Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary

Abstract

We consider a non-Newtonian flow in a thin domain of thickness $\varepsilon$. The flow is described by the 3D incompressible Navier-Stokes (Stokes) system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index $p$.

The bottom of the domain is irregular by the present of slight roughness of amplitude $\varepsilon^\delta$ and period $\varepsilon^\beta$, satisfying the relation $1<\beta<\delta$. Assuming pure slip or partial slip with a friction coefficient $\varepsilon^{-\gamma}$, with $\gamma>0$, on the rough boundary, we consider the limit when domain thickness tends to zero and we obtain different models depending on the magnitude $\delta$ with respect to ${2p-1\over p}\beta-{p-1\over p}$, and the magnitude $\gamma$ with respect to ${1\over p-1}$.