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

Bourgeat, Alain; Mikelić, Andro; Tapiéro, Roland

C. R. Acad. Sci., Paris, Sér. I 316, No. 9, 965-970 (1993).

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

Keywords:

incompressible Navier-Stokes system with a nonlinear viscosity; power law; nonlinear filtration; injection velocity; convergence theorem