On the integral manifold approach to a flame propagation problem: pressure-driven flames in porous media

Abstract

The problem of a pressure‐driven flame in an inert porous medium filled with a flammable gaseous mixture is considered. In the frame of reference attached to an advancing combustion wave and after a suitable non‐dimensionalization the corresponding mathematical description of the problem includes three highly nonlinear ordinary differential equations. The system is rewritten in the form of a singularly perturbed system of ordinary differential equations and is analysed analytically by the geometrical version of the asymptotic method of integral manifolds (MIM). The paper focuses on an analysis of the fine structure of the flame and its velocity on the basis of an asymptotical consideration of an arbitrary trajectory of the considered system in the phase space. It is shown that two different stages of the trajectory correspond to the two various sub‐zones of the flame: the first stage (fast motion from the initial point to the slow integral) is interpreted as a preheat sub‐zone and the second stage of the path corresponds to a reaction sub‐zone. It is shown that an inter‐zone boundary plays an important role in a determination of the flame properties: characteristics of the gaseous mixture at that point determine the flame velocity. The accepted approach of the investigation allows us to gain an analytical expression for the flame velocity. It appears that the velocity formula represents a cubic‐root dependence on the Arrhenius exponent, which in turn contains the parameters of the boundary point. The theoretical predictions are found to coincide rather well with the data of direct numerical simulations.