In this paper, we propose a Poisson-Nernst-Planck-Navier-Stokes-Cahn-Hillard (PNP-NS-CH)model for an electrically charged droplet suspended in a viscous fluid subjected to an external electric field. Our model incorporates spatial variations of electric permittivity and diffusion constants, as well as interfacial capacitance. Based on a time scale analysis, we derive two approximations of the original model, namely a dynamic model for the net charge and a leaky-dielectric model. For the leaky-dielectric model, we conduct a detailed asymptotic analysis to demonstrate the convergence of the diffusive-interface leaky-dielectric model to the sharp interface model as the interface thickness approaches zero. Numerical computations are performed to validate the asymptotic analysis and demonstrate the model's effectiveness in handling topology changes, such as electrocoalescence. Our numerical results of these two approximation models reveal that the polarization force, which is induced by the spatial variation of electric permittivity in the direction perpendicular to the external electric field, consistently dominates the Lorentz force, which arises from the net charge. The equilibrium shape of droplets is determined by the interplay between these two forces along the direction of the electric field. Furthermore, in the presence of the interfacial capacitance, a local variation of effective permittivity leads to an accumulation of counter-ions near the interface, resulting in a reduction in droplet deformation. Our numerical solutions also confirm that the leaky dielectric model serves as a reasonable approximation of the original PNP-NS-CH model when the electric relaxation time is sufficiently short. The Lorentz force and droplet deformation both decrease when the diffusion of net charge is significant.

翻译：本文提出了一种适用于外电场作用下悬浮于粘性流体中带电液滴的泊松-能斯特-普朗克-纳维-斯托克斯-卡恩-希利亚德（PNP-NS-CH）模型。该模型考虑了介电常数与扩散系数的空间变化，以及界面电容特性。基于时间尺度分析，我们推导出原模型的两种近似形式：净电荷动态模型与漏电介质模型。针对漏电介质模型，我们通过详细的渐近分析证明了当界面厚度趋近零时，扩散界面漏电介质模型收敛于尖锐界面模型。通过数值计算验证了渐近分析的正确性，并展示了模型处理拓扑变化（如电聚结）的有效性。两种近似模型的数值结果表明：由垂直外电场方向介电常数空间变化诱导的极化力始终主导源于净电荷的洛伦兹力，液滴平衡形状由这两种力沿电场方向的相互作用决定。此外，在界面电容存在时，有效介电常数的局部变化导致反离子在界面附近积累，从而减小液滴形变。数值解还证实，当电弛豫时间足够短时，漏电介质模型可作为原始PNP-NS-CH模型的合理近似。净电荷的显著扩散会同时降低洛伦兹力与液滴形变程度。