We develop a level-set method in the finite-element framework. The contact line singularity is removed by the slip boundary condition proposed by Ren and E (2007) [6], which has two friction coefficients: ${\beta }_N$ that controls the slip between the bulk fluids and the solid wall and ${\beta }_{CL}$ that controls the deviation of the microscopic dynamic contact angle from the static one. The predicted contact line dynamics from our method matches the Cox theory very well. We further find that the same slip length in the Cox theory can be reproduced by different combinations of $({\beta }_N,{\beta }_{CL})$, based on which we come up with a computational strategy for mesh-independent results that can match the experiments. There is no need to impose the contact angle condition geometrically, and the dynamic contact angle automatically emerges as part of the numerical solution. With a little modification, our method can also be used to compute contact angle hysteresis, where the tendency of contact line motion is readily available from the level-set function. Different test cases, including code validation and mesh-convergence study, are provided to demonstrate the efficiency and capability of our method.

我们在有限元框架中开发了一种水平集方法。Ren和E（2007）[6]提出的滑移边界条件消除了接触线的奇异性，它具有两个摩擦系数：${\beta }_{ñ}$ 控制散装流体和固体壁之间的滑动 ${\beta }_{C\mathrm{大号}}$控制微观动态接触角与静态接触角的偏差。我们方法预测的接触线动力学与Cox理论非常吻合。我们进一步发现，考克斯理论中相同的滑移长度可以通过以下方法的不同组合来再现：$（{\beta }_{ñ}，{\beta }_{C\mathrm{大号}}）$，在此基础上，我们提出了可与实验匹配的网格无关结果的计算策略。无需在几何上施加接触角条件，并且动态接触角会自动出现为数值解的一部分。稍加修改，我们的方法也可以用于计算接触角滞后，其中可以通过水平设置函数轻松获得接触线运动的趋势。提供了不同的测试用例，包括代码验证和网格收敛研究，以证明我们方法的效率和功能。