Morphological instabilities in directional solidification of a binary alloy - End effects

This paper considers the unidirectional solidification of a binary alloy for which the liquid and solid phases are bounded by endwalls. In order to account for the transport of latent heat and solute, thermal and solutal boundary layers must be placed at the solidifying interface. Further, under the assumption of fixed temperature gradients, the presence of the endwalls leads to a velocity of solidification that decreases with time, and hence to an unsteady basic state having a planar interface. From a stability analysis of this state, a nonlinear long-wave evolution equation of Sivashinsky type is derived, with modified coefficients, that shows how the onset of cellular structure is delayed by the presence of endwalls.