A two-scale reaction–diffusion system with micro-cell reaction concentrated on a free boundary

Sebastian A. Meier; Adrian Muntean

doi:10.1016/j.crme.2008.02.012

A two-scale reaction–diffusion system with micro-cell reaction concentrated on a free boundary
[Sur un système de réaction–diffusion avec une frontière libre de réaction pénétrant la micro-structure]

Présenté par : Évariste Sanchez-Palencia

Sebastian A. Meier ¹ ; Adrian Muntean ²

¹ Centre for Industrial Mathematics (ZeTeM), FB3, University of Bremen, Postfach 330 440, 28334 Bremen, Germany
² CASA-Centre for Analysis, Scientific Computing and Applications, Department of Mathematics and Computer Science, Technical University of Eindhoven, PO Box 513, 5600 MB Eindhoven, The Netherlands

Comptes Rendus. Mécanique, Volume 336 (2008) no. 6, pp. 481-486.

Abstract (VO)
Résumé

We discuss the fast-reaction limit of a two-scale reaction–diffusion model. We point out that if the reaction constant a explodes to infinity, then a two-scale PDE system with free boundary at the micro cell is obtained. The aim of this note is to answer the question: Can the same two-scale free-boundary problem be obtained if we first pass to the fast-reaction limit $a \to \infty$ and then take the homogenisation limit $ε \to 0$ that is behind the derivation of the two-scale model? Here ε is the width of a thin two-dimensional strip. Using the method of asymptotic expansions, we show that it does not matter whether we first take $ε \to 0$ and then $a \to \infty$ , or vice-versa. Finally, we illustrate numerically the solution behaviour of the two-scale model in case of a fast reaction.

On considère un modèle de réaction-diffusion à deux échelles, dont la micro-structure contient une réaction rapide. Lorsque la constante de réaction a explose vers l'infini, le modèle à deux échelles converge vers un modèle à frontière libre concentrée dans la micro-structure. Le but de cette Note est de montrer qu'en échangeant la limite d'homogénéisation $ε \to 0$ avec celle de la réaction rapide $a \to \infty$ , on ne change pas le modèle limite. Des résultats numériques sont également presentés.

Reçu le : 2008-01-17
Accepté le : 2008-02-28
Publié le : 2008-04-18

DOI : 10.1016/j.crme.2008.02.012

Keywords: Porous media, Two-scale model, Homogenisation, Fast reaction, Free-boundary problem
Mots-clés : Milieux poreux, Modèle à deux échelles, Homogénéisation, Réaction rapide, Frontière libre

Affiliations des auteurs :

Sebastian A. Meier ¹ ; Adrian Muntean ²

¹ Centre for Industrial Mathematics (ZeTeM), FB3, University of Bremen, Postfach 330 440, 28334 Bremen, Germany
² CASA-Centre for Analysis, Scientific Computing and Applications, Department of Mathematics and Computer Science, Technical University of Eindhoven, PO Box 513, 5600 MB Eindhoven, The Netherlands

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     title = {A two-scale reaction{\textendash}diffusion system with micro-cell reaction concentrated on a free boundary},
     journal = {Comptes Rendus. M\'ecanique},
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     doi = {10.1016/j.crme.2008.02.012},
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Sebastian A. Meier; Adrian Muntean. A two-scale reaction–diffusion system with micro-cell reaction concentrated on a free boundary. Comptes Rendus. Mécanique, Volume 336 (2008) no. 6, pp. 481-486. doi : 10.1016/j.crme.2008.02.012. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.02.012/

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