A three-dimensional level set method for two-phase electrohydrodynamics with finite electric Reynolds number

Abstract

The electrohydrodynamics (EHD) of droplets under electric fields underpins technologies from ink-jet printing and electrosprays to droplet sorting and microfluidics, yet accurate prediction remains challenging because most existing studies are confined to two-dimensional or axisymmetric models and often neglect surface-charge convection, a mechanism that strongly modifies interfacial stresses and breakup. To address this gap, we develop a fully three-dimensional (3D) level-set computational framework for leaky–dielectric two-phase flows that resolves bulk charge conservation, interfacial surface-charge convection, and topology change over a wide range of electric Reynolds numbers $R{e}_E$ (the ratio of charge-relaxation to convection time) and electric capillary numbers $C{a}_E$ (the ratio of electric stress to surface tension). Unlike existing three-dimensional studies that either neglect surface-charge convection or are restricted to small deformations without breakup, our framework provides a comprehensive 3D treatment of finite-$R{e}_E$ charge convection, topology change, and breakup mapping. The method is carefully verified (mass conservation error $<0.5\text{\%}$) and validated against Taylor’s small-deformation theory and silicone–castor oil experiments, confirming quantitative accuracy. Our simulations demonstrate that surface-charge convection redistributes interfacial charges, weakens EHD circulation, suppresses oblate deformation, and enhances prolate deformation; three-dimensional charge maps and two-dimensional cross-sectional contours quantify these effects in detail. For prolate drops, we capture and classify breakup transitions in full 3D — from end-pinching to conic cusping and ultimately tip streaming — and construct a comprehensive $(C{a}_E,R{e}_E)$ phase diagram. By integrating finite-$R{e}_E$ effects, 3D surface-charge diagnostics, and breakup mapping in a validated computational method, this study establishes a novel predictive framework for electric-field-driven droplet technologies.