Derivation of the two-dimensional carreau law for a quasi-newtonian fluid flow through a thin slab

Abstract

We consider the polymer flow through a slab of small thickness The flow is described by the three-dimensional incompressible steady state Navier-Stokes system with a non-linear viscosity following the Carreau Law which is in common use for polymeric fluids. We study the limit when the thickness tends to zero and prove that the averaged limit velocity satisfies according to the order of Reynolds Number and the power r of the Carreau Law, a linear or non-linear two-dimensional Reynold's Law of Power or Carreau type. We prove a convergence theorem for velocity and pressure in appropriate functional spaces.