On the steady and breathing propagation of 2D flame-balls

In this article, we examine the combustion of 2D flame-balls inside of the Hele-Shaw cell. Experiments have shown that the propagation of such structures occurs in a plethora of different regimes. Most of them remain unexplained. Here, we investigate the steady and cyclic dynamics numerically. First, we use a parametric study to numerically derive the stable manifold in the plane defined by the imposed inflow velocity and the heat-loss intensity. The conditions of observability are identified, involving a minimal velocity of propagation. Based on these results, the topology of the space of phases around these conditions is explored utilizing an eigenvalue analysis. It was concluded that cyclic solutions are a transient regime. In addition, its appearance results from a subcritical Hopf bifurcation. These conclusions are also sustained through qualitative analytic discussions.