Homogenization of the Navier-Stokes equations in perforated domains in the inviscid limit

We study the solution \(u_\varepsilon\) to the Navier-Stokes equations in \(\mathbb R^3\) perforated by small particles centered at \((\varepsilon \mathbb Z)^3\) with no-slip boundary conditions at the particles. We study the behavior of \(u_\varepsilon\) for small \(\varepsilon\), depending on the...

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Bibliographic Details
Published in:arXiv.org 2024-10
Main Author: Höfer, Richard M
Format: Article
Language:English
Subjects:
Alpha rays
Boundary conditions
Darcys law
Euler-Lagrange equation
Fluid flow
Macroscopic equations
Mathematical analysis
Navier-Stokes equations
Reynolds number
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Summary:We study the solution \(u_\varepsilon\) to the Navier-Stokes equations in \(\mathbb R^3\) perforated by small particles centered at \((\varepsilon \mathbb Z)^3\) with no-slip boundary conditions at the particles. We study the behavior of \(u_\varepsilon\) for small \(\varepsilon\), depending on the diameter \(\varepsilon^\alpha\), \(\alpha > 1\), of the particles and the viscosity \(\varepsilon^\gamma\), \(\gamma > 0\), of the fluid. We prove quantitative convergence results for \(u_\varepsilon\) in all regimes when the local Reynolds number at the particles is negligible. Then, the particles approximately exert a linear friction force on the fluid. The obtained effective macroscopic equations depend on the order of magnitude of the collective friction. We obtain a) the Euler-Brinkman equations in the critical regime, b) the Euler equations in the subcritical regime and c) Darcy's law in the supercritical regime.
ISSN:2331-8422