Progress and Challenges in Computational Rheology

Abstract

The mathematical models and numerical techniques that provide the theoretical foundation of classical Computational Fluid Dynamics have a range of validity limited to gases or low molecular weight Newtonian liquids. The provocative flow phenomena observed with polymeric fluids, for example, cannot at all be predicted by the Navier-Stokes equations. Non-Newtonian behavior has many facets, the most striking being memory effects associated with the elasticity of the material. The theoretical challenge is to translate the complex rheological behavior of non-Newtonian fluids into suitable constitutive equations, and to develop accurate numerical procedures for use of these models in the analysis of complex flows.The aim of this lecture is to review the field of large-scale non-Newtonian flow simulations as it stands in 1990, with a special emphasis put on the modeling of viscoelastic effects. We begin with a discussion of the mathematical models that are of current use in numerical work. These models combine ellipticity and hyperbolicity in a rather subtle way. We then introduce a classification of numerical approaches which is used to describe the entire spectrum of available numerical techniques. Some of the published simulations that predict significant viscoelastic effects are reviewed. Finally, we focus on the considerable mathematical, numerical and algorithmic challenges associated with the prediction of viscoelastic flows.