Comparison of computational efficiency of flow simulations with multimode constitutive equations: integral and differential models

Abstract

A decoupled finite-element method (INT/FEM) is presented for calculation of two-dimensional viscoelastic flows with integral constitutive models. The momentum and continuity equations are solved by Galerkin's method with the viscoelastic stress treated as a fixed body force. The viscoelastic stress is computed by using the stream function to track fluid particles upstream, integrating a system of ordinary differential equations that govern the displacement-gradient tensor, and evaluating the integral constitutive equation by numerical quadrature. The quasi-linear upper-convected Maxwell and Oldroyd-B models, as well as the nonlinear model of Papanastasiou, Scriven and Macosko (PSM), are used in the simulations. The efficiency of the integral method is compared to that of the recently developed finite-element method (EVSS/FEM) for differential constitutive models. Convergence and accuracy of the INT/FEM are shown by calculations for flow between eccentric cylinders. The upper limit in De attainable by using the INT/FEM is comparable to values for the EVSS/FEM only for constitutive models with a shear-thinning ratio of the first normal stress difference to the shear stress and a large solvent contribution to the solution viscosity. The INT/FEM becomes the more efficient technique for simulation with this type of constitutive equation when three or more relaxation modes are included in the memory function.