Hydrodynamic stability of the melt during the solidification of a binary alloy with small segregation coefficient

We consider the convective flow present during the directional solidification of a dilute binary alloy having small segregation coefficient k. The constitutional undercooling is set to zero, so that morphological instabilities of the melt-crystal interface are suppressed and the interface is planar. In the linearized stability theory, the critical wave number of the convective mode a_c is found to behave like a_c ≈ k ^{1 4} as k → 0, which is identical in behavior to the morphological mode. The weakly nonlinear theory gives rise to a Sivashinsky equation governing the spatial structure of the convection. Unlike previous long-wave evolution equations governing convective processes, this one arises in a geometrically unbounded domain with system boundaries that are neither fixed heat flux nor small Biot number; here the equivalent Biot number is of unit order.