Differentiation of a two-component melt in a crystallizing magmatic chamber
We propose a model of solidifying melt cooled from above. The following system of equations is solved: equations of conservation of mass for two melt components and crystal fraction, equation of heat transfer, equation of phase equilibrium, and relationship between relative and absolute settling velocity. The solution consists of six functions: the depth dependence of two concentrations, crystal fraction, temperature, and the velocities of melt and crystals. The counterflow of melt produced by crystal settling is included in the model as well as the flow due to the density difference between melt and growing crystals. These effects are essential in the dynamics of crystal settling. First, they give rise to discontinuities resulting from changes of settling velocities; second, they yield a new solution where a boundary layer splits into two parts. The splitting follows from the difference between two velocities, $U_s$ (the settling velocity) and $U_b$ (the velocity of the solidification front). When crystal fraction near the front increases, $U_s$ becomes less than $U_b$, whereas outside of the layer, $U_s>U_b$. As a result, rapid cooling from above reduces the flow of the solid phase from the upper surface.