Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T22:34:22.106Z Has data issue: false hasContentIssue false

A NOTE ON TRAVELLING WAVES IN COMPETITIVE REACTION SYSTEMS

Published online by Cambridge University Press:  26 November 2013

LAWRENCE K. FORBES*
Affiliation:
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia
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.

This note considers an established reaction–diffusion model for a combustion system, in which there are competing endothermic and exothermic reaction pathways. A combustion front is assumed to move at constant speed through the medium. An asymptotic theory is presented for solid fuels in which material diffusion is ignored, and it allows a simple and complete analysis of the approximate system in the phase plane. Both the adiabatic and nonadiabatic cases are discussed.

MSC classification

Type
Research Article
Copyright
Copyright ©2013 Australian Mathematical Society 

References

Ball, R., McIntosh, A. C. and Brindley, J., “Thermokinetic models for simultaneous reactions: a comparative study”, Combust. Theory Model. 3 (1999) 447468; doi:10.1088/1364-7830/3/3/302.CrossRefGoogle Scholar
Brindley, J., Griffiths, J. F., McIntosh, A. C. and Zhang, J., “Initiation of combustion waves in solids, and the effects of geometry”, ANZIAM J. 43 (2001) 149163; doi:10.1017/S1446181100011482.CrossRefGoogle Scholar
Forbes, L. K., “A two-dimensional model for large-scale bushfire spread”, J. Aust. Math. Soc. Ser. B 39 (1997) 171194; doi:10.1017/S0334270000008791.CrossRefGoogle Scholar
Forbes, L. K. and Derrick, W., “A combustion wave of permanent form in a compressible gas”, ANZIAM J. 43 (2001) 3558; doi:10.1017/S144618110001141X.CrossRefGoogle Scholar
Gray, B. F., Kalliadasis, S., Lazarovici, A., Macaskill, C., Merkin, J. H. and Scott, S. K., “The suppression of an exothermic branched-chain flame through endothermic reaction and radical scavenging”, Proc. R. Soc. Lond. A 458 (2002) 21192138; doi:10.1098/rspa.2002.0961.CrossRefGoogle Scholar
Gray, P. and Scott, S. K., Chemical oscillations and instabilities: non-linear chemical kinetics (Clarendon Press, Oxford, 1994).Google Scholar
Hmaidi, A., McIntosh, A. C. and Brindley, J., “A mathematical model of hotspot condensed phase ignition in the presence of a competitive endothermic reaction”, Combust. Theory Model. 14 (2010) 893920; doi:10.1080/13647830.2010.519050.CrossRefGoogle Scholar
Matkowsky, B. J. and Sivashinsky, G. I., “Propagation of a pulsating reaction front in solid fuel combustion”, SIAM J. Appl. Math. 35 (1978) 465478; doi:10.1137/0135038.CrossRefGoogle Scholar
Murray, J. D., Mathematical biology (Springer, Berlin, 1989).CrossRefGoogle Scholar
Sharples, J. J., Sidhu, H. S., McIntosh, A. C., Brindley, J. and Gubernov, V. V., “Analysis of combustion waves arising in the presence of a competitive endothermic reaction”, IMA J. Appl. Math. 77 (2012) 1831; doi:10.1093/imamat/hxr072.CrossRefGoogle Scholar
Weber, R. O., Mercer, G. N., Sidhu, H. S. and Gray, B. F., “Combustion waves for gases ($Le= 1$) and solids ($Le\rightarrow \infty $)”, Proc. R. Soc. Lond. A 453 (1997) 11051118; doi:10.1098/rspa.1997.0062.CrossRefGoogle Scholar
Weber, R. O., Mercer, G. N. and Sidhu, H. S., “Combustion leftovers”, Math. Comput. Model. 36 (2002) 371377; doi:10.1016/S0895-7177(02)00131-0.CrossRefGoogle Scholar
Zel’dovich, Ya. B., Barenblatt, G. I., Librovich, V. B. and Makhviladze, G. M., Mathematical theory of combustion and explosions (Plenum Press, New York, 1985).CrossRefGoogle Scholar