On dynamics of gasless combustion in slowly varying periodic media

In this paper we consider a classical model of gasless combustion in a one dimensional formulation under the assumption of ignition temperature kinetics. We study the propagation of flame fronts in this model when the initial distribution of the solid fuel is a spatially periodic function that varies on a large scale. It is shown that in certain parametric regimes the model supports periodic traveling fronts. An accurate asymptotic formula for the velocity of the flame front is derived and studied. The stability of periodic fronts is also explored, and a critical condition in terms of parameters of the problem is derived. It is also shown that the instability of periodic fronts, in certain parametric regimes, results in a propagation-extinction-conduction-reignition pattern which is studied numerically. Novelty and significance statement: This work provides a closed form asymptotic description of periodic traveling fronts in a gasless combustion model with step-wise ignition temperature kinetics with a slowly varying concentration field. The stability analysis is performed, and the range of applicability of asymptotic formulas is given. A new propagation-extinction-conduction-reignition regime is identified. This regime emerges exclusively due to periodicity of the concentration field.