MATHEMATICAL DERIVATION OF THE POWER-LAW DESCRIBING POLYMER FLOW-THROUGH A THIN SLAB

We consider the polymer flow through a slab of thickness epsilon. The flow is described by 3D incompressible Navier-Stokes system with a non linear viscosity, being a power of a norm of the shear rate (power law ). We consider the limit when epsilon --> 0 and prove that the limit a veraged velocity, averaged over the thickness, satisfies a nonlinear t wo-dimensional Poiseuille's law, with non-linear vicosity depending on the power of the length of the gradient of the pressure. It is found out that the powers in the starting law and in the limit law are conju gate. Furthermore, we prove a convergence theorem for velocity and pre ssure in appropriate functional spaces.