Asymptotic analysis of a two-phase Stefan problem in an annulus with the convective boundary

Abstract

The Stefan problem (whether in its classical form or not) has assumed that the interface moves instantaneously with time as a prescribed initial condition. This assumption is valid in the problems subjected to an isothermal boundary. Yet, it may not represent the physics for a convective boundary since a certain amount of time is needed to cool or melt the convective surface to its fusion depending on the heat transfer coefficient (or Biot number in a dimensionless way). In this study, we formulate a two-phase Stefan problem in an annulus for outward solidification subjected to a convective or Robin boundary condition while not assuming that the interface moves right away at time t=0. A comprehensive asymptotic analysis is performed by expanding around a small Stefan number; four spatial and five temporal scales are characterized based on the Stefan problems involving a convective surface. The method of property averaging is also employed at the scale where the equilibrium freezing occurs. The developed asymptotic solution is verified with numerical data generated by the enthalpy method at various Biot and Stefan numbers. The results demonstrate the significance of abandoning the ansatz (i.e., the interface does not move immediately with time), especially towards small Biot numbers. Further, it is found that the presented asymptotic solution considerably extends the valid range of the Stefan number when compared with the conventional asymptotic technique.