Traveling Waves in Natural Counterflow Filtration Combustion and Their Stability

We consider two-dimensional filtration combustion in a porous medium in which an exothermic reaction takes place between the solid and a pure gaseous oxidant which is delivered to the reaction zone by filtration through the pores of the medium. As a result of the reaction, oxidant is consumed and a solid product is formed. The consumption of gas in the reaction causes a pressure gradient which drives filtration. Since no external forcing is required, this arrangement is termed natural filtration combustion. The samples are assumed to be open to gas permeation at one end with ignition at the other end so that gas flow is opposite to the direction of reaction propagation. This configuration is termed counterflow, so we study natural counterflow filtration combustion. This reaction scheme and configuration describe conditions of self-propagating high-temperature synthesis (SHS), in which combustion waves are employed to synthesize advanced materials. Asymptotic solutions describing traveling waves are determined using a thin reaction zone approximation valid for large activation energy. Two regimes, distinguished by whether gas or solid reactant is the deficient component of the reaction, are analyzed. Linear stability analysis of the traveling wave solutions predicts three types of instabilities. In the case of gas deficient combustion, a cellular instability is found when the Zeldovich number Z exceeds a critical value Z_c^g. For the case of solid deficient combustion, a pulsating instability, similar to that in gasless solid fuel combustion, arises for Z > Z_p^s. For solid deficient combustion with very little excess gas, i.e., close to stoichiometric conditions, we also find a cellular instability, which occurs for Z < Z_c^s, with Z_c^s large. Direct numerical simulation of the time-dependent system validates the asymptotic solution and the linear stability results. Moving beyond the stability boundary into the unstable region yields uniformly propagating cellular structures. In addition to single-front cellular structures, we find an apparently new two-front cellular structure. The number of cells increases with the cross section of the sample, with the activation energy, and with a decrease in the ambient gas pressure.