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One-dimensional pattern formation in a model of burning

Published online by Cambridge University Press:  17 February 2009

Lawrence K. Forbes
Lawrence K. Forbes
Affiliation:
Dept of Mathematics, University of Queensland, St. Lucia, Queensland, 4072, Australia.
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Abstract

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We discuss a model of a burning process, essentially due to Sal'nikov, in which a substrate undergoes a two-stage decay through some intermediate chemical to form a final product. The second stage of the process occurs at a temperature-sensitive rate, and is also responsible for the production of heat. The effects of thermal conduction are included, and the intermediate chemical is assumed to be capable of diffusion through the decomposing substrate. The governing equations thus form a reaction-diffusion system, and spatially inhomogeneous behaviour is therefore possible.

This paper is concerned with stationary patterns of temperature and chemical concentration in the model. A numerical method for the solution of the governing equations is outlined, and makes use of a Fourier-series representation of the pattern. The question of the stability of these patterns is discussed in detail, and a linearised solution is presented, which is valid for patterns of very small amplitude. The results of accurate solutions to the fully non-linear equations are discussed, and compared with the predictions of the linearised theory. Parameter regions in which there exists genuine nonuniqueness of solutions are identified.

Type
Research Article
Information
The ANZIAM Journal , Volume 35 , Issue 2 , October 1993 , pp. 145 - 173
Copyright
Copyright © Australian Mathematical Society 1993

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