CAPILLARY ELECTROHYDROSTATICS OF CONDUCTING DROPS HANGING FROM A NOZZLE IN AN ELECTRIC-FIELD

A hybrid boundary element/finite element method is used to determine the axisymmetric equilibrium shapes and stability of a conducting drop hanging from a nozzle of length H2 that is attached to the top plate of a parallel-plate capacitor. The finite element method is used to solve the Young-Laplace equation for drop shape and the boundary element method is used to solve an integral equation for the electric field distribution on the drop and solid surfaces. When the relative importance of interfacial tension force is large compared to electrical and gravitational forces, equilibrium drop shapes are segments of spheres and are conveniently parametrized by a single parameter −1 ≤ D ≤ 1: D = 0 corresponds to a hemisphere, as D → 1 the drop approaches a sphere, and as D → −1 the drop vanishes. The results show that equilibrium families of fixed drop volume, i.e., fixed D, lose stability at turning points with respect to field strength. When H2 = 0 (case of no nozzle), drop shapes at the stability limit are always conical. However, when H2 > 0 (case with a nozzle), drop shapes at the stability limit are conical only when D > D1, where D1 is a negative number. By contrast, very “skinny” drops, D < D2 < D1, take on a two-lobed or a dog-boned appearance at the stability limit. In between over a very narrow range of D2 < D < D1 values, drop shapes at the stability limit resemble nipples on a baby bottle. The new results suggest that immediately upon loss of equilibrium some pendant drops jet from their tips whereas others go unstable by ejecting annular jets from their periphery. The temporal evolution of the instability or jetting phenomena cannot of course be determined by the present static analysis. By repeatedly increasing the number of elements on drop surfaces, which would be prohibitively costly if the electric field were calculated by finite element or difference methods, the apparent cone angle at the tip of drops tending toward conical is determined. Although the apparent cone angle is a function of D, this angle is shown to lie between 40° and 50° when D = 0: it increases monotonically with nozzle length and is bound above by the Taylor limit of 49.3°. Moreover, the variation of the critical field strength for instability ℰc with plate spacing H and nozzle length H2and the asymptotic values approached by ℰc as H and H2 grow without bound are also evaluated. These results are of primary importance in capillary electrohydrostatics and in the development and design of electrodispersion apparatus finding use in the production of ceramic precursor powders and in chemical separations. © 1993 by Academic Press, Inc.