Stability of a spherical flame ball in a porous medium

Gaseous flame balls and their stability to symmetric disturbances are studi ed numerically and asymptotically, for large activation temperature, within a porous medium that serves only to exchange heat with the gas. Heat losse s to a distant ambient environment, affecting only the gas, are taken to be radiative in nature and are represented using two alternative models. One of these treats the heat loss as being constant in the burnt gases and line arizes the radiative law in the unburnt gas (as has been studied elsewhere without the presence of a solid). The other does not distinguish between bu rnt and unburnt gas and is a continuous dimensionless form of Stefan's law, having a linear part that dominates close to ambient temperatures and a fo urth power that dominates at higher temperatures. Numerical results are found to require unusually large activation temperatu res in order to approach the asymptotic results. The latter involve two bra nches of solution, a smaller and a larger flame ball, provided heat losses are not too high. The two radiative heat loss models give completely analog ous steady asymptotic solutions, to leading order, that are also unaffected by the presence of the solid which therefore only influences their stabili ty. For moderate values of the dimensionless heat-transfer time between the solid and gas all flame balls are unstable for Lewis numbers greater than unity. At Lewis numbers less than unity, part of the branch of larger flame balls becomes stable, solutions with the continuous radiative law being st able over a narrower range of parameters. Tn both cases, for moderate heat- transfer times, the stable region is increased by the heat capacity of the solid in a way that amounts, simply, to decreasing an effective Lewis numbe r for determining stability, just as if the heat-transfer time was zero.