Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

Abstract

We prove the homogenization to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. The fluid satisfies a no-slip boundary condition at the holes. The balls generating the holes have centres distributed according to a Poisson point process and i.i.d. unbounded radii satisfying a suitable moment condition. We stress that our assumption on the distribution of the radii does not exclude that, with overwhelming probability, the holes contain clusters made by many overlapping balls. We show that the formation of these clusters has no effect on the limit Brinkman equations. In contrast with the case of the Poisson equation studied in [A. Giunti, R. Höfer, and J.J.L. Velázquez, Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes], the incompressibility condition requires a more detailed study of the geometry of the random holes generated by the class of probability measures considered.in a domain D ε , that is obtained by removing from a bounded set D ⊆ R d , d > 2, a random number of small balls having random centres and radii. More precisely, for ε > 0, we define 2)1 similar size. We emphasize, however, that it neither prevents the balls generating H ε from overlapping, nor it implies a uniform upper bound on the number of balls of very different size which combine into a cluster (see Section 6). The main technical effort of this paper goes into developing a strategy to deal with these geometric considerations and succeed in controlling the term in (1.3). We refer to Subsection 2.3 for a more detailed discussion on our strategy. We also mention that, to avoid further technicalities, we only treat the case where the centres of the balls in (1.2) are distributed according to a homogeneous Poisson point process. It is easy to check that our result applies both to the case of periodic centres and to any (short-range) correlated point process for which the results contained in Appendix C hold.After Brinkman proposed the equations (1.3) in [3] for the fluid flow in porous media, an extensive literature has been developed to obtain a rigorous derivation of (1.3) from (1.1) in the case of periodic configuration of holes [2,15,20,16]. We take inspiration in particular from [1], where the method used in [5] for the Poisson equations is adapted to treat the case of the Stokes equations in domains with periodic holes of arbitrary and identical shape. In [1], by a compactness argument, the same techniques used for the Stokes equations also provide the analogous result in the case of the stationary Navier-Stokes equations. The same is true also in our setting (see Remark 2.2 in Section 2). In [6], with methods similar to [1] and [5], the homogenization of stationary Stokes and Navier-Stokes equations has been extended also to the case of spherical holes where different and constant Dirichlet boundary conditions are prescribed at the boundary of each ball. This corre...

Convergence rates for the homogenization of the Poisson problem in randomly perforated domains

“…In the case of the Stokes operator, the limit equation contains an additional zero-th order term similar to C 0 in (1.4). Under the same assumptions of this paper, the analogue of the homogenization result contained in [13] has been proven for a Stokes system in [12,11]. We believe that techniques similar to the one of this paper may be used to prove the same result of (1.6) also in the case of a Stokes system.…”

A Unified Homogenization Approach for the Dirichlet Problem in Perforated Domains

“…Moreover, a corresponding framework for quantification was established by Kacimi and Murat [21]. The framework was later extended by Allaire to Stokes and Navier-Stokes problems [1,2], to the obstacle problems by Caffarelli and Mellet [8], and to random settings by Hoàng [18,17]; see also [13] and see [16,15] for randomly perforated domains based on Poisson piont processes.…”

Derivation of Darcy’s law in randomly perforated domains

“…For holes that are not periodic, the extremal regimes α ∈ {1, 3} have been rigorously studied both in deterministic and random settings. For α = 3 we mention, for instance [6,9,[16][17][18][23][24][25] and refer to the introductions in [12] and [14] for a detailed overview of these results. We stress that the homogenization of (0.3) and (0.4) when H ε is as in (0.1) with α = 3 has been studied in the series of papers [12][13][14].…”

“…For α = 3 we mention, for instance [6,9,[16][17][18][23][24][25] and refer to the introductions in [12] and [14] for a detailed overview of these results. We stress that the homogenization of (0.3) and (0.4) when H ε is as in (0.1) with α = 3 has been studied in the series of papers [12][13][14]. These works prove the convergence to the effective equation under the minimal assumption that H ε has finite averaged capacity density.…”

“…We are not aware of analogous results when the holes H ε are not periodic. The present paper considers this problem when H ε is random and, in the same spirit of [12,14], the balls in H ε may overlap and cluster.…”