Freckle formation in a solidifying binary alloy

The effect of solutal convection on the unidirectional solidification of a binary alloy cooled from below is considered. Soon after the onset of solidification a mushy region often forms, and is accompanied by vigorous convection in the molten alloy above. In contrast, the fluid in the mush appears unaffected by the neighbouring flow and remains essentially quiescent. This work considers the nonlinear convective stability of the mush and determines a criterion for channelling in the mush to occur. The basic state is a similarity solution, so that a quasi-static approximation must be applied in order to apply conventional stability theory. Moreover, although the model for solidification is relatively simple, an analytical expression for linear stability is not available. Thus the series of equations arising from the nonlinear stability analysis lead to a complicated set of symbolic and numerical calculations. Stable finite-amplitude solutions are found for Rayleigh numbers larger than critical for all values of the chosen superheat. The nonlinear solutions demonstrate the possibility of channelling within the mush dependent upon the strength of convection. These finite-amplitude solutions are extended further by the calculation of a numerical solution of the model equations. The evolution of the stream function and the mass fraction is followed, the onset of convection and freckling can then be deduced from the numerical simulation. The onset of convection in the mush is found in terms of the mush Rayleigh number, and compares favourably with linear stability theory and experimental data. The onset of freckling is also given in terms of a Rayleigh number, but is sensitive to the initial conditions. This appears to explain the large disagreement found in experiments aimed at finding a criterion for freckling.