Double diffusive instabilities of chemical fronts

Gravitational Hele-Shaw fingering of an autocatalytic reaction diffusion interface is investigated theoretically. Dimensional analysis based on reaction diffusion length, time, and velocity scales reveal the dependence on the Lewis number $Le$, and thermal and concentration Rayleigh numbers $R_T$ and $R_c$. Linear stability analysis of a planar upward propagating (against the gravitational acceleration) interface results in an eigenvalue problem for each wavenumber $k$ which we solve using a Chebyshev pseudospectral method. A novel light over heavy instability of an endothermic reaction was found when $Le>1$. It is shown that this instability is equivalent to that of a downward propagating exothermic wave. Nonlinear second-order Crank-Nicolson, finite volume simulations are in agreement with linear theory and also show the docile nature of the interface breakup. A displaced particle argument confirms that this unexpected instability is local, that it is subdued by a region of local stability, and elucidates its dependence on the underlying reaction diffusion mechanism.