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(e). We recover the reduced macroscopic equations from [21] 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.