相场模型、尖锐界面极限及接触线动力学的数值格式
Phase-Field Models, Sharp Interface Limits, and Numerical Schemes for Contact Line Dynamics
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中文总结 AI 辅助
研究固体基底上液滴接触线动力学的相场和尖锐界面模型，通过匹配渐近展开恢复相应尖锐界面极限，识别出一致梯度流结构，基于最小化运动原理开发数值格式，为接触线运动提供统一变分原理解释及相关结果。
中文摘要 AI 辅助
我们在统一变分框架下研究固体基底上液滴接触线动力学的相场和尖锐界面模型。接触线处液、气、固三相交汇，因无滑移流体动力学的经典应力奇异性给连续介质建模带来根本困难。相场模型通过引入厚度为的薄过渡层并在基底上用壁能增强的金兹堡 - 朗道自由能来编码界面效应，从而正则化此奇异性。从总自由能\(E = E_b + E_w\)出发，分析了两个相场模型：艾伦 - 卡恩方程和卡恩 - 希利尔方程。利用匹配渐近展开当\(\delta \to 0\)时恢复其相应的尖锐界面极限。在艾伦 - 卡恩情形下，极限产生平均曲率运动且接触线定律由动态接触角与杨氏角的偏差驱动。在卡恩 - 希利尔情形下，极限导致具有相同形式接触线动力学的穆林斯 - 塞克卡问题。这项工作的一个核心结果是识别出两个模型一致的梯度流结构。艾伦 - 卡恩动力学对应\(L^2\)梯度流，卡恩 - 希利尔动力学对应\(H^{-1}\)梯度流，且两者都收敛到保持相同能量耗散结构的尖锐界面演化。这为接触线运动提供了基于单一变分原理的统一解释。最后，基于最小化运动原理开发了能量稳定的数值格式，并建立了全离散问题的离散能量耗散和适定性。数值例子证实两种格式都趋向于相同的稳态尖锐界面解，同时它们的动力学反映了不同的耗散机制。
英文摘要
We study phase-field and sharp-interface models for contact line dynamics of a liquid droplet on a solid substrate within a unified variational framework. The motion of the contact line, where liquid, gas, and solid phases meet, poses a fundamental difficulty in continuum modeling due to the classical stress singularity of no-slip hydrodynamics. Phase-field models regularize this singularity by introducing a thin transition layer of thickness and encoding interfacial effects through a Ginzburg-Landau free energy augmented by a wall energy on the substrate. Starting from the total free energy $E = E_b + E_w$, we analyze two phase-field models: the Allen-Cahn equation and the Cahn-Hilliard equation. Using matched asymptotic expansions as $δ\to 0$, we recover their corresponding sharp interface limits. In the Allen-Cahn case, the limit yields motion by mean curvature with a contact line law driven by deviations of the dynamic contact angle from Young's angle. In the Cahn-Hilliard case, the limit leads to a Mullins-Sekerka problem with the same form of contact line dynamics. A central result of this work is the identification of consistent gradient-flow structures across both models. The Allen-Cahn dynamics correspond to an $L^2$-gradient flow, while the Cahn-Hilliard dynamics correspond to an $H^{-1}$-gradient flow, and both converge to sharp-interface evolutions that preserve the same energy-dissipation structure. This provides a unified interpretation of contact line motion as a consequence of a single variational principle. Finally, we develop energy-stable numerical schemes based on the minimizing movement principle and establish discrete energy dissipation and well-posedness of the fully discrete problem. Numerical examples confirm that both schemes relax toward the same stationary sharp interface solution while their dynamics reflect the different dissipation mechanisms.
