Thermosolutal convection during dendritic solidification of alloys: Part i. Linear stability analysis

This paper describes the simulation of thermosolutal convection in directionally solidified (DS) alloys. A linear stability analysis is used to predict marginal stability curves for a system that comprises a mushy zone underlying an all-liquid zone. In the unperturbed and nonconvecting state . e.}, the basic state), isotherms and isoconcentrates are planar and horizontal. The mushy zone is realistically treated as a medium with a variable volume fraction of liquid that is con-sistent with the energy and solute conservation equations. The perturbed variables include tem-perature, concentration of solute, and both components of velocity in a two-dimensional system. As a model system, an alloy of Pb-20 wt pct Sn, solidifying at a velocity of 2 X 10^-3 cm s^-1 was selected. Dimensional numerical calculations were done to define the marginal stability curves in terms of the thermal gradient at the dendrite tips, G _L, vs the horizontal wave number of the perturbed quantities. For a gravitational constant of 1 g, 0.5 g, 0.1 g, and 0.01 g, the marginal stability curves show no minima; thus, the system is never unconditionally stable. Nevertheless, such calculations quantify the effect of reducing the gravitational constant on reducing convection and suggest lateral dimensions of the mold for the purpose of suppressing convection. Finally, for a gravitational constant of 10^-4 g, calculations show that the system is stable for the thermal gradients investigated (2.5 ≤G _L ≤ 100 K-cm^-1).