We investigate the one-dimensional growth of a solid into a liquid bath, starting from a small crystal, using the Guyer-Krumhansl and Maxwell-Cattaneo models of heat conduction. By breaking the solidification process into the relevant time regimes we are able to reduce the problem to a system of two coupled ordinary differential equations describing the evolution of the solid-liquid interface and the heat flux. The reduced formulation is in good agreement with numerical simulations. In the case of silicon, differences between classical and non-classical solidification kinetics are relatively small, but larger deviations can be observed in the evolution in time of the heat flux through the growing solid. From this study we conclude that the heat flux provides more information about the presence of non-classical modes of heat transport during phase-change processes.

中文翻译：

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具有非傅立叶热传导的一维 Stefan 问题

我们从一个小晶体开始，使用 Guyer-Krumhansl 和 Maxwell-Cattaneo 热传导模型研究了固体在液体浴中的一维生长。通过将凝固过程分解为相关的时间范围，我们能够将问题简化为两个耦合常微分方程系统，描述固液界面和热通量的演变。简化的公式与数值模拟非常吻合。在硅的情况下，经典和非经典凝固动力学之间的差异相对较小，但在通过生长固体的热通量随时间的演变中可以观察到较大的偏差。