Homogenization of the Navier–Stokes equations in perforated domains in the inviscid limit

Abstract

We study the solution u ɛ to the Navier–Stokes equations in R 3 perforated by small particles centered at ( ε Z ) 3 with no-slip boundary conditions at the particles. We study the behavior of u ɛ for small ɛ, depending on the diameter ɛ α , α > 1, of the particles and the viscosity ɛ γ , γ > 0, of the fluid. We prove quantitative convergence results for u ɛ in all regimes when the local Reynolds number at the particles is negligible. Then, the particles approximately exert a linear friction force on the fluid. The obtained effective macroscopic equations depend on the order of magnitude of the collective friction. We obtain (a) the Euler–Brinkman equations in the critical regime, (b) the Euler equations in the subcritical regime and (c) Darcy’s law in the supercritical regime.