A numerical study of chemical front propagation in a Hele-Shaw flow under buoyancy effects

Abstract

We consider the propagation of chemical fronts in a Hele-Shaw flow where the front is assumed to propagate with a curvature dependent velocity. The motivation is to model some recent experiments that employ aqueous autocatalytic chemical reactions in such a device. The density change across the front in such experiments is quite small so the Boussinesq approximation can be used, and the flow field generated is exclusively due to buoyancy effects. We derive a free boundary formulation based on Darcy’s law and potential theory, and describe the evolution in terms of the θ−L formulation, in which the tangent angle and the perimeter of the closed front are followed in time. Numerical solutions are obtained for this formulation with a rising and expanding bubble. As observed in the experiments, a fingering phenomenon which is different from the surface tension associated phenomenon appears in our calculations. The mechanisms that control the wavelength selection of the fingers, and a comparison with the result of a linear stability analysis for flat fronts are discussed.