Homogenization of the Stefan problem, with application to maple sap exudation

Abstract

The technique of periodic homogenization with two-scale convergence is applied to the analysis of a two-phase Stefan-type problem that arises in the study of a periodic array of melting ice bars. For this "reduced model" we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, which requires solving a macroscale problem for the global temperature field and a reference cell problem at each point in space which captures the underlying phase change process occurring on the microscale. We state a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. The same homogenized temperature equations are then applied to solve a much more complicated problem involving multi-phase flow and heat transport in trees, where the sap is present in both frozen and liquid forms and a third gas phase is also present. Our homogenization approach has the advantage that the global temperature field is a solution of the same reduced model equations, while all the remaining physics are relegated to the reference cell problem. Numerical simulations are performed to validate our results and draw conclusions regarding the phenomenon known as sap exudation, which is of great importance in sugar maple trees and few other related species.