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

Abstract

In this note, we consider the homogenization of the compressible Navier-Stokes equations in a periodically perforated domain in R 3 . Assuming that the particle size scales like ε 3 , where ε > 0 is their mutual distance, and that the Mach number decreases fast enough, we show that in the limit ε → 0, the velocity and density converge to a solution of the incompressible Navier-Stokes equations with Brinkman term. We strongly follow the methods of Höfer, Kowalczik and Schwarzacher [arXiv:2007.09031], where they proved convergence to Darcy's law for the particle size scaling like ε α with α ∈ (1, 3).

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].…”

Homogenization of the full compressible Navier-Stokes-Fourier system in randomly perforated domains

“…For large particles with α ∈ (1, 3), Darcy's law was just recently treated in [16] for a low Mach number limit. Their methods can also be used to treat the critical case α = 3 [4]. The case of small particles (α > 3) was treated in [9,11,18] for different growing conditions on the pressure.…”