On a wide class of nonlinear models for non-Newtonian fluids with mixed boundary conditions in thin domains

Abstract

The behaviour of Newtonian and non-Newtonian flows through a thin three-dimensional domain are widely studied in the literature. Usually, authors deal with special models related to particular concrete fluids. In this work, our aim is to present a general model, governing the behaviour of a large class of Newtonian and non-Newtonian fluids. Moreover, we deal with mixed boundary conditions, which are not often studied in the literature related to flows in thin domains. We consider a nonlinear model of a flow in a thin three-dimensional domain, and we study its behaviour when the thickness in one direction tends to zero. At the limit, we obtain a quasilinear two-dimensional problem for the pressure, a nonlinear Reynolds's law for the velocity and a nonlinear Darcy's law for the averaged velocity. Finally, we check that our results hold for a large class of non-Newtonian fluids by producing concrete examples.