Onset of convection for autocatalytic reaction fronts in a vertical slab

A fully three-dimensional linear stability analysis shows that ascending autocatalytic reaction fronts in vertical slabs are unstable to convection for large-wavelength perturbations at all finite values of the dimensionless driving parameter S=δ${\mathit{ga}}^3$/ν${\mathit{D}}_{\mathit{C}}$. This parameter involves a fractional density difference δ between the unreacted and reacted fluids, the acceleration of gravity g, the slab width a, the kinematic viscosity ν, and the catalyst molecular diffusivity ${\mathit{D}}_{\mathit{C}}$. Buoyancy dominates over the competing curvature dependence of the front velocity in a band 0<q<${\mathit{q}}_{\mathit{c}}$ of unstable dimensionless wave numbers, with ${\mathit{q}}_{\mathit{c}}$→S/24 as S→0 and ${\mathit{q}}_{\mathit{c}}$→(S/4${)}^{1/3}$ as S→∞. As S→0, the perturbation with wave number ${\mathit{q}}_{\mathit{m}}$=${\mathit{q}}_{\mathit{c}}$/2 has the maximum dimensionless growth rate ${\mathrm{\sigma }}_{\mathit{m}}$=${\mathit{D}}_{\mathit{C}}$${\mathit{S}}^2$/48ν. For general S, the cutoff wave number ${\mathit{q}}_{\mathit{c}}$ is calculated using exact analytical solutions for the perturbed fluid velocity. The calculated results should be observable in experiments.