Gradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface
Gradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface
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DOI：
10.4171/ifb/451
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发表时间：
2020-01
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影响因子：
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通讯作者：
Yuan Gao;Jian‐Guo Liu
中科院分区：
文献类型：
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作者：
Yuan Gao;Jian‐Guo Liu
我们研究在没有惯性和粘性应力影响的情况下，液滴在倾斜的粗糙表面上移动的动力学。在这种情况下，就润湿域和毛细管表面而言，液滴的动力学是纯粹的几何运动。使用单个图形表示，我们将这种几何运动解释为希尔伯特流形上的梯度流。我们提出了无条件稳定的一阶/二阶数值方案来模拟液滴的这种几何运动，这是使用平均曲率与移动接触线相结合的运动来描述的。该方案基于（i）显式移动边界，它将接触线和毛细管表面的动态更新解耦，（ii）移动网格上的半拉格朗日方法，以及（iii）具有高达二阶精度的非线性椭圆求解器的预测校正器方法。对于数值格式中具有连续空间变量的准静态动力学的情况，我们证明了一阶/二阶数值格式的稳定性和收敛性。为了证明所提出方案的准确性和长期验证，构建了几个具有挑战性的计算示例，包括呼吸液滴、不均匀粗糙表面上的液滴和准静态开尔文悬垂液滴，并与通过去奇异化微分代数方程组（DAE）获得的准静态动力学的精确解进行比较。
We study the dynamics of a droplet moving on an inclined rough surface in the absence of inertial and viscous stress effects. In this case, the dynamics of the droplet is a purely geometric motion in terms of the wetting domain and the capillary surface. Using a single graph representation, we interpret this geometric motion as a gradient flow on a Hilbert manifold. We propose unconditionally stable first/second order numerical schemes to simulate this geometric motion of the droplet, which is described using motion by mean curvature coupled with moving contact lines. The schemes are based on (i) explicit moving boundaries, which decouple the dynamic updates of the contact lines and the capillary surface, (ii) a semi-Lagrangian method on moving grids and (iii) a predictor-corrector method with a nonlinear elliptic solver upto second order accuracy. For the case of quasi-static dynamics with continuous spatial variable in the numerical schemes, we prove the stability and convergence of the first/second order numerical schemes. To demonstrate the accuracy and long-time validation of the proposed schemes, several challenging computational examples - including breathing droplets, droplets on inhomogeneous rough surfaces and quasi-static Kelvin pendant droplets - are constructed and compared with exact solutions to quasi-static dynamics obtained by desingularized differential-algebraic system of equations (DAEs).