Abstract
A combustion model which captures the interacting among nonlinear convection, chemical reaction and radiative heat transfer is studied. New phenomena are found with radiative heat transfer present. In particular, there is a detonation wave solution in which there is a sonic point inside the reaction zone. As a consequence, the traveling wave speed cannot be determined before the problem is solved. The shooting method is used to prove the existence of the traveling wave. The condition we shoot at is the compatibility condition at the sonic point. Furthermore, the speed of the detonation wave decreases as the heat loss coefficient increases, as expected physically. We study the time-asymptotic limit of solutions of initial value problem for the same problem. We prove that the solution exists globally and the solution converges uniformly, away from the shock, to a shifted traveling wave solution as t → + ∞ for certain “compact support” initial data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Bukiet, B. (1989). The effect of curvature on detonation speed. SIAM J. Appl. Math. 49, 1433–1446.
Chen, L.-D. Private communications.
Fickett, W., and Davis, W. C. (1979). Detonation, University of California Press.
Courant, R., and Friedrichs, K. O. (1948). Supersonic Flow and Shock Waves, Springer Verlag.
Coddington, E. A., and Levinson, N. (1952). A boundary value problem for a nonlinear differential equation with a small parameter. Proc. Am. Math. Soc. 3, 73–81.
Lax, P. D. (1957). Hyperbolic systems of conservation laws, II. Comm. Pure Appl. Math. 10, 537–566.
Li, T. (1994). On the initiation problem for a combustion model. J. Diff. E. 112, 351–373.
Liu, T. P. (1985). Nonlinear stability of shock waves for viscous conservation laws. Mem. Am. Math. Soc. No. 328.
Hsu, S., and Liu, T.-P. (1990). Nonlinear singular Sturm-Liouville problems and an application to transonic flow through a nozzle. Comm. Pure App. Math. 43, 31–61.
Rosales, R., and Majda, A. (1983). Weakly nonlinear detonation waves. SIAM J. Appl. Math. 43(5), 1086–1118.
Smith, T. F., Byun, K.-H., and Chen, L.-D. (1988). Effects of radiative and conductive transfer on thermal ignition. Combust. Flame 73, 67.
Ying, L. A., and Teng, Z. H. (1984). Riemann problems for a reacting and convection hyperbolic system. J. Approx. Theory Appl. 1, 95–122.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Li, T. Time-Asymptotic Limit of Solutions of a Combustion Problem. Journal of Dynamics and Differential Equations 10, 577–604 (1998). https://doi.org/10.1023/A:1022629125232
Issue Date:
DOI: https://doi.org/10.1023/A:1022629125232