Roughness-Induced Effect at Main Order on the Reynolds Approximation
Didier Bresch, Catherine Choquet, Laurent Chupin, Thierry Colin, Marguerite Gisclon · Multiscale Modeling and Simulation · 2010
Usually the Stokes equations that govern a flow in a smooth thin domain (with thickness of order $\varepsilon$) are related to the Reynolds equation for the pressure $p_{\mathrm{smooth}}$. In this paper, we show that for a rough thin domain (with rugosities of order $\varepsilon^2$) the flow is governed by a modified Reynolds equation for a pressure $p_{\mathrm{rough}}$. Moreover, we find the relation $p_{\mathrm{rough}}=K\,p_{\mathrm{smooth}}$, where K is an explicit coefficient depending only on the form of the rugosities and on the viscosity of the fluid. In some sense, we see that the flow may be accelerated using adequate rugosity profiles on the bottom. The limit system is mathematically justified through a variant of the notion of two-scale convergence, the originality and difficulty being the anisotropy in the height profile.