In this talk, we will present the study of a model for particle suspensions in a non-Newtonian Ostwald-DeWaele fluid with a potentially degenerate viscosity coefficient.

The analysis of problems associated with such systems is a very active research topic, the source of many recent results, particularly in the case of particles that sediment in a Newtonian fluid (see, for example, the works of D. Cobb, R. Höfer, A. Mecherbet, R. Schubert, F. Sueur). We will consider the case where particles are suspended in a non-Newtonian fluid. From a mathematical point of view, this is characterized by a Stokes-Transport equation with the particularity that the Stokes equation is nonlinear, the viscosity term being expressed as a p-Laplacian for the symmetrized gradient. After briefly contextualizing the problem, we will present our result: the existence of global weak energy solutions.

In order to show that we do define an active scalar equation, i.e. one in which the relative density of the particles suspended in the fluid gives meaning to a solution of the system through an inverse mapping, it is then necessary to use monotonicity methods in conjunction with techniques derived from DiPerna-Lions theory to establish the existence of suitable weak solutions. We will therefore present the main ideas for establishing the existence of such ones.