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

M.F. NATALE; E.A. SANTILLAN MARCUS; D. A. TARZIA

NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS

PERGAMON-ELSEVIER SCIENCE LTD

Año: 2010 vol. 11 p. 1946 - 1952

1468-1218

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(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.