TY - JOUR
T1 - Classical two-phase Stefan problem for spheres
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
JO - PROC R SOC A
SP - 2055
LP - 2076
M3 - 10.1098/rspa.2007.0315
VL - 464
IS - 2096
AU - McCue, Scott W
AU - Wu, Bisheng
AU - Hill, James M
Y1 - 2008/08/08
UR - http://rspa.royalsocietypublishing.org/content/464/2096/2055.abstract
N2 - The classical Stefan problem for freezing (or melting) a sphere is usually treated by assuming that the sphere is initially at the fusion temperature, so that heat flows in one phase only. Even in this idealized case there is no (known) exact solution, and the only way to obtain meaningful results is through numerical or approximate means. In this study, the full two-phase problem is considered, and in particular, attention is given to the large Stefan number limit. By applying the method of matched asymptotic expansions, the temperature in both the phases is shown to depend algebraically on the inverse Stefan number on the first time scale, but at later times the two phases essentially decouple, with the inner core contributing only exponentially small terms to the location of the solidâ€“melt interface. This analysis is complemented by applying a small-time perturbation scheme and by presenting numerical results calculated using an enthalpy method. The limits of zero Stefan number and slow diffusion in the inner core are also noted.
ER -