SOME PROPERTIES OF THE UPPER CONVECTED MAXWELL MODEL FOR VISCOELASTIC FLUID-FLOW

We have studied in two dimensions the upper convected Maxwell constitutive equation for a viscoelastic fluid together with the usual equations for conservation of mass and momentum. A small degree of compressibility is allowed. This can be considered as an artificial compressibility. We show convergence from the compressible to the incompressible solution for the periodic one-dimensional Cauchy problem. Given smooth initial data, an initially positive definite modified stress tensor tau = tau + (eta/lambda)I is shown to be a sufficient condition for short time well-posedness, i.e. existence of a unique solution, smooth both in time and space, for the periodic two-dimensional Cauchy problem. Hyperbolicity will be lost if the eigenvalues of tau go to zero but, in finite time, this is only possible if the velocity gradients go to infinity in finite time.