An asymptotic theory for hot spot formation and transition to detonation for reactive granular materials

The transition to detonation in energetic granular materials is a complex multiphase process. The idea is proposed and developed that, under appropriate conditions, the resonant interaction of a fast-moving compaction burning front with a gas acoustic mode is an essential process in triggering the transition to detonation. The approach involves the study of solutions of a simple quantitative asymptotic model consisting of two nonlinear coupled partial differential equations that has been derived recently in an asymptotic limit from the full reactive multiphase equations. These equations retain some of the resonant mechanisms involving the gas acoustics and the compaction burn front. An analysis of the properties of solutions of the simplified system is developed. This analysis, involving simple resonant mechanisms, is able to qualitatively capture a number of phenomena observed in experiments and large-scale simulations for the equations for multiphase flow. The phenomena documented here include development of the transition to detonation in the region trailing the burning front, development of the resonant gas acoustic hot spot within the burning front, and stability of the fast moving burning front without transition. All these cases occur in the model in parameter regimes with physical significance that is identified through the asymptotic procedure.