Combustion fronts in a porous medium with two layers
The paper studies the propagation of a combustion front through a porous medium consisting of two parallel one-dimensional layers having different properties. In each layer, the model considers the temperature, \(u\) and \(v\), and the fuel concentration, \(y\) and \(z\), as a function of time \(t\) and space \(x\). The governing equations are deduced assuming each layer consists of a nonlinear reaction-diffusion-convection system based on balance equations and Darcy's law. Coupling between the layers comes from the heat transfer \(q(u-v)\) from one layer to the other. The authors study travelling waves which correspond to heteroclinics of a system of five first order ordinary differential equation. These joint two critical points and lie in the intersection of the unstable manifold of a hyperbolic saddle point and the stable manifold of a non-hyperbolic critical point. Such solutions correspond to combustion fronts which connect an unburned state (\(y=z=0\)) into a burned state (\(y=z=1\)). A first result concerns a one layer problem, which corresponds to the decoupling of the layers (\(q=0\)). It is proved that a connecting orbit, i.e. a travelling wave, takes place if and only if some parameter \(\sigma\) is larger than some constant. Similar results are obtained in the general case of two coupled layers. For small coupling, i.e. if the heat transfer coefficient \(q\) is small, a perturbation result is obtained. Travelling waves occur for a small range of system parameter near values where the decoupled layers have travelling waves of the same speed. For large coupling \(q\), an asymptotic result is obtained, i.e. for any fixed values of the parameters, travelling waves exist provided \(q\) is large enough. The method is based on a detailed analysis of the geometry of the singular points in order to use geometric singular perturbation theory. In the last section, some numerical results are presented.
scientific article; zbMATH DE number 5076287