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Comptes Rendus Mathématique
Volume 344, n° 8
pages 523-528 (avril 2007)
Doi : 10.1016/j.crma.2007.03.008
Received : 7 January 2007 ;  accepted : 6 Mars 2007
Homogenization of a convection-diffusion model with reaction in a porous medium
Homogénéisation dʼun modèle de convection-diffusion avec réaction en milieu poreux
 

Grégoire Allaire , Anne-Lise Raphael
Centre de mathématiques appliquées, École polytechnique, 91128 Palaiseau cedex, France 

Abstract

We study the homogenization of a convection-diffusion equation with reaction in a porous medium when both the Péclet and Damkohler numbers are large. We prove that, up to a large drift, the homogenized equation is a diffusion equation. Our method is based on a factorization principle and two-scale convergence. The main consequence is that we obtain rigorous definitions of homogenized coefficients which justify heuristic arguments in the method of volume averaging. We perform 2-d numerical computations of the diffusion-dispersion homogenized coefficient which are in very good agreement with previous results obtained by the method of volume averaging. To cite this article: G. Allaire, A.-L. Raphael, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

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Résumé

On étudie lʼhomogénéisation dʼun problème de convection-diffusion avec réaction en milieu poreux lorsque les nombres de Péclet et de Damkohler sont grands. Nous démontrons que, dans un repère dérivant à grande vitesse, lʼéquation homogénéisée est une équation de diffusion. Notre méthode est basée sur un principe de factorisation et sur la convergence à deux échelles. La conséquence pratique la plus importante est que nous obtenons ainsi une définition rigoureuse des coefficients homogénéisés qui justife des arguments heuristiques utilisés dans la méthode de la prise de moyenne. Nous avons effectué des calculs numériques en 2-d du coefficient homogénéisé de diffusion-dispersion qui donnent des valeurs très semblables à celles obtenues par prise de moyenne. Pour citer cet article : G. Allaire, A.-L. Raphael, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

The full text of this article is available in PDF format.


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