Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers

We study the homogenization of a reaction-diffusion-convection system posed in an e-periodic d-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 d>0 of the thin layer, we homogenize the presence of the microstructures (the classical periodic homogenization limit e¿0); (2) In the homogenized problem, we pass to d¿0 (the vanishing limit of the layer's width). In this way, we are preparing the stage for the simultaneous homogenization (e¿0) and dimension reduction limit (d¿0) with d=d(¿). 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. Keywords: Homogenization, dimension reduction, thin layers, filtration combustion, two-scale convergence, anisotropic singular perturbations.