Homogenization of the Navier-Stokes equations in perforated domains in the inviscid limit
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929548294094848 |
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| author | Höfer, Richard M. |
| author_facet | Höfer, Richard M. |
| contents | 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 $\varepsilon^α$, $α> 1$, of the particles and the viscosity $\varepsilon^γ$, $γ> 0$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_06075 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Homogenization of the Navier-Stokes equations in perforated domains in the inviscid limit Höfer, Richard M. Analysis of PDEs 35Q30, 35Q31, 76D07, 76M50, 76S05, 76T25 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 $\varepsilon^α$, $α> 1$, of the particles and the viscosity $\varepsilon^γ$, $γ> 0$, 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. |
| title | Homogenization of the Navier-Stokes equations in perforated domains in the inviscid limit |
| topic | Analysis of PDEs 35Q30, 35Q31, 76D07, 76M50, 76S05, 76T25 |
| url | https://arxiv.org/abs/2209.06075 |