On the route to extinction in nonadiabatic solid flames

We consider nonadiabatic gasless solid fuel combustion employing a reaction sheet model. We derive an integrodifferential equation for the location of the interface separating the fresh fuel from the burned products. There are two parameters in our model, the Zeldovich number Z, related to the activation energy of the exothermic chemical reaction, and the heat loss parameter F. For any value of Z there is an extinction limit Γ _m, so that if Γ > Γ _m, the combustion wave cannot be sustained. For all values of Z and Γ < Γ _m the model admits a uniformly propagating combustion wave. This solution is subject to a pulsating instability for Z sufficiently large. The effect of heat losses is destabilizing in the sense that pulsations occur for smaller values of Z when heat loss is considered. We consider the dynamics of the combustion wave as Γ increases, thus, describing the dynamics of the model on the route to extinction. We consider values of Z below the adiabatic stability limit, so that for Γ = 0 the only stable steady state solution is the uniformly propagating combustion wave. We find that for Z near the adiabatic stability limit, the effect of heat loss is to promote a period doubling cascade leading to chaotic behavior prior to extinction. We also find an interval of laminar behavior within the chaotic window, corresponding to a secondary period doubling sequence. Specifically, we find solutions of period 12T, 24T, 48T. We show that for smaller values of Z the full period doubled sequence does not necessarily occur. Rather, extinction follows after a finite, possibly small, number of periodic solutions.