A Schauder theory for the Stokes equations in rough domains

The vanishing viscosity limit for 2D Navier–Stokes in a rough domain

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.

Usually the Stokes equations that govern a flow in a smooth thin domain (with thickness of order e) are related to the Reynolds equation for the pressure psmooth. We begin by showing that the flow may be accelerated using adequate rugosity profiles on the bottom. Indeed, we prove that for a rough thin domain, with rugosities of order e2, the flow is governed by a modified Reynolds equation for a pressure prough such that Prough = Kpsmooth where K is an explicit coefficient depending only on the form of the rugosities and on the viscosity of the fluid.Next, we explore the effects of more or less rapid changes in the roughness.The limit systems are mathematically justified through a variant of the notion of two‐scale convergence: the originality and difficulty being the anisotropy in the height profile.

We consider an incompressible flow governed by the nonlinear Navier–Stokes equations with periodic boundary conditions in a domain with periodic roughness. Approximations to the flow up to the second order are deduced starting from a modification of the analogous flow in a simpler truncated domain, i.e. its extension to a series with respect to the amplitude of the rough part of the flow domain. The rigorous estimates for the approximations are given.

Existence of suitable weak solutions to the Navier–Stokes equations in time varying domains