Convection in a mushy zone forced by sidewall heat losses

We calculate the convective state due to weak sidewall heat losses in a mushy zone during the steady directional solidification of a binary alloy. The configuration consists of a warm liquid region and a cold solid region separated by a mixed phase region, the mushy zone, which is modeled as a reactive porous matrix. The structure of the convection that arises from horizontal temperature gradients, induced by the heat losses at the sidewalls, is characterized by a set of nondimensional parameters that describe the effects of latent heat, composition, permeability, and thermal and solutal buoyancy. We observe a wide range of behaviors and show that, as the critical Rayleigh number for convection in a horizontally uniform mush is reached, we begin to see the precursors of chimneys near the cooled boundaries. The strength of the cooling plays an important role in determining the strength and degree of localization of the convection near the boundary. We find, in common with other authors, that upflow, in this case caused by lighter fluid being released at the cooled sidewalls, leads to regions of dissolution, which are precursors to chimney formation. Although a treatment of the stability of the steady convective states presented is not considered, we identify the effects of the different physical parameters on the steady states.