Of the two-phase mixture models used to study deflagration-to-detonation transition in granular explosives, the Baer–Nunziato model is the most highly developed. It allows for unequal phase velocities and phase pressures, and includes source terms for drag and compaction that strive to erase velocity and pressure disequilibria. Since typical time scales associated with the equilibrating processes are small, source terms are stiff. This stiffness motivates the present work where we derive two reduced models in sequence, one with a single velocity and the other with both a single velocity and a single pressure. These reductions constitute outer solutions in the sense of matched asymptotic expansions, with the corresponding inner layers being just the partly dispersed shocks of the full model. The reduced models are hyperbolic and are mechanically as well as thermodynamically consistent with the parent model. However, they cannot be expressed in conservation form and hence require a regularization in order to fully specify the jump conditions across shock waves. Analysis of the inner layers of the full model provides one such regularization [Kapila et al., Phys. Fluids 9, 3885 (1997)], although other choices are also possible. Dissipation associated with degrees of freedom that have been eliminated is restricted to the thin layers and is accounted for by the jump conditions.
The two-phase mixture model developed by Baer and Nunziato ͑BN͒ to study the deflagration-to-detonation transition ͑DDT͒ in granular explosives is critically reviewed. The continuum-mixture theory foundation of the model is examined, with particular attention paid to the manner in which its constitutive functions are formulated. Connections between the mechanical and energetic phenomena occurring at the scales of the grains, and their manifestations on the continuum averaged scale, are explored. The nature and extent of approximations inherent in formulating the constitutive terms, and their domain of applicability, are clarified. Deficiencies and inconsistencies in the derivation are cited, and improvements suggested. It is emphasized that the entropy inequality constrains but does not uniquely determine the phase interaction terms. The resulting flexibility is exploited to suggest improved forms for the phase interactions. These improved forms better treat the energy associated with the dynamic compaction of the bed and the single-phase limits of the model. Companion papers of this study ͓Kapila et al., Phys. Fluids 9, 3885 ͑1997͒; Kapila et al., in preparation; Son et al., in preparation͔ examine simpler, reduced models, in which the fine scales of velocity and pressure disequilibrium between the phases allow the corresponding relaxation zones to be treated as discontinuities that need not be resolved in a numerical computation.
An analytical steady-state theory of the detonation ‘diameter effect’ is presented. This theory, which includes the off-axis flow, is a generalization of the Wood-Kirkwood analysis. When the state dependence of the reaction rate is stronger than that of the product of the sound speed squared and the flow divergence, detonation failure can occur. The leading term in the extrapolation of the detonation velocity to infinite charge size is quadratic in the inverse charge size and not linear as popularly believed. When calibrated to the detonation velocity vs. charge-size data, the theory reproduces the limited amount of experimental shock loci to a high degree of accuracy.
The structure of the velocity relaxation zone in a hyperbolic, nonconservative, two-phase model is examined in the limit of large drag, and in the context of the problem of deflagration-to-detonation transition in a granular explosive. The primary motivation for the study is the desire to relate the end states across the relaxation zone, which can then be treated as a discontinuity in a reduced, equivelocity model, that is computationally more efficient than its parent. In contrast to a conservative system, where end states across thin zones of rapid variation are determined principally by algebraic statements of conservation, the nonconservative character of the present system requires an explicit consideration of the structure. Starting with the minimum admissible wave speed, the structure is mapped out as the wave speed increases. Several critical wave speeds corresponding to changes in the structure are identified. The archetypal structure is partly dispersed, monotonic, and involves conventional hydrodynamic shocks in one or both phases. The picture is reminiscent of, but more complex than, what is observed in such ͑simpler͒ two-phase media as a dusty gas.
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