## Derivation of averaged equations describing a non-Newtonian flow through a thin slab. (Dérivation des équations moyennées décrivant un écoulement non newtonien dans un domaine de faible épaisseur.)(French. Abridged English version)Zbl 0777.76006

Summary: We consider the non-Newtonian flow through a thin slab, governed by injection of a fluid. Starting from the $$3D$$ incompressible Navier-Stokes system with a nonlinear viscosity, obeying the power law, we consider the limit when thickness $$\varepsilon$$ of the slab tends to zero. We find that averaged velocity obeys a nonlinear filtration law associated to the generalized $$r'$$-Laplacian of the averaged pressure. On the boundary the normal component of averaged velocity is equal to the normal component of the injection velocity average. Finally, we prove the convergence theorem for velocity and pressure in appropriate functional spaces.

### MSC:

 76A05 Non-Newtonian fluids 76S05 Flows in porous media; filtration; seepage 35Q35 PDEs in connection with fluid mechanics