Abstract
This report concerns the solidification of a “supercooled” liquid, whose temperature is initially below the equilibrium melt temperature, Tm of the solid. A new approach, the phase-field method, will be applied for this Stefan problem with supercooling, which simulates the solidification process of a pure material into a supercooled liquid in a spherical region. The advantage of the phase-field method is that it bypasses explicitly tracking the freezing front. In this approach the solid-liquid interface is treated as diffuse, and a dynamic equation for the phase variable is introduced in addition to the equation for heat flow. Thus, there are two coupled partial differential equations for temperature and phase field. In the reported study, an implicit numerical scheme using finite-difference techniques on a uniform mesh is employed to solve both Fourier phase-field equations and non-Fourier (known as damped wave or telegraph) phase-field equations. The latter gurantees a finite speed of propagation for the solidification front. Both Fourier (parabolic) and non-Fourier (hyperbolic) Stefan problems with supercooling are satisfactorily simulated and their solutions compared in the present work.
State-constrained optimal control of phase-field equations with obstacle