Selection mechanisms for multiple similarity solutions for solidification and melting

Abstract

When a solid phase of uniform temperature T S∞ and composition C S∞ is brought into contact with a liquid phase of uniform temperature T L∞ and composition C L∞, there exist (under the assumption of local equilibrium at the solid–liquid interface) similarity solutions for which the position X of the interface is proportional to the square root of time t, i.e., X=λ t . Recently, we explored situations in which there are multiple similarity solutions, e.g., three values of λ for the same far-field and initial conditions. We examine the stability of these similarity solutions with respect to perturbations which preserve the planar geometry. When there are three similarity solutions, we find that the solution corresponding to the intermediate value of λ is unstable and the other two solutions are stable. We then relax the assumption of local equilibrium by assuming a linear kinetic law in which the interface velocity is proportional to the deviation of interface temperature from the equilibrium temperature. We use an expansion technique, valid at small times, which leads to a single solution with a finite initial velocity. Time-dependent numerical calculations of the complete set of governing equations are used to follow the transition from kinetic control to a similarity solution.