Imbibition with solidification in alumina feeding

Liquid aluminium is produced from alumina by electrolysis in a Hall-H\'{e}roult cell. The process by which the alumina is fed into the cell influences the overall behaviour and efficiency of the cell. We develop a one-dimensional Stefan-type model for the imbibition of molten cryolite into a cold porous lump of alumina. In the small overheat limit, we analyse the small-time behaviour using the method of matched asymptotic expansions and find there to be locally self-similar solutions that describe analytically the competition between imbibition and freezing. Depending on the balance between these effects, the problem may exhibit two self-similar solutions (one being stable and the other unstable) or such a solution may cease to exist (so that imbibition is not possible without the molten cryolite freezing first on the exterior of the porous lump). Our asymptotic predictions are validated by direct numerical simulations that are also used to investigate the late time behaviour. In particular we predict the depth of imbibition before the cryolite freezes.