Optimal and hysteretic fluxes in alloy solidification: Variational principles and chimney spacing

We take a numerical approach to analyze the mechanisms controlling the spacing of chimneys -- channels devoid of solid -- in two-dimensional mushy layers formed by solidifying a binary alloy. Chimneys are the principal conduits through which buoyancy effects transport material out of the mushy layer and into the liquid from which it formed. Experiments show a coarsening of chimney spacing and we pursue the hypothesis that this observation is a consequence of a variational principle: the chimney spacing adjusts to optimize material transport and hence maximize the rate of removal of potential energy stored in the mushy layer. The optimal solute flux increases approximately linearly with the mushy layer Rayleigh number. However, for spacings below a critical value the chimneys collapse and solute fluxes cease, revealing a hysteresis between chimney convection and no flow.