Mathematical Modeling of Measured Recalescence Velocities

ABSTRACT A moving‐boundary problem of synchronous directional and volumetric solidification in undercooled liquid drops is formulated and solved. Directional solidification is caused by the temperature gradient along the spatial direction, whereas volumetric growth of new phase elements is caused by the melt undercooling ahead of the phase transition boundary. As this takes place, the solidification domain is divided into three regions occupied by solid material, two‐phase layer, and liquid phase. The heat and mass transfer model is formulated inside all regions with the corresponding boundary conditions. To solve the problem, we use self‐similar variables and functions, as well as the Laplace method for evaluating the Laplace‐type integral and the technique of small parameter expansion of unknown functions in series. The temperature, impurity concentration, solid phase fraction distributions, and laws of motion of two solidification boundaries are found analytically. Our approximate solution shows that nucleation and growth of particles ahead of the solid phase: two ‐ phase region boundary leads to a U‐shaped curve for the solidification velocity as a function of melt undercooling. The theory under consideration describes real experimental data on the solidification of Al‐Ni melts.