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We study the solution $u_\varepsilon$ to the Navier-Stokes equations in$\mathbb R^3$ perforated by small particles centered at $(\varepsilon \mathbbZ)^3$ with no-slip boundary conditions at the particles. We study the behaviorof $u_\varepsilon$ for small $\varepsilon$, depending on the diameter$\varepsilon^\alpha$, $\alpha > 1$, of the particles and the viscosity$\varepsilon^\gamma$, $\gamma > 0$, of the fluid. We prove quantitativeconvergence results for $u_\varepsilon$ in all regimes when the local Reynoldsnumber at the particles is negligible. Then, the particles approximately exerta linear friction force on the fluid. The obtained effective macroscopicequations depend on the order of magnitude of the collective friction. Weobtain a) the Euler-Brinkman equations in the critical regime, b) the Eulerequations in the subcritical regime and c) Darcy's law in the supercriticalregime.

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