Multiscale systems abound in science and engineering. Their comprehensive treatment requires accurate mathematical modeling of processes at the fine scales and a mathematical framework for flow of information across scales. This proposal addresses this problem in the context of heterogeneous explosives. These have a complex microstructure with fragments of the energetic material, voids and pores existing within the granular aggregate. When subjected to a sufficiently strong shock, a detonation is initiated. Although the crystalline homogeneous explosive has a high ignition threshold, relatively weaker stimuli are sufficient to initiate the heterogeneous aggregate. This is caused by the appearance of discrete sites, or hot spots, where burning commences and then spreads to consume the entire bulk. The multi-scale nature of the system is a daunting obstacle in the way of any attempt at ab-initio modeling of the detonation phenomena, at least at the present time. Two major approaches to mathematical modeling have been proposed. Both generate continuum equations at the macro scale, wherein a certain degree of homogenization is implicit and fine-scale processes have been included as subgrid models. The first approach, typified by the ignition-and-growth model, treats the explosive as a homogeneous mixture of two distinct constituents, the unreacted explosive and the products of reaction, at pressure and temperature equilibrium. To each constituent is assigned an equation of state, and a single reaction-rate law is postulated for the conversion of the explosive to products. The second approach explicitly recognizes the two-phase character of the explosive mixture. The resulting model has separate balance laws of mass, momentum and energy for each phase, plus a rule that allows compaction of the solid phase driven by pressure difference between the phases. Terms representing interfacial exchange of mass, momentum and energy appear, corresponding to the nonequilibrium processes of reaction, drag and heat transfer. These models are systems of hyperbolic partial differential equations that can be considered as generalizations of the Euler equations of gasdynamics. This proposal is aimed at studies of existing continuum models as well as the investigation of fine-scale phenomena. The primary objective will be to study how the strength and distribution of hot spots depend, in a quantitative way, upon the constitutive properties of the explosive and the size of the igniting stimulus, thus providing important information that can be used in attempts at multi-scale descriptions.
Many areas in science and engineering, including weather, combustion, pollution, and biological systems, among others, involve behavior at the scale of observation that is determined by the events occurring at micro scales. A thorough treatment of such systems requires, on the one hand, a fundamental understanding and accurate mathematical modeling of processes at the micro scales, and on the other, development of a theoretical and computational framework that would facilitate flow of information across scales, so that behavior at the scale of observation can be predicted reliably and accurately. Attention to such systems is particularly timely and apt in the context of science-based stewardship of explosive devices, and it is problems from this arena that form the core of this proposal.
