A one-phase Stefan problem with size-dependent thermal conductivity

Abstract

• A one-phase Stefan problem with a size-dependent thermal conductivity is formulated. • The size-dependent thermal conductivity is relevant in the context of solidification at the nanoscale. • Approximate analytical and numerical solutions to the model are found. • The size-dependent thermal conductivity induces a delay in the propagation of the solidification front. • The speed of the front is constant as time goes to 0, in contrast to the solution of the classical Stefan problem. In this paper a one-phase Stefan problem with size-dependent thermal conductivity is analysed. Approximate solutions to the problem are found via perturbation and numerical methods, and compared to the Neumann solution for the equivalent Stefan problem with constant conductivity. We find that the size-dependant thermal conductivity, relevant in the context of solidification at the nanoscale, slows down the solidification process. A small time asymptotic analysis reveals that the position of the solidification front in this regime behaves linearly with time, in contrast to the Neumann solution characterized by a square root of time proportionality. This has an important physical consequence, namely the speed of the front predicted by size-dependant conductivity model is finite while the Neumann solution predicts an infinite and, thus, unrealistic speed as t → 0.