Dynamics of Drop Formation in an Electric Field Patrick K. Notz and Osman A. Basaran 1 Purdue University, School of Chemical Engineering, West Lafayette, Indiana 47907-1283 Received October 19, 1998; accepted January 28, 1999 The effect of an electric field on the formation of a drop of an inviscid, perfectly conducting liquid from a capillary which pro- trudes from the top plate of a parallel-plate capacitor into a surrounding dynamically inactive, insulating gas is studied com- putationally. This free boundary problem which is comprised of the surface Bernoulli equation for the transient drop shape and the Laplace equation forthe velocity potential inside the drop and the electrostatic potential outside the drop is solved by a method of lines incorporating the finite element method for spatial discreti- zation. The finite element algorithm employed relies on judicious use of remeshing and element addition to a two-region adaptive mesh to accommodate large domain deformations, and allows the computations to proceed until the thickness of the neck connecting an about to form drop to the rest of the liquid in the capillary is less than 0.1%of the capillary radius. The accuracy of the com- putations is demonstrated by showing that in the absence of an electric field predictions made with the new algorithm are in excellent agreement with boundary integral calculations (Schul- kes, R. M. S. M. J. Fluid Mech. 278, 83 (1994)) and experimental measurements on water drops (Zhang, X., and Basaran, O. A. Phys. Fluids 7(6), 1184 (1995)). In the presence of an electric field, the algorithm predicts that as the strength of the applied field increases, the mode of drop formation changes from simple drip- ping to jetting to so-called microdripping, in accordance with experimental observations (Cloupeau, M., and Prunet-Foch, B. J. Aerosol Sci. 25(6), 1021 (1994); Zhang, X., and Basaran, O. A. J. Fluid Mech. 326, 239 (1996)). Computational predictions of the primary drop volume and drop length at breakup are reported over a wide range of values of the ratios of electrical, gravitational, and inertial forces to surface tension force. In contrast to previ- ously mentioned cases where both the flow rate in the tube and the electric field strength are nonzero, situations are also considered in which the flow rate is zero and the dynamics are initiated by impulsively changing the field strength from a certain value to a larger value. When the magnitude of the step change in field strength is small, the results of the new transient calculations accord well with those of an earlier stability analysis (Basaran, O. A., and Scriven, L. E. J. Colloid Interface Sci. 140(1), 10 (1990)) and thereby provide yet anothertestament to the accuracy of the new algorithm. © 1999 Academic Press Key Words: electrohydrodynamics (EHD); drop formation; fi- nite element method (FEM). 1. INTRODUCTION The application of electric fields during the formation of drops from capillary tubes, nozzles, or orifices is in use in a number of technological situations including spray coating (6), inkjet printing (7), spraying of agricultural chemicals (8), sep- aration processes (9), and mass spectrometry (10), among others. Electric fields also play an important role in processes of natural occurrence because of their influence on liquid drops in fields as diverse as meteorology (11) and nuclear physics (12). Although the first scientific observations of the dynamics of formation of drops in an electric field date back almost to the beginning of this century to the pioneering studies of Zeleny (13, 14), a comprehensive theoretical understanding of the phenomena have heretofore been lacking. Remedying this sit- uation when the drop liquid is highly conducting is the goal of this paper. The equilibrium shapes and stability of both free and sup- ported—pendant and sessile— drops in an electric field have been studied exhaustively. Early theoretical works focused on situations in which departures from spherical, cylindrical, or planar base states were either (a) arbitrary but infinitesimal in amplitude, as in Rayleigh’s (15) pioneering study of the sta- bility of an isolated, perfectly conducting, charged drop, or (b) finite in amplitude but entailed assumptions made a priori about the symmetry of those deformations, as in the clever surmize made by Taylor (16) in the now celebrated spheroidal approximation he used to determine the deformation and sta- bility of a perfectly conducting, uncharged drop immersed in a uniform external electric field. A century after Rayleigh’s stability analysis which inaugurated the birth of the science of electrohydrodynamics (EHD), Miksis (17), Joffre et al. (18), and Basaran and Scriven (19) developed numerical methods to systematically calculate without any simplifying assumptions the equilibrium shapes and stability of electrified drops. Miksis (17) theoretically determined the equilibrium shapes of free drops of a dielectric liquid surrounded by a dielectric medium that are subjected to an electric field. Joffre et al. (18) theoretically determined the equilibrium shapes of conducting drops that are pendant from a capillary and surrounded by a dielectric fluid in the presence of an electric field. These 1 To whom correspondence should be addressed. Journal of Colloid and Interface Science 213, 218 –237 (1999) Article ID jcis.1999.6136, available online at http://www.idealibrary.com on 218 0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.