Nonlinear Three-Dimensional Electrohydrodynamic Interactions of Viscous Dielectric Drops

Abstract: When a drop of a leaky dielectric fluid is suspended in another fluid and subjected to a uniform DC electric field, it becomes polarized, leading to tangential electric stresses that drive fluid motion both inside and outside the drop. In the presence of a second drop, the dynamics of the first drop are altered due to electrohydrodynamic interactions with the second, causing the drops to translate due to dielectrophoretic forces and hydrodynamic interactions. We present a semi-analytical nonlinear three-dimensional small deformation theory for a pair of identical, widely-separated leaky dielectric drops suspended in a weakly conducting fluid. This theory is valid under conditions of large drop separation, high drop viscosity, and high surface tension, ensuring that the drops remain nearly spherical. For the first time, we develop a model within the Taylor--Melcher leaky dielectric framework that incorporates both transient charge relaxation and convection. This allows the model to capture the transition to Quincke rotation, a symmetry-breaking phenomenon in which drops begin to spontaneously rotate in sufficiently strong fields. We derive and numerically integrate coupled nonlinear ordinary differential equations for the dipole moments, shapes, and positions of the drops. Our results show good quantitative agreement with previous numerical and experimental work in the limit of zero charge relaxation and convection. We also discuss the hysteresis in the onset of Quincke rotation of isolated drops observed in experiments. Various trajectories for pairs of drops undergoing Quincke rotation are presented, along with results for fixed drops. In particular, we show that the onset of Quincke rotation for a pair of drops is qualitatively different from that for an isolated drop due to electrohydrodynamic interactions and a pair of solid spheres due to straining flows present only in drops.