[Sur un système de réaction–diffusion avec une frontière libre de réaction pénétrant la micro-structure]
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 avec celle de la réaction rapide , on ne change pas le modèle limite. Des résultats numériques sont également presentés.
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 and then take the homogenisation limit 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 and then , or vice-versa. Finally, we illustrate numerically the solution behaviour of the two-scale model in case of a fast reaction.
Accepté le :
Publié le :
Mot clés : Milieux poreux, Modèle à deux échelles, Homogénéisation, Réaction rapide, Frontière libre
@article{CRMECA_2008__336_6_481_0,
author = {Sebastian A. Meier and Adrian Muntean},
title = {A two-scale reaction{\textendash}diffusion system with micro-cell reaction concentrated on a free boundary},
journal = {Comptes Rendus. M\'ecanique},
pages = {481--486},
publisher = {Elsevier},
volume = {336},
number = {6},
year = {2008},
doi = {10.1016/j.crme.2008.02.012},
language = {en},
}
TY - JOUR AU - Sebastian A. Meier AU - Adrian Muntean TI - A two-scale reaction–diffusion system with micro-cell reaction concentrated on a free boundary JO - Comptes Rendus. Mécanique PY - 2008 SP - 481 EP - 486 VL - 336 IS - 6 PB - Elsevier DO - 10.1016/j.crme.2008.02.012 LA - en ID - CRMECA_2008__336_6_481_0 ER -
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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Commentaires - Politique
A homogenisation-based two-scale model for reactive transport in media with evolving microstructure
Sebastian Meier
C. R. Méca (2008)
Homogenisation of a chemical degradation mechanism inducing an evolving microstructure
Malte A. Peter
C. R. Méca (2007)