Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers
Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers
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DOI：
10.3934/nhm.2014.9.709
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发表时间：
2014-12
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影响因子：
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通讯作者：
T. Fatima;Ekeoma R. Ijioma;T. Ogawa;A. Muntean
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作者：
T. Fatima;Ekeoma R. Ijioma;T. Ogawa;A. Muntean
我们研究了由双组分（固体-空气）复合材料制成的$\varepsilon$ -周期$\delta$ -薄层中反应-扩散-对流系统的均匀化。微观系统包括热流、扩散和对流，并伴有非线性表面化学反应。我们处理两种不同的渐近情形：(1)对于固定宽度$\delta>0$的薄层，我们均匀化微观结构的存在（经典的周期均匀化极限$\varepsilon\to 0$）；(2)在均质化问题中，我们通过$\delta\to 0$（层宽度的消失极限）。通过这种方式，我们正在准备与$\delta=\delta(\epsilon)$同时均质化（$\varepsilon\to 0$）和降维极限（$\delta\to 0$）的舞台。我们用精确的有效输运系数和反应系数公式恢复了[25]的简化宏观方程。我们用阴燃燃烧的几个模拟案例来补充分析结果。所选择的多尺度场景与各种实际应用相关，从耐火混凝土的火灾响应预测，发动机耐热陶瓷基材料的微观结构设计，到微重力条件下薄多孔样品的阴燃。
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.