Generalized flame balls

We consider generalized flame balls which correspond to stationary spherical flames with a flow of hot inert gas, either a source or a sink, at the origin. Depending on the flow, these flames can have positive, zero, or negative burning speeds, with zero speeds characterizing the Zeldovich flame balls. A full analytical description of these structures and their stability to radial perturbations is provided, using a large activation energy asymptotic approach and a thermo-diffusive approximation. The results are also complemented by a numerical study. The number and stability of the generalized flame balls are identified in various regions of the l-M-h0 space, where l is the (reduced) Lewis number, and M and h0 the flow rate and its enthalpy at the origin, respectively. It is typically found that, when the flow is a source, there is a maximum value of the flow rate Mmax depending on l and h0, above which no stationary solutions exist, and below which there are two solutions characterizing a small stable flame ball and a large unstable flame ball; the implications of these results to the problem of ignition by a hot inert gas stream are discussed. When the flow is a sink, however, there is typically a single unstable solution, except for sufficiently large values of the Lewis number and large negative values of M, where three flame balls exist, the medium one being stable. Finally, the relation between the flame speed, positive or negative, and the flame curvature, small or large, is discussed.