Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation

Abstract

We consider a model of mixture of non-newtonian fluids described with an order parameter defined by the volume fraction of one fluid in the mixture, a mean-velocity field and an extra-stress tensor field. The evolution of the order parameter is given by a Cahn-Hilliard equation, while the velocity satisfies the classical Navier-Stokes equation with non constant viscosity. The non-newtonian extra-stress tensor, which is symmetric, evolves through a constitutive law with time relaxation of Oldroyd type. We derive at first a physical model for incompressible flows (with free-divergence property for the velocity). In fact, the model we consider contains an additional stress diffusion, which derives from a microscopic dumbbell model analysis. The main result of this paper concerns the existence and uniqueness of a local regular solution for this model.