Two-phase viscous modeling of compaction of granular materials

Abstract

An inviscid model for deflagration-to-detonation transition in granular energetic materials is extended by addition of explicit intraphase momenta and energy diffusion so as to (1) enable the use of a straightforward numerical scheme, (2) avoid prediction of structures with smaller length scales than the component grains, and (3) have a model prepared to describe long time scale transients that are present in some slow processes which can lead to detonation. The model is shown to be parabolic, frame invariant, and dissipative. Consideration of the characteristics for cases with and without intraphase diffusion indicate what boundary conditions are necessary for a well posed problem. A simple numerical method, based on a method of lines applied to the nonconservative form of the equations, is shown to predict convergence at the proper rate to unique solutions which agree well with known solutions for an unsteady inviscid shock tube and a steady piston-driven viscous shock. A series of simulations of inert piston-driven subsonic compaction waves in which the additional mechanisms of interphase compaction, drag, and heat transfer are systematically introduced at an order of magnitude suggested by experiments reveals that interphase drag and heat transfer equilibrate velocities and temperatures, and that compaction equilibrates solid and configurational stresses. At higher piston velocities, supersonic shock and compaction waves are induced; comparison of predictions with and without viscosity demonstrate some of the computational advantages of explicit inclusion of diffusion. The local dissipation rates for each mechanism are quantified, and it is seen that dissipation due to compaction dominates that due to intraphase and interphase transport of linear momenta and energy, suggesting that compaction is the key mechanism in inducing the transition to detonation in piston-driven experiments.