Derivation of Darcy’s law in randomly perforated domains

Abstract

We consider the homogenization of a Poisson problem or a Stokes system in a randomly punctured domain with Dirichlet boundary conditions. We assume that the holes are spherical and have random centres and radii. We impose that the average distance between the balls is of size $$\varepsilon $$ ε and their average radius is $$\varepsilon ^{\alpha }$$ ε α , $$\alpha \in (1; 3)$$ α ∈ ( 1 ; 3 ) . We prove that, as in the periodic case (Allaire, G., Arch. Rational Mech. Anal. 113(113):261–298, 1991), the solutions converge to the solution of Darcy’s law (or its scalar analogue in the case of Poisson). In the same spirit of (Giunti, A., Höfer, R., Ann. Inst. H. Poincare’- An. Nonl. 36(7):1829–1868, 2019; Giunti, A., Höfer, R., Velàzquez, J.J.L., Comm. PDEs 43(9):1377–1412, 2018), we work under minimal conditions on the integrability of the random radii. These ensure that the problem is well-defined but do not rule out the onset of clusters of holes.

Homogenization and Low Mach Number Limit of Compressible Navier-Stokes Equations in Critically Perforated Domains

“…This condition was removed by Giunti and Höfer [10], where they showed that for incompressible fluids and randomly distributed holes with random radii, the randomness does not affect the convergence to Brinkman's law. More recently, for large particles, Giunti showed in [9] a similar convergence result to Darcy's law.…”

Homogenization and low Mach number limit of compressible Navier-Stokes equations in critically perforated domains

“…This condition was removed by Giunti and Höfer [12], were they showed that for incompressible fluids and randomly distributed holes with random radii, the randomness does not affect the convergence to Brinkman's law. More recently, for large particles, Giunti showed in [10] a similar convergence result to Darcy's law.…”

Corrector estimates and homogenization error of unsteady flow ruled by Darcy's law

“…In terms of the related research topics, to the authors' best knowledge, the pioneer literatures were contributed by E. Marušić-Paloka, A. Mikelić, L. Paoli [30,33], where they obtained a rate O(ε 1 6 ) for steady Stokes problems and an error O(ε) for non-stationary incompressible Euler's equations, respectively, in the case of d = 2. Homogenization and porous media have been received many studies, without attempting to be exhaustive, we refer the reader to [1,5,9,10,11,12,13,14,19,18,23,24,27,38,42,43,46,50] and the references therein for more results. 7 In stationary case, the same extension could be found in [3,26], and it played a similar role in the qualitative theory for the unsteady case presented by [2,4,31].…”