Morphological stability analysis of directional solidification into an oscillatory fluid layer

We study the stability of a planar solid-melt boundary during directional solidification of a binary alloy when the solid is being periodically vibrated in the direction parallel to the boundary (or equivalently, under a far field uniform and oscillatory flow parallel to the planar boundary). The analysis is motivated by directional solidification experiments under the low level residual acceleration field characteristic of a microgravity environment, and possible effects on crystal growth in space. It is known that periodic modulation of the solid-melt interface under the conditions stated induces second order stationary streaming flows within a boundary layer adjacent to the interface, the thickness of which is the same as the wavelength of the modulation. We derive an effective solute transport equation by averaging over the fast time scale of the oscillatory flow, and obtain the resulting dispersion relation for a small disturbance of a planar interface. We find both regions of stationary and oscillatory instability. For small ratios of the viscous to solutal layer thicknesses, s, the flow generally destabilizes the planar interface. For s ≃ 1, the flow stabilizes the stationary branch, but it can also excite an oscillatory instability. For large s, the effect of the flow is small.