Singularly perturbed homotopy analysis method applied to the pressure driven flame in porous media

In this paper we combined the homotopy analysis method (HAM) and the method of integral manifold (MIM) for the problem of a pressure-driven flame in an inert medium filled with a flammable gaseous mixture. In order to apply the MIM, we have rewritten the non-dimensional model in the form of a singular perturbed system (SPS) of ordinary differential equations. Two different stages of the trajectory correspond to the sub-zones of the flame: (1) fast motion from the initial condition to the slow curve, and its interpretation as a preheat sub-zone and (2) the path that corresponds to a reaction sub-zone.By applying the HAM and the MIM we derived an analytical expression for the condition of ignition and also an analytical expression for the delay time. The assumption that the heat flux is continuous at the point where the fast and slow parts of the trajectory are glued enabled us to obtain an expression for the flame velocity. Our theoretical results are justified by the numerical simulations.An optimal value of the convergence control parameter is given through the square residual error. By minimizing the square residual error, the optimal convergence-control parameters can be obtained. The more quickly the residual error decreases to zero, the faster the corresponding homotopy series solution converges.