Prediction and measurement of leaky dielectric drop interactions

Application of an electric field across the curved interface of two fluids of low but nonzero conductivities, or "leaky dielectrics," can give rise to electric stresses that drive sustained fluid flow. In a uniform dc electric field of sufficiently weak magnitude, the electric and velocity fields around an isolated, neutrally buoyant leaky dielectric drop at zero Reynolds number are fore-aft and azimuthally symmetric about the applied field axis. Consequently, the drop remains stationary. The presence of a second drop breaks these symmetries, resulting in the relative motion of the drop pair. Recently, Sorgentone et al. derived an analytical expression for the relative velocity of a pair of widely separated drops of identical constitution, asymptotic in the inverse separation distance between the drop centroids [C. Sorgentone, J. I. Kach, A. S. Khair, L. M. Walker, and P. M. Vlahovska, Numerical and asymptotic analysis of the three-dimensional electrohydrodynamic interac-tions of drop pairs, J. Fluid Mech. 914, A24 (2021)]. In the present work, we generalize the theory of Sorgentone et al. to interactions of dissimilar drops (of different size or constitution), and the pairwise additive interactions of three or more drops. We perform experiments on silicone oil drops suspended in castor oil, and we compare to asymptotic predictions of the drop pair trajectories. Experimental trajectories of drops with their line of centers initially at an arbitrary angle to the field direction are shown to be qualitatively predicted by our theory. We show results of experiments of dissimilar drops and of three and four drops, again observing qualitative agreement with our theoretical predictions.