Dynamics of smoulder waves near extinction

Smoulder combustion involves a two phase solid/gas reaction inside a porous media that restricts the flow of air. We study opposed flow smoulder, that is, conditions where the smoulder front burns towards an incoming gas that contains oxygen. Thus gas and solid reactants enter the reaction site from the same direction. Extinction occurs when the mass flux of the forced gas is sufficiently high. Asymptotic methods for large activation energy provide us with descriptions of uniformly propagating smoulder waves and their existence (extinction) limit for relevant parameter values. Linear stability analysis shows that the planar uniformly burning smoulder wave can become unstable before extinction. One-dimensional numerical simulations justify these asymptotic results and elucidate the resulting dynamics of the smoulder wave as the incoming gas flux increases. For reasonable parameter values, the system oscillates and then proceeds through a period doubling cascade of bifurcations to chaotic behaviour before extinction occurs. This cascade does not seem to interact with the extinction of dynamic waves as parameter values can be chosen so that extinction occurs anywhere within the cascade.