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

Article Mathematics, Applied

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

Peter Bella et al.

Summary: In this note, the homogenization of the compressible Navier-Stokes equations in a periodically perforated domain in R-3 is investigated. By assuming that the particle size scales with their mutual distance and that the Mach number decreases sufficiently fast, it is shown that in the limit as the size tends to zero, the velocity and density converge to a solution of the incompressible Navier-Stokes equations with Brinkman term. The methods used in the paper closely follow those of Hofer, Kowalczyk, and Schwarzacher [https://doi.org/10.1142/S0218202521500391], where the convergence to Darcy's law was proven for particle size scaling with epsilon(alpha) and alpha ∈ (1,3).

JOURNAL OF MATHEMATICAL FLUID MECHANICS (2022)

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Article Mathematics, Applied

Derivation of Darcy's law in randomly perforated domains

A. Giunti

Summary: The study focuses on the homogenization of a Poisson problem or a Stokes system in a randomly punctured domain with spherical holes having random centers and radii. It is proven that the solutions converge to Darcy's law with minimal conditions on the integrability of random radii, despite the possibility of cluster formation.

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS (2021)

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Article Mathematics, Applied

Darcy's law as low Mach and homogenization limit of a compressible fluid in perforated domains

Richard M. Hoefer et al.

Summary: The study focuses on the homogenization limit of the compressible and barotropic Navier-Stokes equations in a three-dimensional domain with periodically distributed identical particles. By examining the particle sizes and distances, the volume fraction of particles approaching zero while their resistance density tends to infinity is considered. Unlike previous results, the study estimates the deficit related to the pressure approximation using the Bogovskii operator, allowing for more flexible estimates of pressure in Lebesgue and Sobolev spaces and convergence results for all barotropic exponents gamma > 3/2.

MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES (2021)

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Article Mathematics, Applied