Explicit solutions for one-dimensional two-phase free boundary problems with either shrinkage or expansion

Abstract

We consider a one-dimensional solidification of a pure substance which is initially in liquid state in a bounded interval. Initially, the liquid is above the freezing temperature, and cooling is applied at x= 0 while the other end x= l is kept adiabatic. At the time t = 0, the temperature of the liquid at x= 0 comes down to the freezing point and solidification begins, where x=s(t) is the position of the solid-liquid interface. As the liquid solidifies, it shrinks (0 < r < 1) or expands (r < 0) and appears a region between x=0 and x= rs(t), with r < 1. Temperature distributions of the solid and liquid phases and the position of the two free boundaries (x= rs(t) and x= s(t)) in the solidification process are studied. For three different cases, changing the condition on the free boundary x= rs(t) (temperature boundary condition, heat flux boundary condition and convective boundary condition) an explicit solution is obtained. Moreover, the solution of each problem is given as a function of a parameter which is the unique solution of a transcendental equation and for two of the three cases a condition on the parameter must be verified by data of the problem in order to have an instantaneous phase-change process. In all the cases, the explicit solution is given by a representation of the similarity type.