Heat Conduction With Melting or Freezing in a Corner

Abstract

This paper presents an analytical solution to the two-dimensional free boundary problem of solidification of a liquid, initially at a uniform temperature and filling the quarter-space x, y > 0, subject to a constant wall temperature. The problem is the two-dimensional analog of Neumann’s freezing problem. The solution is characterized by similarity in the variables x/t1/2, y/t1/2 and is obtained by treating the heat of solidification as a moving heat source. A nonlinear, singular, integro-differential equation for the solid-liquid interface is thereby derived and used to establish superhyperbolas to approximate the interface position. Results are presented for a range of the two dimensionless parameters of the problem. The accuracy of the superhyperbolic representation of the interface position is determined by comparison with a finite-difference solution. Equations are given for the calculation of the temperature fields in the solid and liquid regions that are valid for all time (i.e., they are not necessarily short- or long-time solutions).