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

Presented by: 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
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.

Received: 2008-01-17
Accepted: 2008-02-28
Published online: 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

Author's affiliations:

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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     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},
}

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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/

References
Cited by

[1] Y. Nishiura Far-from-Equilibrium Dynamics, Translations of Mathematical Monographs, vol. 209, Amer. Math. Soc., Providence, RI, 2002

[2] S.A. Meier, M.A. Peter, A. Muntean, M. Böhm, J. Kropp, A two-scale approach to concrete carbonation, in: Proc. Int. RILEM Workshop on Integral Service Life Modeling of Concrete Structures, 2007, Guimares, Portugal, pp. 3–10

[3] T.L. van Noorden, I.S. Pop, A Stefan problem modelling crystal dissolution and precipitation, IMA J. Appl. Math., in press

[4] A. Friedman; A.T. Tzavaras A quasilinear parabolic system arising in modeling of catalytic reactors, J. Differential Equations, Volume 70 (1987), pp. 167-196

[5] M.A. Murad; J.H. Cushman Multiscale flow and deformation in hydrophilic swelling porous media, Int. J. Engrg. Sci., Volume 34 (1996) no. 3, pp. 313-338

[6] T. Arbogast; J. Douglas; U. Hornung Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., Volume 21 (1990) no. 4, pp. 823-836

[7] U. Hornung; W. Jäger; A. Mikelić Reactive transport through an array of cells with semi-permeable membranes, RAIRO Modél. Math. Anal. Numér., Volume 8 (1994) no. 1, pp. 59-94

[8] A. Fasano; M. Primicerio; R. Ricci Limiting behaviour of some problems in diffusive penetration, Rend. Mat. Ser. VII, Volume 10 (1990), pp. 39-57

[9] D. Hilhorst; R. van der Hout; L.A. Peletier The fast reaction limit for a reaction–diffusion system, J. Math. Anal. Appl., Volume 199 (1996), pp. 349-373

[10] A. Muntean, S.A. Meier, Existence of weak solutions to a two-scale reaction–diffusion system with micro-cell reaction concentrated on a free boundary, in preparation

[11] D. Ciorănescu; P. Donato An Introduction to Homogenization, Oxford University Press, 1999

[12] E. Sanchez-Palencia Non-Homogeneous Media and Vibration Theory, Springer-Verlag, Berlin, 1980

[13] T. van Noorden, Crystal precipitation and dissolution in a thin strip, CASA report 07-30, Eindhoven University of Technology, 2007

[14] C. Eck; P. Knabner; S. Korotov A two-scale method for the computation of solid–liquid phase transitions with dendritic microstructure, J. Comput. Phys., Volume 178 (2002), pp. 58-80

[15] M.A. Peter Homogenisation of a chemical degradation mechanism inducing an evolving microstructure, C. R. Mecanique, Volume 335 (2007) no. 11, pp. 679-684

[16] S.A. Meier, Two-scale models of reactive transport in porous media involving microstructural changes, PhD thesis, University of Bremen, Germany, in preparation

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