Mathematical analysis of a two-phase continuum mixture theory. (English) Zbl 0760.76096
Summary: We study the mathematical structure of a continuum reactive mixture model of the combustion of granular energetic materials. We obtain and classify the wave fields associated with this description. This analysis shows that this system of hyperbolic equations becomes degenerate when the relative flow is locally sonic. We derive the corresponding Riemann invariants and construct simple wave solutions. We also discuss special discontinuous solutions of the system of equations. For fixed upstream conditions, different downstream states are possible when the relative velocities exceed the speed of the sound in gas.
MSC:
| 76V05 | Reaction effects in flows |
| 76T99 | Multiphase and multicomponent flows |
| 80A25 | Combustion |
| 80A17 | Thermodynamics of continua |
| 35L67 | Shocks and singularities for hyperbolic equations |
Keywords:
reactive mixture model; granular energetic materials; wave fields; Riemann invariants; speed of the soundReferences:
| [1] | Truesdell, C. and R. Toupin: The Classical Field Theories, Handbuch der Physik (Ed. S. Flugge), Vol. II/1, p. 226, Berlin: Springer 1960 |
| [2] | Truesdell, C.; N. Noll: The Nonlinear Field Theories of Mechanics, Handbuch der Physik (Ed., S. Flugge), Vol. III/3r. Berlin: Springer 1965 · Zbl 0779.73004 |
| [3] | Baer, M.; Nunziato, J.: A Two-Phase Mixture Theory for the Deflagrationto-Detonation Transition (DDT) in Reactive Granular Materials. Int. J. Multiphase Flow 12 (1986) 861-889 · Zbl 0609.76114 · doi:10.1016/0301-9322(86)90033-9 |
| [4] | Stewart, H.; Wendroff, B.: Two-phase flow: models and methods. J. Comp. Phys. 56 (1984) 363-409 · Zbl 0596.76103 · doi:10.1016/0021-9991(84)90103-7 |
| [5] | Hicks, D.: Well-posedness of the two-phase flow problem, part 2: stability analyses and microstructural models. Report SAND 80-1276, Sandia National Lab.: Albuquerque 1980 |
| [6] | Embid, P.; Baer, M.: Modeling two-phase flow of reactive granular materials. IMA Volumes Math. Applications 29 (1991) 58-67 · Zbl 0726.76101 |
| [7] | Embid, P.; Hunter, J.; Majda, A.: Simplified asymptotic, equations for the transition to detonation in reactive granular materials. SIAM J. Appl. Math (to appear) · Zbl 0778.76093 |
| [8] | Embid, P.; Majda, A.: An asymptotic theory for hot spot formation and transition to detonation for reactive granular materials. Combust. Flame (to appear) · Zbl 0778.76093 |
| [9] | Baer, M.; Gross, R.; Nunziato, J.; Igel, E.: An experimental and theoretical study of deflagration-to-detonation transition (DDT) in the granular explosive CP. Combust. Flame 65 (1986) 15-30 · doi:10.1016/0010-2180(86)90068-4 |
| [10] | Baer, M.; Nunziato, I.; Embid, P.: Deflagration to detonation transition in reactive granular materials. Prog. Astronaut. Aeronaut. 135 (1991) 481-512 |
| [11] | Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. New York Springer 1984 · Zbl 0537.76001 |
| [12] | Lax, P.: Hyperbolic System of Conservation Laws II. Comm. Pure Appl. Math. 10 (1957) 537-566 · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 |
| [13] | Courant, R.; Friedrichs, K.: Supersonic Flow and Shock Waves. New York: Springer 1976 · Zbl 0365.76001 |
| [14] | Jeffrey, A.; Taniuti, T.: Non-Linear Wave Propagation. New York: Academic Press 1964 · Zbl 0117.21103 |
| [15] | Majda, A.: Theory of Conservation Laws, unpublished notes, Princeton University 1984 · Zbl 0581.35050 |
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