On the mathematical analysis of thick fluids

In chemical engineering models, shear-thickening or dilatant fluids converge in the limit case to a class of incompressible fluids with a maximum admissible shear rate, the so-called thick fluids. These non-Newtonian fluids may be obtained, in particular, as the power limit of Ostwald-de Waele fluids, and may be formulated as a new class of evolution variational inequalities, in which the shear rate is bounded by a positive constant or, more generally, by a bounded positive function. We prove the existence, uniqueness and continuous dependence of solutions to this general class of thick fluids with variable threshold on the absolute value of the deformation rate tensor, which solutions belong to a time dependent convex set. For sufficiently large viscosity, we also show the asymptotic stabilization towards the unique steady state.