The steady-state solidification scenario of ternary systems: Exact analytical solution of nonlinear model

A mathematical model describing the steady-state solidification of ternary systems with mushy layers (primary and cotectic) is formulated: solidification along a liquidus surface is characterized by a primary mushy layer, and solidification along a cotectic line is characterized by a secondary (cotectic) mushy layer. Exact analytical solutions of the model under consideration are found in a parametric form (thicknesses of mushy layers, growth rate of their boundaries, temperature and composition fields, solid fractions are determined in an explicit form). The velocity of solidification is completely determined by temperature gradients in the solid and liquid phases. This velocity coincides with similar expressions describing binary melt solidification with a planar front or a mushy layer. It is shown that the liquid composition of the main component decreases in the cotectic and primary layers, whereas the second (cotectic) composition increases in the cotectic layer, attains a maximum point and decreases in the primary layer.