非扩散Stefan问题的解析解和数值解
Analytical and numerical solutions to the non-diffusive Stefan problem
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中文总结 AI 辅助
研究非扩散Stefan问题，用麦克斯韦-卡塔尼奥-韦尔诺特方程建模，通过微扰级数展开求解，提出修正策略，研究了Stefan数、无量纲热弛豫时间和热扩散率对模型误差的影响。
中文摘要 AI 辅助
在这项工作中，麦克斯韦-卡塔尼奥-韦尔诺特（MCV）方程被用于在小Stefan数（Ste $\ll$ 1）极限下对一维双曲Stefan问题进行建模。通过将时间表示为固液界面位置的函数的重新表述，用微扰级数展开近似求解。首先在考虑相变界面处扩散热传递的框架下导出解析解，为便于分析。提出两种修正策略来解决该公式中的渐近发散：重新缩放的内解与外解结合得到复合解，以及与尺寸相关的热物理系统参数以更好地捕捉相变界面处的双曲效应。所得界面轮廓呈现出类似抛物线的特征形状，与扩散Stefan问题的结果一致，在较大热弛豫时间下有明显的早期双曲效应。对无量纲系统中的三个相关变量：Stefan数（$\mathrm{Ste}$）、无量纲热弛豫时间（$\widetilde \tau$）和热扩散率（$\alpha$）进行了参数研究。研究表明，模型误差与Stefan数的缩放符合微扰展开的理论截断误差。此外，较大的$\widetilde \tau$值会放大早期双曲效应，从而增加模型误差，而较大的$\alpha$会扩展这些双曲效应显著的相对时间域，也对应于模型误差的增加。
英文摘要
In this work, the Maxwell--Cattaneo--Vernotte (MCV) equation is used to model the one-dimensional hyperbolic Stefan problem in the limit of a small Stefan number (Ste $\ll$ 1). The solutions are approximated with perturbation series expansions using a reformulation in which time is expressed as a function of the solid-liquid interface position. The first proposed solution is derived in a framework that considers diffusive heat transfer at the phase change interface, for analytic tractability. Two rectification strategies are proposed to address the asymptotic divergence present in this formulation: a rescaled inner solution which is then combined with the outer solution to yield a composite solution, and size-dependent thermo-physical system parameters for better capture of hyperbolic effects at the phase change interface. The resulting interface profiles exhibit a characteristic parabolic-like shape, consistent with diffusive Stefan problem findings, with pronounced early-time hyperbolic effects at larger thermal relaxation times. Parametric studies are done over three pertinent variables in the dimensionless system: the Stefan number ($\mathrm{Ste}$), the dimensionless thermal relaxation time ($\widetilde τ$), and the thermal diffusivity ($α$). The studies suggest that model error scales with the Stefan number in accordance with the theoretical truncation error of the perturbation expansion. Additionally, larger values of $\widetilde τ$ amplify early-time hyperbolic effects, thereby increasing model error, while larger $α$ extends the relative temporal domain over which these hyperbolic effects remain significant, also corresponding to an increase in model error.
发表机构
- McGill University（麦吉尔大学）
- University of Toronto（多伦多大学）
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