Complex dynamics in a simple model of solid flame microstructure

Jeffrey Beck, Northwestern University

We present a simple model of condensed phase combustion which attempts to elucidate the effects of spatially localized reaction sites on the frontal propagation of a combustion wave.  Heat transfer is assumed to be uniform but the exothermic reactions which drive the wave are allowed to occur only at discrete but evenly distributed locations.  Combined with ignition temperature kinetics this results is a model similar to that of integrate-and-fire neural networks.  Since micro-structure is of interest we do not apply the continuum assumption, but rather investigate the stability of waves which propagate through the discrete medium. '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, the dynamics of this system can be related to a history dependent implicit map $\vec{f}:\Rset^\infty \rightarrow \Rset^\infty$.  Iteration of this map demonstrates that average wave speed undergoes a period doubling cascade in one dimension and a period doubling bifurcation with regions of bi-stability and chaos in two dimensions.  We also elucidate a technique which allows for a linear stability analysis of the discrete system capable of obtaining stability boundaries for both steady and periodic waves as well as a discrete "wave number" associated with various growth modes.  The effects of boundary locations which break the symmetry of the grid are also discussed.