Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary

We consider a non-Newtonian fluid flow in a thin domain with thickness ${\eta }_{\epsilon }$ and an oscillating top boundary of period $\epsilon$. The flow is described by the 3D incompressible Navier-Stokes system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index $p$, with $9\mathrm{/}5\le p<+\mathrm{\infty }$. We consider the limit when the thickness tends to zero and we prove that the three characteristic regimes for Newtonian fluids are still valid for non-Newtonian fluids, i.e. Stokes roughness (${\eta }_{\epsilon }\approx \epsilon$), Reynolds roughness (${\eta }_{\epsilon }\ll \epsilon$) and high-frequency roughness (${\eta }_{\epsilon }\gg \epsilon$) regime. Moreover, we obtain different nonlinear Reynolds type equations in each case.