Convection in directionally solidifying alloys under inclined rotation

In an experiment on binary alloys directionally solidifying from below, Sample and Hellawell (1984) showed that the plume convection can be successfully prohibited by rotating the cooling tank around an inclined axis. In the present paper we interpret their experimental observation by an analytical approach. Results show that there is a flow induced by the inclination. The induced flow in the fluid layer is a parallel shear flow consisting of three parts: the thermal boundary-layer flow, the solute boundary-layer flow, and the Ekman-layer flow. In the mush, the induced flow is also a parallel flow but of much smaller velocity, consisting of two flows of opposite directions. The induced velocity in the fluid layer increases with inclination angle and decreases with the effective Taylor number T(e). The induced velocity in the mush also increases with inclination angle but remains virtually the same on varying the speed of rotation. The linear stability analysis of the mushy layer shows that, due mostly to the reduction of buoyancy, the mush becomes more stable as the inclination angle increases. In the precession-only case, the most-unstable mode of instability is the longitudinal mode, which propagates in a direction perpendicular to the induced flow. In the spin (with or without precession) case, the instability modes propagating in different directions are of equal stability. Because the induced flow changes direction with a frequency equal to the spin angular velocity, the flow scans over all the directions of the system and stabilizes equally the modes in different directions. We conclude on the basis of the present results and from the practical point of view that spin-only rotation is more effective than the precession-only rotation in stabilizing the convection during solidification.