The study of the shape of droplets on surfaces is an important problem in the physics of fluids and has applications in multiple industries, from agrichemical spraying to microfluidic devices. Motivated by these real-world applications, computational predictions for droplet shapes on complex substrates -- rough and chemically heterogeneous surfaces -- are desired. Grid-based discretisations in axisymmetric coordinates form the basis of well-established numerical solution methods in this area, but when the problem is not axisymmetric, the shape of the contact line and the distribution of the contact angle around it are unknown. Recently, particle methods, such as pairwise force smoothed particle hydrodynamics (PF-SPH), have been used to conveniently forego explicit enforcement of the contact angle. The pairwise force model, however, is far from mature, and there is no consensus in the literature on the choice of pairwise force profile. We propose a new pair of polynomial force profiles with a simple motivation and validate the PF-SPH model in both static and dynamic tests. We demonstrate its capabilities by computing droplet shapes on a physically structured surface, a surface with a hydrophilic stripe, and a virtual wheat leaf with both micro-scale roughness and variable wettability. We anticipate that this model can be extended to dynamic scenarios, such as droplet spreading or impaction, in the future.

翻译：表面液滴形状研究是流体物理学中的重要课题，在从农业化学喷洒到微流控设备等多个工业领域具有应用价值。受这些实际应用的驱动，人们需要针对复杂基底（粗糙且化学非均匀表面）上液滴形状的计算预测。轴对称坐标系中的网格离散化构成了该领域成熟数值求解方法的基础，但当问题不具备轴对称性时，接触线形状及其周围接触角分布均属未知。近年来，粒子方法（如成对力光滑粒子流体动力学）已被用于便捷地避免接触角的显式强制实施。然而，成对力模型远未成熟，文献中对于成对力分布函数的选择尚未形成共识。我们提出了一对具有简明物理动机的多项式力分布函数，并通过静态与动态测试验证了PF-SPH模型。通过计算物理结构化表面、亲水条纹表面以及兼具微尺度粗糙度与可变润湿性的虚拟小麦叶片上的液滴形状，我们展示了该模型的性能。我们预期该模型未来可扩展至动态场景，如液滴铺展或撞击过程。