PHYSICAL REVIEW E 86, 036310 (2012) Electrohydrodynamics of a liquid drop in confined domains Asghar Esmaeeli * and Ali Behjatian Department of Mechanical Engineering & Energy Processes, Southern Illinois University, Carbondale, Illinois 62901, USA (Received 17 May 2012; revised manuscript received 26 July 2012; published 17 September 2012) The steady-state electrohydrodynamics of a leaky dielectric drop in confined domains is investigated analytically. The governing electrohydrodynamic equations are solved for Newtonian and immiscible fluids in the framework of leaky dielectric theory and for the creeping flow regime. The domain confinement strengthens or weakens the electric field, depending on R> 1 or R< 1, respectively, where R = σ i /σ o is the ratio of electric conductivity of the drop to that of the surrounding fluid. Similarly, the flow intensity decreases for R< 1, but it remains unchanged or increases for R> 1, depending on the interplay of electric and hydrodynamic effects. An expression for the drop deformation for small distortion from the spherical shape is found using the domain perturbation technique. It is shown that below a threshold domain size the confinement effect will lead to the reversal of the tendency of the net normal hydrodynamic stress in deforming the drop to an oblate or a prolate shape, and that below a critical domain size the necessary condition for having an oblate drop will be opposite to the classical one for an unbounded domain. DOI: 10.1103/PhysRevE.86.036310 PACS number(s): 47.65.−d, 82.70.−y, 83.50.−v, 47.57.−s I. INTRODUCTION The behavior of a liquid drop in an externally applied electric field has been a problem of long-standing interest because of its relevance in a broad range of natural and industrial process. Examples include disintegration of rain drops in thunderstorm [1], electric breakdown of insulating dielectric liquids due to the presence of small water droplets [2], enhancement of heat and mass transfer [3,4], and enhanced coalescence and demixing in emulsions [5]. The electric field provides a well-known means for manipulation of the drops through induced interfacial stresses that can deform, burst, or set the drop in motion. Currently there is a renewed interest in the subject in the context of micro- and bio-fluidics applications, such as manipulation of droplets by continuous electrowetting (electrocapillarity) [6], protein transfection into cells by collision of droplets [7], and enhancement of heat and mass transfer by electric-field-driven chaotic mixing [8], to name a few. Early analytical studies on the subject were done in the framework of “electrohydrostatic” theory [2,9–13], where the drop and the ambient fluid are treated as both being perfect dielectrics, or as a perfectly conducting fluid in a perfect dielectric fluid. In either case, the electrohydrostatic theory predicts that the “net” electric stress will be normal to the interface, pointing from the fluid of higher electric permittivity to the one with lower permittivity, and the drop will always elongate in the direction of the electric field, producing a prolate spheroid. Furthermore, since the theory entails con- tinuity of tangential electric stress at the interface, it precludes the existence of fluid flow at the equilibrium. However, the experiments of Allan and Mason [11] for a wide range of fluid systems showed that conducting drops deformed into prolate spheroids, in agreement with the theory, while some perfect dielectric drops elongated in the direction perpendicular to the electric field, becoming an oblate spheroid. Motivated by the anomalous observations of Allan and Mason [11], * Corresponding author: esmaeeli@engr.siu.edu Taylor [14] pointed out that interfacial hydrodynamic stresses are consequential for nonprolate deformation. To account for these stresses, therefore, the fluids should not be treated as perfect dielectrics; rather they should be considered as having slight conductivity to allow for accumulation of free charge at the interface. The action of electric field on this charge will then lead to an imbalance in the tangential interfacial electric stresses, which in turn leads to hydrodynamic interfacial shear stresses that must develop to balance the electrical shear stresses. Taylor [14] solved the steady-state axisymmetric electro- hydrodynamic equations for the fluids inside and outside of a spherical drop in the creeping flow regime. The domain was unbounded and an electric field E ∞ , uniform at large distance, was applied to the drop. He showed that the relative importance of the ratios of electric conductivities R = σ i /σ o and permittivities S = ǫ i /ǫ o , of the fluid in the drop to the ambient fluid, is the key parameter in setting the senses of drop deformation and fluid circulation. Specifically, he showed that the electric field establishes a circulatory flow in the drop, consisting of four vortices of equal strengths that are matched by counterpart vortices in the ambient fluid. For R<S , the direction of the ambient flow is from the poles (aligned in the direction of the electric field) to the equator, while for R>S the flow direction is the opposite. For R = S , there is no fluid flow since the interface is free of charge. He also found a characteristic function  to predict the sense of the drop deformation. For < 0, the drop deforms to an oblate spheroid while for > 0 it deforms to a prolate spheroid.  = 0 represents a zero deformation state which is a possibility for leaky dielectric fluids because of intricate interplay of electric and hydrodynamic stresses, despite the distorting effect of electric field. Since the seminal work of Taylor [14], there have been several major analytical and numerical studies concerning the dynamics of a leaky dielectric drop. In what follows we only refer to the more relevant ones. The deformation of a drop D was calculated by Vizika and Saville [15] who used Taylor’s solution [14] and balanced the normal stresses at the drop surface posteriorly. The authors used Taylor’s 036310-1 1539-3755/2012/86(3)/036310(13) ©2012 American Physical Society