On the detonation of a combustible gas
HTML articles powered by AMS MathViewer
- by Robert A. Gardner PDF
- Trans. Amer. Math. Soc. 277 (1983), 431-468 Request permission
Abstract:
This paper is concerned with the existence of detonation waves for a combustible gas. The equations are those of a viscous, heat conducting, polytropic gas coupled with an additional equation which governs the evolution of the mass fraction of the unburned gas (see (1)). The reaction is assumed to be of the simplest form: $A \to B$, i.e., there is a single product and a single reactant. The main result (see Theorem 2.1) is a rigorous existence theorem for strong, and under certain conditions, weak detonation waves for explicit ranges of the viscosity, heat conduction, and species diffusion coefficients. In other words, a class of admissible "viscosity matrices" is determined. The problem reduces to finding an orbit of an associated system of four ordinary differential equations which connects two distinct critical points. The proof employs topological methods, including Conley’s index of isolated invariant sets.References
- Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
- Charles C. Conley and Joel A. Smoller, Shock waves as limits of progressive wave solutions of higher order equations, Comm. Pure Appl. Math. 24 (1971), 459–472. MR 283414, DOI 10.1002/cpa.3160240402
- Charles C. Conley and Joel A. Smoller, On the structure of magnetohydrodynamic shock waves, Comm. Pure Appl. Math. 27 (1974), 367–375. MR 368586, DOI 10.1002/cpa.3160270306
- R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948. MR 0029615
- Paul C. Fife, Propagating fronts in reactive media, Nonlinear problems: present and future (Los Alamos, N.M., 1981) North-Holland Math. Stud., vol. 61, North-Holland, Amsterdam-New York, 1982, pp. 267–285. MR 675637
- Andrew Majda, A qualitative model for dynamic combustion, SIAM J. Appl. Math. 41 (1981), no. 1, 70–93. MR 622874, DOI 10.1137/0141006
- Rodolfo R. Rosales and Andrew Majda, Weakly nonlinear detonation waves, SIAM J. Appl. Math. 43 (1983), no. 5, 1086–1118. MR 718631, DOI 10.1137/0143071
- J. A. Smoller and R. Shapiro, Dispersion and shock-wave structure, J. Differential Equations 44 (1982), no. 2, 281–305. Special issue dedicated to J. P. LaSalle. MR 657783, DOI 10.1016/0022-0396(82)90018-3 F. Williams, Combustion theory, Addison-Wesley, Reading, Mass., 1965. Ia. B. Zeldovich and A. S. Kampaneets, The theory of detonation, Academic Press, New York, 1960.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 277 (1983), 431-468
- MSC: Primary 35L67; Secondary 58E07, 76L05
- DOI: https://doi.org/10.1090/S0002-9947-1983-0694370-1
- MathSciNet review: 694370