Theory of interaction between a solid inclusion in a liquid with nonuniform viscosity with an advancing solidification front

Abstract

A theory of behavior of foreign solid inclusions in a liquid being crystallized is developed. As the temperature of the liquid is nonuniform near the solidification front and the viscosity of the liquid depends on temperature, a dependence of viscosity on the distance from the solid-crystal surface is taken into account in the proposed theory. The dependence of the inclusion/liquid interface specific surface free energy at the same distance from the crystal surface has major importance in the present theory. A model is used in which the weight of the particle, its buoyancy, a force due to the gradient of interfacial tension, and that of viscous drag are important. Attention is paid to the case when the velocities of the inclusion and of the solidification front move in the same direction. A peculiarity in the action of viscous drag is ascertained; namely, the drag force diminishes with the movement of the inclusion towards the region of decreasing viscosity in a liquid with nonuniform viscosity. A correction factor for the Stokes formula is derived which comprises the influences of the presence of the solidification front and of the nonuniformity of viscosity. The analysis of inclusion behavior is performed in terms of three dimensionless variables; one of these variables is the modified Weber number; the other two are dimensionless complexes having the following meanings: (1) the relation of the drag force in a homogeneous viscous medium to the buoyancy force, and (2) the relation of the reduction in the viscous drag force to the buoyancy force (the former decreases with distance as a result). The known experimental data are explained in terms of the relationships between inclusion dimensions, the so-called “critical velocity” for engulfment by the solidification front, and the existence of the variability of viscosity with distance ahead of the solidification front.