The Formation of Freckles in Binary Alloys

This paper presents a synopsis of some recent work, still in progress, aimed at elucidating a quantitative explanation of the processes by which flow chimneys form when certain types of alloys are directionally solidified. If (for example) light fluid is released at the liquid-solid "mushy" (dendrite) zone, and cooling is from below, then the intermediate fluid flow undergoes convection through the porous dendrite mass. This can lead to an "instability" of the form of the mushy zone, such that upwelling light fluid flows preferentially in channels within the dendrite mass. What we seek to develop here, is a mathematical basis by which this phenomenon may be properly understood. Accordingly, a mathematical model is developed, simplified, and partially analysed, and as a result we are able to make one specific prediction concerning a criterion for the onset of convection and freckling. This prediction is equivalent to the classical Rayleigh number condition for convective instability.