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 $\eta_\varepsilon$ and an oscillating top boundary of period $\varepsilon$. 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 $9/5\leq p<+\infty$. 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 ($\eta_\varepsilon\approx \varepsilon$), Reynolds roughness ($\eta_\varepsilon\ll \varepsilon$) and high-frequency roughness ($\eta_\varepsilon\gg \varepsilon$) regime. Moreover, we obtain different nonlinear Reynolds type equations in each case.