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A system of nonlinear partial differential equations, which describe the combustion of a gas passing through a porous medium, is examined. This model provides a bridge between recent porous-medium combustion studies and more classical combustion models. In particular, the effect of solid conversion on the downstream temperature is determined for travelling-wave solutions to the system. In cases for which the solid matrix is a perfect catalyst, solely enhancing the exothermic chemical reaction, with zero mass exchange, it is proved that the downstream temperature must always exceed the upstream temperature. However, when the solid is allowed to react with the gas, travelling-wave solutions for which the up- and down-stream temperatures are equal may be realised. A key result of this work is then the investigation of physical processes and related parameter ranges that give rise to travelling wave solutions with equal up- and down-stream temperatures. Specifically, two critical wavespeeds Ci with 0 < c1 < c2 are identified. For c < c1 the downstream temperature always exceeds that upstream, whilst for c > c2 no bounded travelling-wave solutions exist. Behaviour in the region c1 < c < c2 is new: here solutions having equal up- and down-stream temperatures are realised. The two factors upon which this result depends are the inclusion into the model of solid conversion effects and the distinction between solid and gas temperatures. Thus the inclusion of a new physical process, that of heat storage in the solid varying as the reaction proceeds, allows the existence of a new type of travelling-wave solution. Further phenomena studied include the appearance of nonunique solutions (which are of a type that may be related to ignition processes) and degenerate solutions which terminate precisely when all the solid and gaseous fuel is used up. Parameter regimes for the existence of these various types of solutions are found and the stability of such solutions discussed.

Original publication




Journal article


Journal of Engineering Mathematics

Publication Date





321 - 342