Liquid droplet dynamics are widely used in biological and engineering applications, which contain complex interfacial instabilities and pattern formation such as droplet merging, splitting, and transport. This paper studies a class of mean field control formulations for these droplet dynamics, which can be used to control and manipulate 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. As an example, a lubrication equation for a thin volatile liquid film laden with an active suspension is developed, with control achieved through its activity field. Lastly, we 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：液滴动力学在生物与工程应用中广泛存在，涉及复杂的界面不稳定性与模式形成，如液滴合并、分裂和输运。本文研究了一类针对此类液滴动力学的平均场控制模型，可用于应用中液滴的控制与操控。我们首先将液滴动力学表述为具有非线性迁移率的修正最优输运度量下自由能泛函的梯度流。随后，我们为这些梯度流设计了一个最优控制问题。作为一个示例，我们推导了含有活性悬浮颗粒的挥发性薄液膜的润滑方程，并通过其活性场实现控制。最后，我们应用原始-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了数值模拟。在二维空间域上的数值算例，包括液滴形成、珠状聚集/铺展、输运以及合并/分裂，验证了所提出的平均场控制机制的有效性。