Recent advances in the mathematical theory of steady flow of viscoelastic fluids

Abstract

The paper reviews recent existence theorems for steady flows of viscoelastic fluids. The results concern small perturbations of either the rest state or a flow with uniform velocity. Differential as well as integral models are discussed. For differential models, we also discuss the boundary conditions which need to be prescribed at an inflow boundary; it turns out that there is a difference between Maxwell- and Jeffreys-type models in this respect. For Newtonian fluids, existence results which go much farther are known; the data need not be small and no regularity of the boundary is required. A number of problems are pointed out which make similar results for viscoelastic fluids difficult and in some cases unlikely.