D. Riley, S. H. Davis
Feb 1, 1990
Siam Journal on Mathematical Analysis
Long-wave instabilities in a directionally-solidified binary mixture may occur in several limits. Sivashinsky identified a small-segregation-coefficient limit and obtained a weakly-nonlinear evolution equation governing subcritical two-dimensional bifurcation. Brattkus and Davis identified a near-absolute-stability limit and obtained a strongly-nonlinear evolution equation governing supercritical two-dimensional bifurcation. The present investigation identifies a third, strongly-nonlinear, evolution equation, arising in the small-segregation-coefficient, large-surface-energy limit. This equation links both of the former and describes the change from the sub- to super-critical bifurcations. This study sets the previous long-wave analyses into a logical framework.