Liquid droplet dynamics are widely used in biological and engineering applications, which contain complex interfacial instabilities and pattern formulation such as droplet merging, splitting, and transport. This paper studies a class of mean field control formulation towards these droplet dynamics. They are used to control and maintain the manipulation of droplets in applications. We first formulate the droplet dynamics as gradient flows of free energies in modified optimal transport metrics with nonlinear mobilities. We then design an optimal control problem for these gradient flows. We lastly apply the primal-dual hybrid gradient algorithm with high-order finite element methods to simulate the proposed mean field control problems. Numerical examples, including droplet formation, bead-up/spreading, transport, and merging/splitting on a two-dimensional spatial domain, demonstrate the effectiveness of the proposed mean field control mechanism.

翻译：液滴动力学广泛应用于生物与工程领域，其包含复杂的界面不稳定性和图案形成过程，如液滴合并、分裂与输运。本文研究针对这类液滴动力学的一类平均场控制公式，用于控制并维持应用中液滴的操控操作。我们首先将液滴动力学表述为具有非线性迁移率的改进最优输运度量下自由能的梯度流，进而针对这些梯度流设计最优控制问题，最后采用原始-对偶混合梯度算法结合高阶有限元方法对提出的平均场控制问题进行数值模拟。二维空间域上的数值算例（包括液滴形成、隆起/铺展、输运、合并与分裂）验证了所提出的平均场控制机制的有效性。