Onset of convection for autocatalytic reaction fronts in a vertical slab.

Abstract

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=\ensuremath{\delta}${\mathit{ga}}^{3}$/\ensuremath{\nu}${\mathit{D}}_{\mathit{C}}$. This parameter involves a fractional density difference \ensuremath{\delta} between the unreacted and reacted fluids, the acceleration of gravity g, the slab width a, the kinematic viscosity \ensuremath{\nu}, and the catalyst molecular diffusivity ${\mathit{D}}_{\mathit{C}}$. Buoyancy dominates over the competing curvature dependence of the front velocity in a band 0q${\mathit{q}}_{\mathit{c}}$ of unstable dimensionless wave numbers, with ${\mathit{q}}_{\mathit{c}}$\ensuremath{\rightarrow}S/24 as S\ensuremath{\rightarrow}0 and ${\mathit{q}}_{\mathit{c}}$\ensuremath{\rightarrow}(S/4${)}^{1/3}$ as S\ensuremath{\rightarrow}\ensuremath{\infty}. As S\ensuremath{\rightarrow}0, the perturbation with wave number ${\mathit{q}}_{\mathit{m}}$=${\mathit{q}}_{\mathit{c}}$/2 has the maximum dimensionless growth rate ${\mathrm{\ensuremath{\sigma}}}_{\mathit{m}}$=${\mathit{D}}_{\mathit{C}}$${\mathit{S}}^{2}$/48\ensuremath{\nu}. 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.