Numerical computations of the dynamics of the disintegration of a drop situated in an electric field

Abstract

The marker-and-cell technique, which has recently been developed for modelling the dynamics of incompressible fluid flow by means of a high-speed computer, has been applied to a study of the instability of an uncharged liquid drop of radius R and surface tension T situated in an electric field of strength E . This problem, which is of considerable importance in certain cloud physical situations, was previously treated analytically by Taylor who assumed that the drop retained a spheroidal shape throughout the period of deformation until the instability point was attained. His calculated instability criteria, namely that E(R/T) ½ = 1.625 when the ratio of the semi-major to semi-minor axes a/b = 1.9, agree well with experimental measurements. The present numerical calculations permit a quantitative assessment to be made of the validity of the spheroidal assumption and, of greater importance, provide a description of the dynamics of the disintegration of a drop subjected to intense electrical forces. In order to conserve computer time the initial condition was assumed to be that a spheroidal drop of undistorted radius 0.2 cm and surface tension 70 dyn cm –1 , possessing a degree of deformation represented by a/b = 1.9, was introduced into a field of strength E = 9500 V cm –1 , which is 4% greater than the critical value deduced on the basis of the spheroidal assumption. Computed cross-sections through the axis of the drop at appropriate intervals of time illustrate the onset of instability at the poles of the drop and demonstrate that the spheroidal assumption provides an extremely accurate representation of the shape of a drop situated in an electric field. This latter conclusion is reinforced by the calculated distribution of pressure over the surface of a spheroidal drop introduced into a critical field. Calculated values of the outward velocity of the fluid at the poles of the disintegrating drop show that capillary waves generated on the surface increase with amplitude until the final stage of instability is initiated, whereupon the velocity increases extremely rapidly, culminating in the ejection of fluid from each pole of the drop, probably in the form of a jet which subsequently breaks up to produce a number of droplets. The corresponding inward velocities at the equatorial points undergo much less variation than the polar velocity and do not exhibit a particularly pronounced increase at the time of instability. The computations indicate that the velocity of ejection of fluid from the poles of the drops is of the order of 100 cm s –1 . This value is in excellent agreement with experimental measurements made by several workers.