[论文解读] Fixed-grid sharp-interface numerical solutions to the three-phase spherical Stefan problem

Yavkreet Swami, Jacob Barajas+1 more|arXiv (Cornell University)|Feb 9, 2026

Phase Change Materials Research被引用 0

一句话总结

本文将固定网格尖界面方法扩展到有限尺寸粒子的三相球形Stefan问题，纳入密度和动能跃变，验证结果与两相结果一致，且表明在纳米尺度粒子中动能效应显著，但随粒子尺寸增大而减弱。

ABSTRACT

Many metal manufacturing processes involve phase change phenomena, which include melting, boiling, and vaporization. These phenomena often occur concurrently. A prototypical 1D model for understanding the phase change phenomena is the Stefan problem. There is a large body of literature discussing the analytical solution to the two-phase Stefan problem that describes only the melting or boiling of phase change materials (PCMs) with one moving interface. Density-change effects that induce additional fluid flow during phase change are generally neglected in the literature to simplify the math of the Stefan problem. In our recent work [1], we provide analytical and numerical solutions to the three-phase Stefan problem with simultaneous occurrences of melting, solidification, boiling, and condensation in Cartesian coordinates. Our current work builds on our previous work to solve a more challenging problem: the three-phase Stefan problem in spherical coordinates for finite-sized particles. There are three moving interfaces in this system: the melt front, the boiling front, and the outer boundary which is in contact with the atmosphere. Although an analytical solution could not be found for this problem, we solved the governing equations using a fixed-grid sharp-interface method with second-order spatio-temporal accuracy. Using a small-time analytical solution, we predict a reasonably accurate estimate of temperature (in the three phases) and interface positions and velocities at the start of the simulation. Our numerical method is validated by reproducing the two-phase nanoparticle melting results of Font et al. [2]. Lastly, we solve the three-phase Stefan problems numerically to demonstrate the importance of kinetic energy terms during phase change of smaller (nano) particles. In contrast, these effects diminish for large particles (microns and larger).

研究动机与目标

• 将三相Stefan问题建模从平面几何扩展到有限尺寸粒子所需的球形几何。
• 在所有界面保留密度和动能跃变条件。
• 开发并应用固定网格尖界面浸没边界方法于球坐标系。
• 使用小时间解析解对初始移动界面进行正确启动。
• 结合已知的两相结果进行方法验证，并分析不同粒子尺度下动能项的作用。

提出的方法

• 给出固相、液相和气相的守恒方程（包括密度、能量和动量，以及界面跃变和Diracδ项）。
• 对控制方程进行无量纲化，并推导初始初始化所需的小时间解析解。
• 在球坐标下实现二阶固定网格尖界面方法，并利用浸没边界追踪移动界面。
• 在低/高密度比极限下进行小时间分析，以获得初始界面位置、速度和温度场。
• 在有限大小的球体上求解三相球形Stefan问题，研究界面演化以及密度和动能项的影响。
• 通过数值验证再现Font等人两相纳米颗粒熔化结果。

(a) Two-phase spherical Stefan problem

实验结果

研究问题

• RQ1 三相球形Stefan问题在有限尺寸粒子中同时发生熔化和沸腾前沿时如何演化？
• RQ2 密度跃变与动能跃变对不同粒子尺度的界面位置、速度与熔/沸时间有何影响？
• RQ3 在球坐标下固定网格尖界面方法对多相Stefan问题的准确性如何？
• RQ4 动能项对纳米尺度粒子与微米尺度粒子的熔化时间有何不同影响？

主要发现

• 动能项对较小（纳米级）粒子的熔化时间有显著影响。
• 对于较大粒子（微米及以上），动能项的影响减弱。
• 该固定网格尖界面方法在空间和时间上达到二阶精度。
• 通过再现Font等人两相纳米颗粒熔化结果对方法进行数值验证。
• 三相模拟量化了在同时熔化与沸腾过程中各相之间密度和动能项的作用，显示在小粒子中它们的重要性。

(b) Three-phase spherical Stefan problem