Rayleigh-Benard instability generated by a diffusion flame

We investigate the Rayleigh-Benard convection problem within the context of a diffusion flame formed in a horizontal channel where the fuel and oxidizer concentrations are prescribed at the porous walls. This problem seems to have received no attention in the literature. When formulated in the low-Mach-number approximation the model depends on two main non-dimensional parameters, the Rayleigh number and the Damkohler number, which govern gravitational strength and reaction speed respectively. In the steady state the system admits a planar diffusion flame solution; the aim is to find the critical Rayleigh number at which this solution becomes unstable to infinitesimal perturbations. In the Boussinesq approximation, a linear stability analysis reduces the system to a matrix equation with a solution comparable to that of the well-studied non-reactive case of Rayleigh-Benard convection with a hot lower boundary. The planar Burke-Schumann diffusion flame, which has been previously considered unconditionally stable in studies disregarding gravity, is shown to become unstable when the Rayleigh number exceeds a critical value. A numerical treatment is performed to test the effects of compressibility and finite chemistry on the stability of the system. For weak values of the thermal expansion coefficient alpha, the numerical results show strong agreement with those of the linear stability analysis. It is found that as alpha increases to a more realistic value the system becomes considerably more stable, and also exhibits hysteresis at the onset of instability. Finally, a reduction in the Damkohler number is found to decrease the stability of the system.