Simplified asymptotic equations for the transition to detonation in reactive granular materials

In this paper a simplified asymptotic system of equations to describe combustion in reactive granular materials is derived. In contrast with one-phase flow, the continuum equations for two-phase flow develop resonant behavior at choked flow states, where one of the gas-acoustic speeds equals the solid particle speed. The asymptotic model is valid near those choked flow states and near ignition temperature conditions, and incorporates nonlinear resonant effects of gas acoustic waves, compaction of the granular material, and grain burning. The asymptotic system is derived with a combination of nonlinear geometrical optics and large activation energy asymptotics near the resonant point. A general abstract derivation of the asymptotic system is provided, and the results are then specialized to the continuum equations for reactive granular materials. Thus, the derivation developed here should prove useful in developing simplified asymptotic equations in other contexts in multiphase flow as well as other physical problems involving the coupling of several complex hyperbolic systems. For the special case of reactive granular materials, a detailed discussion of the physical connections between the asymptotic model and the continuum system is also developed.