Effect of thermal expansion on the linear stability of planar premixed flames for a simple chain-branching model: The high activation energy asymptotic limit

The linear stability of freely propagating, adiabatic, planar premixed ames is investigated in the context of a simple chain-branching chemistry model consisting of a chain-branching reaction step and a completion reaction step. The role of chain-branching is governed by a crossover temperature. Hydrodynamic effects, induced by thermal expansion, are taken into account and the results compared and contrasted with those from a previous purely thermal-di�usive constant density linear stability study. It is shown that when thermal expansion is properly accounted for, a region of stable ames predicted by the constant density model disappears, and instead the ame is unstable to a long-wavelength cellular instability. For a pulsating mode, however, thermal expansion is shown to have only a weak e�ect on the critical fuel Lewis number required for instability. These e�ects of thermal expansion on the two-step chain-branching ame are shown to be qualitatively similar to those on the standard one-step reaction model. Indeed, as found by constant density studies, in the limit that the chain-branching crossover temperature tends to the adiabatic ame temperature, the two-step model can be described to leading order by the one-step model with a suitably de�ned e�ective activation energy.