A simple model of two-dimensional solid flame microstructure

In Beck and Volpert (2003 Physica D 182 86), we presented a model of condensed phase combustion that attempts to elucidate the effects of spatially localized reaction sites on the propagation of a combustion wave. Heat transfer was assumed to be uniform but the exothermic reactions are allowed to occur only at evenly distributed locations. Under certain simplifying assumptions we showed that the propagation of a one-dimensional combustion wave through such a medium may be described by a history dependent iterative map →f : R∞ → R∞ that relates the times of all previous particle ignitions to the time of the next ignition. In this paper, we consider the propagation of a two-dimensional wave through a channel. For a propagating wave which is semi-planar, we show that the dynamics of the continuous system may be represented by a history dependent map similar to →f. Iteration of this map demonstrates the existence of a wide variety of travelling waves. Furthermore, a linear stability analysis similar to that of Beck and Volpert shows that steady planar waves undergo a period doubling bifurcation as the system transitions to a region of bi-stability in which chaotic behaviour is observed. Stochastic perturbations to particle location are also discussed.