbolic in stress. Three finite element algorithms are presented that exploit this elliptic behavior by expressing the momentum equation in different differential forms. The first is based on the viscous elliptic operator that arises naturally with the introduction of a Newtonian solvent viscosity; the second is based on the explicitly elliptic momentum equation formulation developed for the upper-convected Maxwell model; and the third is based on an elastic-viscous splitting of the momentum equation. Finite element discretizations are created by using Galerkin's method for the momentum and continuity equations and the streamline-upwind Petrov-Galerkin method for the components of the constitutive equation. In the latter two methods, additional interpolants are introduced so as to maintain continuous representations of velocity derivatives across element boundaries; this is a requirement if only those higher-order velocity terms which are explicitly elliptic are integrated by parts. Calculations for flow between eccentric cylinders and through a corrugated tube demonstrate the numerical stability and accuracy of each of the formulations. The robustness of each algorithm for calculation of flows at high Deborah numbers depends on the value of the ratio β ≡ η S/gh 0, where η s is the solvent viscosity and η 0 is the viscosity of the solution. The algorithm based on the elastic-viscous splitting is the most robust across the full range 0 ≤ β ≤ 1.