Point-Source Initiation of Lean Spherical Flames of Light Reactants: An Asymptotic Theory

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Abstract
We study the dynamics of small spherical flame kernels whose evolution is triggered in a lean mixture of a light mobile reactant by a time-dependent point- source of energy. For simplicity, the analysis is conducted in the framework of a one-reactant flame model, with a one-step Arrhenius overall kinetics. Using the method of matched asymptotic expansion for large activation energies, we show that: The concentration and temperature fields split into quasi-steady regions, inside and around the flame kernel, and an unsteady far-field; in both regions convection effects may be neglected. Match ing the near - and far-fields furnishes a parameter-free, non-linear evolution equation for the flame radius ; in addition to memory effects of diffusive origin, it includes explicitly the functional form of the chemical rate and the instan taneous power of the point source of energy we used as ignition device. Through a numerical integration of this equation, flame front trajectories and critical energies are obtained for a 3-parameter family of pulse-like energy inputs. There is an optimum pulse duration, for which the critical energy is minimum; its value is comparable to the spontaneous growth time obtained by the linear stability analyses of steady, adiabatic spherical flames corresponding to the same mixture.

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