Effective boundary condition for a quasi-Newtonian viscous fluid at a slightly rough boundary starting from a Navier condition

We consider the quasi-Newtonian flow in a domain with a periodic rough bottom Gamma(epsilon) of period of order the small parameter epsilon and amplitude delta(epsilon), such that delta(epsilon) << epsilon. The flow is described by the 3D incompressible non-Newtonian Navier-Stokes system where the viscosity is given by the non linear power law which is widely used for dilatant fluids ( shear thickening). Assuming that the fluid satisfies the Navier slip condition on Gamma(epsilon) and letting epsilon -> 0, we obtain three different macroscopic models depending on the magnitude of delta(epsilon) with respect to epsilon(2p-1/p), with p > 2. In the case delta(epsilon) >> epsilon(2p-1/p) the effective boundary condition in the limit epsilon = 0 is the no-slip condition, while if delta(epsilon) << epsilon(2p-1/p) there is no roughness-induced effect. In the critical case when delta(epsilon) similar to epsilon(2p-1/p) we provide a more accurate effective boundary condition of Navier type. Finally, we also obtain corrector result for the pressure and velocity in every cases. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim