[Paper Review] Homogenization of the Stefan problem, with application to maple sap exudation

Isabell Graf, John M. Stockie|arXiv (Cornell University)|Nov 12, 2014

Advanced Mathematical Modeling in Engineering22 references3 citations

TL;DR

This paper develops a periodic homogenization framework using two-scale convergence to model heat transfer and phase change in periodic microstructures, applying it first to a simplified Stefan problem with melting ice bars and then to the complex phenomenon of maple sap exudation. The key contribution is a reduced macroscale model for temperature evolution that captures effective thermal behavior, with all microscale physics encoded in a reference cell problem, enabling efficient numerical simulation of sap pressure generation in trees during spring thaw.

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.

Motivation & Objective

• To develop a rigorous mathematical framework for homogenizing the Stefan problem in periodic media with phase change.
• To model the complex multi-phase flow and heat transport in maple tree xylem, including liquid, frozen, and gaseous sap phases.
• To isolate the macroscale temperature dynamics from microscale physics via a reference cell problem, enabling efficient numerical simulation.
• To validate the model through simulations and provide insight into the physical mechanism of sap exudation in sugar maple trees.

Proposed method

• Applies periodic homogenization with two-scale convergence to a reduced Stefan problem involving periodic arrays of ice bars in water.
• Introduces a two-scale formulation with a macroscale temperature field and a microscale reference cell problem in the periodic unit cell $Y$.
• Solves a coupled system where the macroscale problem governs global temperature, and the cell problem captures local phase change dynamics.
• Uses weak and strong formulations of the limit problem to derive a numerically tractable algorithm for solving the homogenized equations.
• Derives a priori estimates and proves existence, uniqueness, and convergence of the two-scale limit solution using energy methods and Gronwall's inequality.
• Applies the same homogenized model to the full maple sap exudation problem, treating all microscale physics (multiphase flow, osmosis, gas effects) through the reference cell problem.

Experimental results

Research questions

• RQ1 How can periodic homogenization be rigorously applied to a two-phase Stefan problem with temperature-dependent diffusion coefficients?
• RQ2 What is the structure of the two-scale limit system for heat transfer and phase change in a periodic medium with phase transitions?
• RQ3 How does the homogenized model capture the essential physics of sap exudation in maple trees while reducing computational complexity?
• RQ4 Can the same homogenized equations be applied to both a simplified model and the full biological problem with consistent physical interpretation?
• RQ5 What numerical algorithm can efficiently solve the homogenized system while preserving the physical fidelity of the microscale processes?

Key findings

• The two-scale limit solution exists, is unique, and converges strongly in the appropriate function spaces, as proven via energy estimates and Gronwall's inequality.
• The homogenized macroscale problem governs the effective temperature field, while all microscale phase change dynamics are encoded in a reference cell problem on the periodic unit cell $Y$.
• The model successfully reduces the complexity of the full multiphase flow problem in tree xylem to a single set of equations for the global temperature, with microscale effects captured in the cell problem.
• Numerical simulations confirm the model's ability to reproduce key features of sap exudation, including pressure build-up during the spring thaw.
• The approach enables efficient simulation of sap pressure generation without resolving the fine-scale cellular structure, making it suitable for studying biological phenomena at the whole-organism level.
• The homogenized equations are robust and consistent across both the reduced model and the full maple sap exudation problem, validating the method's generality.