Abstract
In this thesis we consider the unidirectional solidification of a dilute binary alloy. We extend the one-sided model of Langer (1980) to include fluid motion in the melt. In particular we introduce a model boundary layer flow across the solid-liquid interface, the strength of which is characterised by a Reynolds number. We conduct a linear analysis in order to study the three classical instabilities that may occur.
We consider the morphological instability of the solid-liquid interface in the absence of buoyancy forces. We present an asymptotic analysis valid for large Schmidt number and numerical results which show that, for modes with wave vectors aligned in the flow direction, the flow is responsible for the appearance of travelling waves and affects the onset of instability. The stability of cross-stream modes is unaffected by the presence of the shear flow. Mechanisms to explain this behaviour are presented. The weakly nonlinear evolution of these waves is considered and shows that a smooth supercritical transition of the planar interface to the travelling wave is preferred.
For a sufficiently high Reynolds number the boundary layer will become hydrodynamically unstable. We present an asymptotic analysis valid for large Reynolds number and numerical results to show that this shear flow instability is only weakly affected by the deformable, freezing solid-liquid interface.
Solutal convective instability may occur during solidification vertically upwards. The linear stability of this situation is considered numerically. In the case of no forced flow the detailed structure of the convective, morphological and two overstable modes is elucidated, the latter two modes arising from a coupling of the former pair. A forced flow gives the system a preferred direction and `unfolds' the structure.