Nonlinear dynamics in a simple model of solid flame microstructure

We present a model of condensed phase combustion which attempts to elucidate the effects of spatially localized reaction sites on the propagation of a combustion wave. Heat transfer is assumed to be uniform but exothermic reactions are allowed to occur only at evenly distributed locations, that we refer to as reactant particles. Thus, combustion wave propagation manifests itself as a process of sequential ignition and burning of particles. Green's functions are used to show that "steady" wave speed is related to particle ignition temperature, particle geometry and the ratio of heat diffusion to reaction times through a single transcendental equation. Furthermore, for the one-dimensional case, the dynamics of this system can be related to a history dependent implicit map f→:R ^∞→R^∞ which determines time to the next ignition. Iteration of this map demonstrates that average wave speed undergoes a period doubling bifurcation to chaos and subsequent extinction. A linear stability analysis of this map is performed to determine the stability boundaries for period 2ⁿ orbits. Additionally, temperature profiles are shown to be in qualitative agreement with experiments which describe a transition from the so-called quasi-homogeneous to relay-race regimes.