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

We consider a one-dimensional solidification of a pure substance which is initially in liquid state in a bounded interval $[0,l]$. 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 $x=s\left(t\right)$ 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 $x=rs\left(t\right)$, with $r<1$. Temperature distributions of the solid and liquid phases and the position of the two free boundaries ($x=rs\left(t\right)$ and $x=s\left(t\right)$) in the solidification process are studied. For three different cases, changing the condition on the free boundary $x=rs\left(t\right)$ (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.