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Analysis of a differential equation occurring in the theory of flame fronts

Published online by Cambridge University Press:  17 February 2009

R. Grimshaw  and
J. Gan
J. Gan
Affiliation:
Department of Mathematics, Monash University, Clayton, Vic. 3168, Australia
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Abstract

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Ronney and Sivashinsky [2] and Buckmaster and Lee [1] have proposed a certain non-autonomous first order ordinary differential equation as a simple model for an expanding spherical flame front in a zero-gravity environment. Here we supplement their preliminary numerical calculations with some analysis and further numerical work. The results show that the solutions either correspond to quenching, or to steady flame front propagation, or to rapid expansion of the flame front, depending on two control parameters. A crucial component of our analysis is the construction of a barrier orbit which divides the phase plane into two parts. The location of this barrier orbit then determines the fate of orbits in the phase plane.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 1 , July 1994 , pp. 1 - 16
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Buckmaster, J. and Lee, C. J., “The effects of confinement and heat loss on outwardly propagating spherical flames”, 1992, preprint.CrossRefGoogle Scholar
[2]Ronney, P. D. and Sivashinsky, G. I., “A theoretical study of propagation and extinction of nonsteady spherical flame fronts”, SIAMJ. Appl. Math. 49 (1989) 1029–1046.CrossRefGoogle Scholar