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The vanishing viscosity limit in the presence of a porous medium

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Abstract

We consider the flow of a viscous, incompressible, Newtonian fluid in a perforated domain in the plane. The domain is the exterior of a regular lattice of rigid particles. We study the simultaneous limit of vanishing particle size and distance, and of vanishing viscosity. Under suitable conditions on the particle size, particle distance, and viscosity, we prove that solutions of the Navier–Stokes system in the perforated domain converges to solutions of the Euler system, modeling inviscid, incompressible flow, in the full plane. That is, the flow is not disturbed by the porous medium and becomes inviscid in the limit. Convergence is obtained in the energy norm with explicit rates of convergence.

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Acknowledgments

A. M. would like to acknowledge the hospitality and financial support of Université Paris-Diderot (Paris 7), Université Pierre et Marie Curie (Paris 6), and the financial support of Paris City Hall and the Fondation Sciences Mathématiques de Paris, to conduct part of this work. The authors are also grateful to the anonymous referee for his valuable comments on the first version of this article.

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Authors and Affiliations

  1. Univ Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Univ Paris 06, 75013, Paris, France

    Christophe Lacave

  2. Penn State University, University Park, PA, 16802, USA

    Anna L. Mazzucato

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  1. Christophe Lacave
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  2. Anna L. Mazzucato
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Correspondence to Christophe Lacave.

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C. Lacave is partially supported by the Agence Nationale de la Recherche, Project DYFICOLTI Grant ANR-13-BS01-0003-01, and by the project Instabilities in Hydrodynamics funded by Paris City Hall (program Emergences) and the Fondation Sciences Mathématiques de Paris. A. Mazzucato was partially supported by the US National Science Foundation Grants DMS-1009713, DMS-1009714, and DMS-1312727.

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Lacave, C., Mazzucato, A.L. The vanishing viscosity limit in the presence of a porous medium. Math. Ann. 365, 1527–1557 (2016). https://doi.org/10.1007/s00208-015-1313-x

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  • DOI: https://doi.org/10.1007/s00208-015-1313-x

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