The geometric motion of small droplets placed on an impermeable textured substrate is mainly driven by the capillary effect, the competition among surface tensions of three phases at the moving contact lines, and the impermeable substrate obstacle. After introducing an infinite dimensional manifold with an admissible tangent space on the boundary of the manifold, by Onsager's principle for an obstacle problem, we derive the associated parabolic variational inequalities. These variational inequalities can be used to simulate the contact line dynamics with unavoidable merging and splitting of droplets due to the impermeable obstacle. To efficiently solve the parabolic variational inequality, we propose an unconditional stable explicit boundary updating scheme coupled with a projection method. The explicit boundary updating efficiently decouples the computation of the motion by mean curvature of the capillary surface and the moving contact lines. Meanwhile, the projection step efficiently splits the difficulties brought by the obstacle and the motion by mean curvature of the capillary surface. Furthermore, we prove the unconditional stability of the scheme and present an accuracy check. The convergence of the proposed scheme is also proved using a nonlinear Trotter-Kato's product formula under the pinning contact line assumption. After incorporating the phase transition information at splitting points, several challenging examples including splitting and merging of droplets are demonstrated.
翻译:小滴滴子的几何运动被放置在透透质基底部上,主要受以下因素的驱动:毛细效应、移动接触线三个阶段表面紧张状态之间的竞争以及不可渗透基质障碍。在根据Onsager关于障碍问题的原则,在柱子边界上引入一个可允许的分流空间的无限维度多元体之后,我们得出相关的抛物线变异性不平等。这些差异性不平等可以用来模拟接触线的动态,由于不可渗透障碍而不可避免地合并和分解滴子。为了有效解决抛物体变异性不平等,我们提议了一个无条件稳定的明确边界更新计划,并辅之以一个预测方法。对明确边界的更新有效地分解了以毛细表面和移动接触线中的平均曲线来计算运动。同时,预测步骤有效地分割了障碍带来的困难和毛细表面的平均曲性运动。此外,我们证明了这个计划无条件的稳定,并提出了准确的检查。为了有效解决抛物变异性差异,我们提议的方案将无条件稳定、明确的边界更新计划与预测方法结合起来,同时提出一个预测方法。对运动面表面表面和分解式的分解模型的假设也证明是具有挑战性的。