Selection mechanisms for multiple similarity solutions for solidification and melting

When a solid phase of uniform temperature T-S infinity and composition C-S infinity is brought into contact with a liquid phase of uniform temperature T-L infinity and composition C-L infinity there exist (under the assumptio n of local equilibrium at the solid-liquid interface) similarity solutions for which the position X of the interface is proportional to the square roo t of time t, i.e., X = lambda root t. Recently, we explored situations in w hich there are multiple similarity solutions, e.g., three values of lambda, for the same far-field and initial conditions. We examine the stability of these similarity solutions with respect to perturbations which preserve th e planar geometry. When there are three similarity solutions, we find that the solution corresponding to the intermediate value of lambda, is unstable and the other two solutions are stable. We then relax the assumption of lo cal equilibrium by assuming a linear kinetic law in which the interface vel ocity is proportional to the deviation of interface temperature from the eq uilibrium temperature. We use an expansion technique, valid at small. times , which leads to a single solution with a finite initial velocity. Time-dep endent numerical calculations of the complete set of governing equations ar e used to follow the transition from kinetic control to a similarity soluti on. (C) 1999 Elsevier Science B.V. All rights reserved.