Spiral Gasless Condensed Phase Combustion

This paper studies combustion fronts propagating on and in a thin disk composed of a solid combustible mixture after ignition in a cylindrical region of radius r₀. A basic solution is constructed that describes an outward propagating circular front of radius R( τ ) on the upper surface of the disk. Since the scales of the combustion process (diffusion and reaction) are much smaller than even r₀, a perturbation analysis is employed for largeR. This analysis corresponds to the situation in which the conditions of ignition no longer affect propagation. It is found, to leading order, that the front propagates at constant velocity dR/dτ, and the temperature distribution on the front is independent of the angle. Correction terms proportional to the curvature of the front are calculated. A stability analysis of the basic solution indicates, in accordance with experimental observations, that, when the expanding circular front attains a critical size, the basic solution becomes unstable and a hot spot (temperature maximum) appears, which then travels along the front. To leading order, the velocity of the hot spot along the front is constant. The hot spot thus traces out a trajectory, which, to leading order, is an Archimedean spiral. As the front expands yet further, more and more hot spots appear, which may ultimately form a luminous circle. Spiral trajectories, as well as the appearance of an increasing number of hot spots and luminous circles, have been experimentally observed. A brief discussion is presented on the effect of axial perturbations, to describe propagation throughout the disk. In this discussion, the experimentally observed fact that the patterns on the upper and lower surfaces of the disk are the same except for a possible phase shift is described.