A Analysis of Mush-Chimney Structure

When a multi-component liquid is cooled and solidified, commonly, the solid phase advances from the cold boundary into the liquid as a branching forest of dendritic crystals. This creates a region of mixed solid and liquid phases, referred to as a mushy zone, in which the solid forms a rigidly connected framework with the liquid occurring in the intercrystalline gaps. When the fluid seeps through the dendrites, further freezing occurs which fills in pores of the matrix and reduces its permeability to the liquid flow. In particular, if a binary alloy (for example, NH_4Cl -H_2O solution) is cooled at bottom and a dense component (for example, NH_4 Cl) is solidified, buoyant material released during freezing in the pores returns to the melt only through thin, vertical, but widely separated, 'chimneys', the flow through the matrix between them being organized to supply these chimneys. We presented photos of a mush-chimney system obtained from the ammonium chloride experiment, and we studied how convection with horizontal divergence affects the structure and flow of the mush-chimney system. We use a simple ODE system in the mush derived by assuming that the temperature depends on vertical coordinate only. We find that the mass fraction of solid increases and the depth of a mush decreases when the strength of convection increases. We present an axisymmetric model containing only one chimney to analyze the structure of the mush-chimney system. We find solutions of the temperature, the solid fraction, and the pressure in the chimney wall. In particular, the pressure expression shows that the fluid flow needs a huge pressure in order to pass through the chimney wall if its permeability is very small. We assume that a ratio of composition is large, which allows us to neglect the pressure contribution of the chimney wall. We use the knowledge of the variables in the mush, evaluated on the chimney wall, to find the fluid flow in the chimney and the radius of chimney. Our procedure employs the von Karman-Pohlhausen technique for determining chimney flow (Roberts & Loper, 1983) and makes use of the fact that the radius of the chimney is much less than the thickness of the mush. We find a relation between a parameter measuring the ratio of viscous and buoyancy forces in the chimney and the vertical velocity component on the top of the mush, and estimate numerically the value of this velocity measuring the strength of convection. The results obtained show reasonably good agreement with theoretical and experimental works (Roberts & Loper (1983), Chen & Chen (1991), Tait & Jaupart (1992), Hellawell etc. (1993), Worster (1991)).