A heat and mass transfer problem for the dissolution of an alumina particle in a cryolite bath

We investigate the spherically symmetric dissolution of an initially cold alumina particle in a bath of molten cryolite. The cryolite initially freezes on the particle, forming a shell that must melt before the particle can dissolve. We derive asymptotic solutions valid in the limits of small-superheat and of small Stefan number. In the small-superheat limit, the evolution of the boundary exhibits a two-scale behaviour. In the small Stefan number limit, we find that the behaviour of a particle could be limited by either the dissolution (in the case where the temperature differences are small) or by heat transfer (when the latent heat is large and the temperature gradients are large). Our asymptotic predictions are validated by a front fixing numerical scheme that we initiate using the early-time asymptotics.