Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-29T05:49:35.704Z Has data issue: false hasContentIssue false

On the existence of Chapman-Jouguet detonation waves

Published online by Cambridge University Press:  17 April 2009

Mahmoud Hesaaraki
Affiliation:
Department of Mathematics, Sharif University of Technology, P.O. Box 11365–9415, Tehran, Iran, e-mail: [email protected]
Abdolrahman Razani
Affiliation:
Department of Mathematics, Faculty of Science, Tarbiat Modarres University, P.O. Box 14155–4838, Tehran, Iran, e-mail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The existence of travelling wave solutions to equations of a viscous, heat-conducting combustible fluid is proved. The reactions are assumed to be one step exothermic reactions with a natural discontinuous reaction rate function. The problem is studied for a general gas. Instead of assuming the ideal gas conditions we consider a general thermodynamics which is described by a fairly mild set of hypotheses. The existence proof of travelling waves for Chapman-Jouguet detonation reduces to finding specific heteroclinic orbits of a discontinuous system of ordinary differential equations: these heteroclinic orbits connect a rest point corresponding to unburnt state to that of the burnt state. The existence proof for heteroclinic orbits corresponding to Chapman-Jouguet detonation waves is carried out by some general topological arguments in ordinary differential equations theory.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Gardner, R., ‘On the detonation of a combustible gas’, Trans. Amer. Math. Soc. 277 (1983), 431468.CrossRefGoogle Scholar
[2]Gasser, Z. and Szmolyan, P., ‘A geometric singular perturbation analysis of detonation and deflagration waves’, SIAM J. Math. Anal. 24 (1993), 968986.CrossRefGoogle Scholar
[3]Hesaaraki, M., ‘The structure of shock waves in magnetohydrodynamics’, Mem. Amer. Math. Soc. 302 (1984).Google Scholar
[4]Hesaaraki, M., ‘The structure of shock waves in magnetohydrodynamics for purely transverse magnetic fields’, SIAM J. Appl. Math. 51 (1991), 412428.CrossRefGoogle Scholar
[5]Hesaaraki, M., ‘The structure of MFD shock waves in a model of two fluids’, Nonlinearity 6 (1993), 124.CrossRefGoogle Scholar
[6]Hesaaraki, M., ‘The structure of MFD shock wavesPures. Appl. 72 (1993), 377404.Google Scholar
[7]Hesaaraki, M., ‘The structure of MFD shock waves for rectilinear motion in some models of plasma’, Trans. Amer. Math. Soc. 347 (1995), 34233452.CrossRefGoogle Scholar
[8]Smoller, J.A., Shock waves and reacting diffusion equations (Springer-Verlag, Berlin, Heidelberg, New York, 1994).Google Scholar
[9]Vick, J.W., Homology theory. An introduction to algebraic topology, Pure and Appl. Math. 53 (Academic press, Inc., New York, London, 1973).Google Scholar
[10]Wagner, D., ‘The existence and behaviour of viscous structure for plane detonation waves’, SIAM J. Math. Anal. 20 (1989), 10351054.CrossRefGoogle Scholar
[11]Williams, F.A., Combustion theory (Benjamin/Cummings, Menlo Park, CA, 1985).Google Scholar