Reynolds equations for a large class of non-Newtonian fluids

We consider a non-Newtonian flow through a thin slab Omega(delta) subset of R(3). The flow is governed by a Navier-Stokes system with Neumann's boundary conditions on a part of the slab boundary. The viscosity of the fluid depends on the second invariant of the strain tensor and on the velocity norm in L(4)(Omega(delta)). We prove convergence results when thickness delta tends to zero. At the limit we obtain a nonlinear Reynolds law for velocity, a nonlinear Darcy law for averaged velocity and a nonlinear two-dimensional Dirichlet's problem for the pressure. Finally we verify that our results are valid for a large class of non-Newtonian fluids by construction of examples obtained by perturbation of classical laws.