A note on the existence of a solution to a problem of Stefan

Abstract

When certain metals are heated slowly, the temperature rises until it reaches a critical temperature at which the structure of the metal changes from one crystalline form to another. As for example, iron changes from a to (3 crystals at 1643°F. Accompanying this change of crystalline form is a latent heat of recry&tallization. In order to study the process we investigate the associated mathematical problem, which requires the solution of a partial differential equation in a region with an undetermined boundary. Our analysis establishes the existence and uniqueness of the solution. In a previous paper2 this problem is treated from the point of view of computing the solution. Suppose a metal slab having two infinite parallel faces is brought uniformly to the critical temperature and then heated by a uniform source covering the front face while an insulator covers the back face. Under these conditions, the new crystals are first formed at the front face, and the interface between the new and old crystals travels from the front face to the back face. Mathematically the problem can be stated as follows, where u = 0 is taken as the critical temperature: Find the temperature, u = u(x, t), and the curve, x = x(t), which satisfy the following conditions ut = auxx for 0 < x 0 (3) x(0) = 0 (4) ux(0, t) =-g (5) where g is a constant > 0. In this notation u, = du/dt, uxx-d2u/dx2, x'(t) = dx{t)/dt, a is the coefficient of thermal diffusivity, A-pH/k where p is the density of the metal, H is the latent heat of recryst-allization, and k is the coefficient of thermal conductivity. We simplify the notation by introducing new variables as follows: v(y, t) = u(x, t)/Aa y = gx/Aa (6) r = g2t/AW;