Article Contents

2014, Volume 9, Issue 4: 709-737. Doi: 10.3934/nhm.2014.9.709

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Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers

1.
Department of Mathematics and Computer Science, CASA - Center for Analysis, Scientific computing and Applications, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven
2.
Graduate School of Advanced Mathematical Science, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525
3.
CASA - Centre for Analysis, Scientific computing and Applications, Department of Mathematics and Computer Science, Institute of Complex Molecular Systems, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven

Received: February 2014

Revised: September 2014

Published: December 2014

Abstract / Introduction Related Papers Cited by

Abstract

We study the homogenization of a reaction-diffusion-convection system posed in an $\varepsilon$-periodic $\delta$-thin layer made of a two-component (solid-air) composite material. The microscopic system includes heat flow, diffusion and convection coupled with a nonlinear surface chemical reaction. We treat two distinct asymptotic scenarios: (1) For a fixed width $\delta>0$ of the thin layer, we homogenize the presence of the microstructures (the classical periodic homogenization limit $\varepsilon\to 0$); (2) In the homogenized problem, we pass to $\delta\to 0$ (the vanishing limit of the layer's width). In this way, we are preparing the stage for the simultaneous homogenization ($\varepsilon\to 0$) and dimension reduction limit ($\delta\to 0$) with $\delta=\delta(\epsilon)$. We recover the reduced macroscopic equations from [25] with precise formulas for the effective transport and reaction coefficients. We complement the analytical results with a few simulations of a case study in smoldering combustion. The chosen multiscale scenario is relevant for a large variety of practical applications ranging from the forecast of the response to fire of refractory concrete, the microstructure design of resistance-to-heat ceramic-based materials for engines, to the smoldering combustion of thin porous samples under microgravity conditions.

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Mathematics Subject Classification: Primary: 35B27; Secondary: 76S05, 74E30, 80A25, 35B25, 35B40.

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