Analysis of the roughness regimes for micropolar fluids via homogenization

We study the asymptotic behavior of micropolar fluid flows in a thin domain of thickness ${\eta }_{\epsilon }$ with a periodic oscillating boundary with wavelength $\epsilon$. We consider the limit when $\epsilon$ tends to zero and, depending on the limit of the ratio of ${\eta }_{\epsilon }\mathrm{/}\epsilon$, we prove the existence of three different regimes. In each regime, we derive a generalized Reynolds equation taking into account the microstructure of the roughness.