# American Institute of Mathematical Sciences

February  2003, 3(1): 45-68. doi: 10.3934/dcdsb.2003.3.45

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

 1 Mathématiques Appliquées de Bordeaux, Université Bordeaux 1, 351, cours de la Libération 33405 Talence Cedex, France

Received  June 2001 Revised  August 2002 Published  November 2002

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.
Citation: L. Chupin. Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 45-68. doi: 10.3934/dcdsb.2003.3.45
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