Combustion of porous samples with melting and flow of reactants

We formulate and analyze a model describing the combustion of porous condensed materials in which a reactant melts and spreads through the pores of the sample. Thus there is liquid motion relative to the porous solid matrix. Our model describes the cases when the melt either fills all the pores or when some gas remains in the pores. in each case the melt occupies a prescribed volume fraction of the mixture. We employ both analytical and numerical methods to find uniformly propagating combustion waves, to analyze their stability and to determine behavior in the instability region. The principal physical conclusion which follows from our analysis is that the flow of the melted component can result in nonuniform composition of the product. Unlike models which do not take into account the relative motion of the components, this model exhibits a dependence of the structure of the product on the mode of propagation of the combustion front. Thus, if the initial mixture is uniform, models which do not allow for relative motion necessarily lead to uniform structure of the product, while in the model employed here the structure can be nonuniform. We observe that the structure of uniformly propagating combustion waves depends on whether the refractory or melting component is in excess in the initial mixture. We determine how various parameters of the system affect stability and find a pulsating instability of the uniformly propagating solutions. We also perform numerical simulations in order to (i) study the dynamical behavior of the combustion wave in the instability region, (ii) obtain a description of the melt flow on the scale of the entire sample rather than on the scale of the combustion wave, i.e. to study the evolution of the liquid melt layer which may occupy only a part of the product region. We show, in particular, that a transition to relaxation oscillations may occur closer to the threshold of instability than in gasless solid fuel combustion. Our numerical and analytical results are in qualitative agreement.