Thermal-diffusive ignition and flame initiation by a local energy source

The ignition and flame initiation in a gaseous reacting mixture subject to a local source of thermal energy is analysed by means of large activation energy asymptotics. The ignition transient is assumed to be long enough for heat conduction to be the dominant cooling mechanism. We show the existence of a critical value of the Damkohler number, defined as the ratio of appropriate characteristic times of conduction and chemical reaction, such that ignition only occurs for supercritical values. Additional conditions are required to ensure self-propagation of a flame after ignition. These are obtained, with the thermal-diffusive model, for a source of energy represented by an instantaneous point, line or planar source. The analysis, involving an unsteady free-boundary problem, shows that the initial flame kernel evolves to a self-propagating flame only if the energy released by the source is greater than a critical value.