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Steady travelling ware solutions of a one-dependent model of porous medium combustion are sought. There are four dependent variables, the solid and gas temperatures and concentrations, and three important dimensionless parametersin this simplified model which still retains the essence of the physical and chemical processes. By assuming that the reaction is confined activated and extinguished. For example, exhaustion of either the solid or gaseous fuel may terminate the reaction. Numerical simulations are performed to partition the key parameter space into distinct regions which are characterised by the types of solution occurring inside them, and are separated by degeneracy surfaces, on which two, or more, solution-types co-exist. Asymptomatic analysis, performed for solutions possessing short reaction zones (0 < L ≪ 1, say), complements the numerical results, indicating how the interplay between the dominant physical mechanisms affects the existence of travelling wave solutions. Thus, for a weak reaction ((net heat production) ∼ L ≪ 1), only one type of travelling wave solution may exist when each of the key parameters are O(1), whereas for a special limit of the parameters, which corresponds to a slow inlet gas velocity, all four solution types are realised. By focussing on stronger reaction rates, the manner in which the reaction rate is approximated is shown to affect the existence and uniqueness of the travelling wave solutions of the model. In particular, by increasing the strength of the reaction rate from O(1) to O(L-1), so that (net heat production) ∼ O(1), uniqueness of the travelling wave solutions is maintained. In contrast, by taking the limit L → 0 and approximating the reaction rate with a delta-function, so that, once again, (net heat production) ∼ O(1), under certain circumstances a continuum of degenerate solutions is predicted (Byrne and Norbury, 1994b). Since, in real applications, only a finite number of solutions are realised, it is suggested that the limiting reaction rate provides a beller approximation to the underlying chemistry than the delta-function rate.


Journal article


Mathematical Engineering in Industry

Publication Date





39 - 62