An asymptotic study of the transition from slow to fast burning in narrow channels

Motivated by recent theoretical developments in the dynamics of slow isobaric flames evolving in narrow channels (thick flames), the present study is concerned with non-isobaric flames. A reduced Darcy-like one-dimensional model capable of capturing an abrupt transition from slow conduction-driven to fast compression-driven burning is derived through an appropriate asymptotic procedure. Specifically, the adopted approach is based on the distinguished limit where the scaled width of the channel is deemed comparable to the system's Mach number, used as a parameter of expansion. The transition is caused by the friction induced gradual pile up of the pressure and hence preheating of the unburned gas adjacent to the advancing flame. The preheating leads to an increase of the flame speed, which in turn promotes further precompression, and so on. Upon some induction period this development leads to a thermal runaway within the reaction zone, triggering the transition.