COLLOCATION AND GALERKIN FINITE-ELEMENT METHODS FOR VISCOELASTIC FLUID-FLOW .1. DESCRIPTION OF METHOD AND PROBLEMS WITH FIXED GEOMETRY

Collocation and Galerkin finite element methods are developed for viscoelastic fluid flow in a fixed geometry. The collocation methods use Hermite cubic polynomials with a global coordinate transformation to permit irregular geometry. The Galerkin method uses isoparametric elements (transformed element by element) with bilinear polynomials for pressure and quadratic polynomials for velocity. Both methods are applied to two-dimensional flow in planar geometry and the Galerkin method is applied to axisymmetric cylindrical geometries as well. The fluid model is a nonlinear Maxwell model but is limited to small elastic components. The two methods are applied to several test problems. Entry-length problems test the ability to model pressure singularities are velocity discontinuities. Stick-slip problems test the ability to model pressure singularities and stress discontinuities. Both test problems have analytic or accurate numerical solutions for Newtonian fluids so that the accuracy of the two methods is compared. © 1979.