Abstract
We extend to a non-periodic framework the classical homogenization result permitting to derive the Darcy law from the Stokes or Navier-Stokes system posed in a perforated domain. Mathematically, we study the asymptotic behavior when epsilon tends to zero of the stationary Stokes system posed in a sequence of varying domains Omega(epsilon) = Omega \ boolean OR T-k is an element of N(epsilon)k where Omega is a smooth bounded open subset of R-3 and T-epsilon(k) are closed sets such that each of them is at a distance of order epsilon of the remaining. Moreover boolean OR T-k is an element of N(epsilon)k is at a distance of order at most epsilon of any point of R-3. Each set T-epsilon(k) is non-empty, smooth and has a size of order delta(epsilon) with epsilon(3) << delta(epsilon) <= r epsilon for some r < 1. In the classical periodic case, the sets T-epsilon(k) are obtained by repeating periodically with period epsilon the set delta T-epsilon with T a non-empty smooth closed set in R-3. As in this periodic case we show that the limit problem corresponds to a Darcy system. However, even when the sets T-epsilon(k) have all the same shape, we show that for delta(epsilon) << epsilon some strong convergence results for the velocity and some capacity formulae for the limit system do not extend from the periodic framework to the non-periodic one. (c) 2021 Elsevier B.V. All rights reserved.
NSTL
Elsevier