Stefan melting problems involve the tracking of a sharp melt front during the heat conduction controlled melting of a solid. A feature of this problem is a jump discontinuity in the heat flux across the melt interface. Time fractional versions of this problem introduce fractional time derivatives into the governing equations. Starting from an appropriate thermodynamic balance statement, this paper develops a new sharp interface time fractional Stefan melting problem, the memory-enthalpy formulation. A mathematical analysis reveals that this formulation exhibits a natural regularization in that, unlike the classic Stefan problem, the flux is continuous across the melt interface. It is also shown how the memory-enthalpy formulation, along with previously reported time fractional Stefan problems based on a memory-flux, can be derived by starting from a generic continuity equation and melt front condition. The paper closes by mathematically proving that the memory-enthalpy fractional Stefan formulation is equivalent to the previous memory-flux formulations. A result that provides a thermodynamic consistent basis for a widely used and investigated class of time fractional (memory) Stefan problems.

中文翻译：

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锐界面记忆通量 Stefan 问题的焓公式

Stefan 熔化问题涉及在固体的热传导控制熔化过程中跟踪尖锐的熔化前沿。该问题的一个特点是熔体界面上热通量的跳跃不连续性。该问题的时间分数版本将分数时间导数引入控制方程。本文从适当的热力学平衡陈述出发，提出了一种新的尖锐界面时间分数Stefan熔化问题，即记忆-焓公式。数学分析表明，该公式表现出自然的正则化，与经典的 Stefan 问题不同，通量在熔体界面上是连续的。它还展示了如何通过从通用连续性方程和熔体前沿条件出发，推导出记忆焓公式以及先前报告的基于记忆通量的时间分数 Stefan 问题。论文最后用数学方法证明了记忆-焓分数 Stefan 公式与之前的记忆-通量公式是等价的。这一结果为广泛使用和研究的时间分数（记忆）Stefan 问题提供了热力学一致的基础。