Directional solidification near minimum c: Two-dimensional isolas and multiple solutions

Mullins and Sekerka showed, using linear-stability theory, that for fixed temperature gradient, the planar interface is stable for all pulling speeds V when the solute concentration c<c*. When c>c*, there is a low-speed instability when the melt is constitutionally undercooled and a high-speed stabilization due to surface energy when the absolute-stability criterion is reached. The two branches meet at c*. In this paper two-dimensional weakly nonlinear instabilities are studied for small -c-c*- so both the low- and high-speed bifurcations can be simultaneously described. When the thermal-conductivity ratio n=kS/kL is small (large), both bifurcations are supercritical (subcritical) and shallow (deep) cells emerge. For intermediate n there is complex behavior exhibiting multiple transitions, separated branches (isolas), and multiple cellular states.