Homogenization approach to filtration through a fibrous medium

Mohamed Belhadj; Eric Cancès; Jean-Frédéric Gerbeau; Andro Mikelić; Mohamed Belhadj; Eric Cancès; Jean-Frédéric Gerbeau; Andro Mikelić

doi:10.3934/nhm.2007.2.529

Networks and Heterogeneous Media

2007, Volume 2, Issue 3: 529-550. doi: 10.3934/nhm.2007.2.529

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Homogenization approach to filtration through a fibrous medium

1.
INRIA Rocquencourt, B.P. 105, F-78153, Le Chesnay Cedex
2.
CERMICS, École Nationale des Ponts et Chaussées, 6 et 8, avenue Blaise Pascal, Cité Descartes - Champs sur Marne, F-77455 Marne La Vallée Cedex 2
3.
INRIA- Project M3N, Rocquencort B.P. 105, F-78153 Le Chesnay Cedex
4.
Université de Lyon, Lyon, F-69003, UFR Mathématiques, Site de Gerland, Bat. A, 50, avenue Tony Garnier, 69367 Lyon Cedex 07

Received: 01 May 2007 Revised: 01 June 2007
76M50, 76S05, 74Q15, 35B27.

We study the flow through fibrous media using homogenization techniques. The fibre network under study is the one already used by M. Briane in the context of heat conduction of biological tissues. We derive and justify the effective Darcy equation and the permeability tensor for such fibrous media. The theoretical results on the permeability are illustrated by some numerical simulations. Finally, the low solid fraction limit is considered. Applying results by G. Allaire to our setting, we justify rigorously the leading order term in the empirical formulas for the effective permeability used in engineering. The results are also confirmed by a direct numerical calculation of the permeability, in which the small diameter of the fibres requires high accuracy approximations.
- Homogenization,
- Filtration,
- Fibrous Porous Media,
- Quasi-Periodic,
- Darcy Equation,
- Permeability,
- Low Solid Fraction
Citation: Mohamed Belhadj, Eric Cancès, Jean-Frédéric Gerbeau, Andro Mikelić. Homogenization approach to filtration through a fibrous medium[J]. Networks and Heterogeneous Media, 2007, 2(3): 529-550. doi: 10.3934/nhm.2007.2.529

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Abstract

We study the flow through fibrous media using homogenization techniques. The fibre network under study is the one already used by M. Briane in the context of heat conduction of biological tissues. We derive and justify the effective Darcy equation and the permeability tensor for such fibrous media. The theoretical results on the permeability are illustrated by some numerical simulations. Finally, the low solid fraction limit is considered. Applying results by G. Allaire to our setting, we justify rigorously the leading order term in the empirical formulas for the effective permeability used in engineering. The results are also confirmed by a direct numerical calculation of the permeability, in which the small diameter of the fibres requires high accuracy approximations.
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© 2007 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)