Nonlinear analysis of pulsating instabilities in diffusion flames

Recent experiments of diffusion flames have identified fascinating nonlinear behavior, especially near the extinction limit. In this work, we carry out a systematic analysis of pulsating instabilities in diffusion flames. We consider the simple geometric configuration of a planar diffusion flame situated in a channel at the interface between a fuel being supplied from below, and oxidant diffusing in from a stream above. Lewis numbers are assumed greater than unity in order to focus on pulsating instabilities. We employ the asymptotic theory of Cheatham and Matalon, and carry out a bifurcation analysis to derive a nonlinear evolution equation for the amplitude of the perturbation. Our analysis predicts three possible flame responses. The planar flame may be stable, such that perturbations decay to zero. Second, the amplitude of a perturbation can eventually become unbounded in finite time, indicating flame quenching. And finally, our amplitude equation possesses time-periodic (limit cycle) solutions, although we find that this regime cannot be realized for parameter values typical of combustion systems. The various possible burning regimes are mapped out in parameter space, and our results are consistent with available experimental and numerical data.