2374 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 47, NO. 6, NOVEMBER/DECEMBER 2011 Dynamics of Water Droplet Distortion and Breakup in a Uniform Electric Field Kazimierz Adamiak, Fellow, IEEE, and Jerzy M. Floryan Abstract—Distortion of a free droplet in a uniform electric field is studied numerically using the boundary element method. It is assumed that the droplet is made of an ideally conducting liquid. There exists a critical magnitude of the electric field intensity. The droplet oscillates for weaker electric fields and elongates until a thin jet emanating from the droplet tip is formed for stronger electric fields. Numerical predictions agree reasonably well with the available experimental data. Index Terms—Droplet distortion, electric field, electrohydro- dynamics (EHDs), numerical simulation. I. I NTRODUCTION I NTERACTION between an electric field (both dc and ac) and droplets is a classical problem of electrohydrodynamics (EHDs) with long record of experimental and numerical in- vestigations. Earlier interests in distortion and disintegration of water droplets by an electric field were related to the importance of this phenomenon in generations of thunderstorms. Modern studies of this problem are driven by its practical significance. In many industrial processes, liquid droplets are charged, so that it is possible to control their motion using electric field. However, during this process, droplets can be distorted, which affects their trajectories. Too strong electric field can also result in ejecting a large number of very small but highly charged droplets. Electromechanics of such droplets is quite different than the original ones. Numerical simulation and optimization of processes using liquid droplets exposed to an electric field need to include a simple, but reliable, algorithm for predicting the droplet shape and breakup characteristics. Initial experimental observations of droplet behavior in elec- tric field were performed by Zeleny [1]. Even though his theoretical explanations were false, as they were based on the assumption that the internal and external pressures are equal, they showed for the first time the formation of a narrow jet em- anating from the droplet tip. Taylor theoretically analyzed the droplet stability conditions assuming that the distorted droplet Manuscript received November 3, 2010; revised April 11, 2011; accepted August 13, 2011. Date of publication September 19, 2011; date of current version November 18, 2011. Paper 2010-EPC-447.R1, presented at the 2010 Industry Applications Society Annual Meeting, Houston, TX, October 3–7, and approved for publication in the IEEE TRANSACTIONS ON I NDUSTRY APPLICATIONS by the Electrostatic Processes Committee of the IEEE Industry Applications Society. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. The authors are with The University of Western Ontario, London, ON N6A 5B9, Canada (e-mail: kadamiak@eng.uwo.ca; mfloryan@eng.uwo.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIA.2011.2168797 shape remains spheroidal. He was successful in predicting the onset of instability, which occurs when E(R/T ) 1/2 =1.625, where E is the electric field magnitude (in electrostatic units), T is the surface tension, and R is the droplet radius. The critical droplet deformation, expressed in terms of semiaxis ratio, was 1.86. Taylor also discovered that shortly before the jet formation, the surface forms a conical tip with the angle of 98.6 ◦ [1]. Experimental investigations with soap film and oil–water interface confirmed the validity of these predictions. Brazier-Smith calculated the exact equilibrium shape of iso- lated droplets and pairs of droplets placed in uniform electric fields of various strengths [2]. This analysis validated the spheroidal approximation for the droplet shape at weak electric fields. Taylor’s predictions for the onset of instabilities and maximum deformation ratio have been also confirmed. The stability parameter decreases for a pair of droplets and depends on the droplet separation. Systematic investigations of droplet deformations under different conditions were undertaken by Basaran and Scriven [3], [4]. Their approach was based on the finite-element method (FEM) solution of the Young–Laplace equation for the droplet shape and the Laplace equation for the electric field distribution. Stability of sessile droplets was defined in terms of two nondimensional numbers: electrical and gravitational Bond numbers. It was shown that it strongly depends on the contact conditions. For a fixed contact angle of 90 ◦ , the value of the critical electrical Bond number (equal to 0.32) shows excellent agreement with Taylor’s value (0.321) and that of experimental investigations (0.322). The critical aspect ratio was determined numerically as 1.82, which well agrees with the experimental value of 1.84. For a droplet with the fixed contact line, the critical Bond number increases to 0.393, and the critical aspect ratio decreases to 1.37. Shapes and stability of free charged droplets were also investigated [3]. The problem of droplet distortion is much more complicated when the droplet is suspended in another immiscible fluid. Assuming that the ambient fluid is ideally dielectric and the droplet is conducting, the external electric field produces an electric force, which is normal to the droplet surface, so the droplet is elongated in the direction of electric field; however, in steady state, the droplet and ambient fluids remain motionless. The problem is much more complicated when the ambient fluid is nonideal and it has some electrical conductivity (such materi- als are sometimes called leaky dielectrics). Finite conductivity of the fluid can lead to accumulation of electric charge on the droplet surface, resulting in a tangential stress and fluid motion. The problem was numerically investigated in detail by Feng and Scott, who used again the FEM methodology [5]. The value of the Reynolds number, electrical properties of fluids, and electric 0093-9994/$26.00 © 2011 IEEE