A dielectric fluid drop in an electric field

Abstract

In this paper an appropriate extension of the virial method developed by Chandrasekhar is used to systematically re-examine the equilibrium and stability of an incompressible dielectric fluid drop situated in a uniform electric field. The equilibrium shapes are initially assumed to be ellipsoidal; and it is shown that only prolate spheroids elongated in the direction of the applied field are compatible with the moment equations of lowest order. The relation between the equilibrium elongation a/b and the dimensionless parameter x = FR 1/2 /T 1/2 , where F is the applied field, R = ( ab 2 ) 1/2 , and T is the surface tension, is obtained for every dielectric permeability e . This relation is monotonic if e ≤ 20.801; but if 20.801 < e < ∞, there exist as many as three different equilibrium elongations (configurations) for some values of x less than 2.0966. Conditions for the onset of instability are obtained from an examination of the characteristic frequencies of oscillation associated with secondharmonic deformations of the equilibrium configurations. Dielectrics having e > 20.801 exhibit instability while those having e ≤ 20.801 do not. In the former case, where there are three different equilibrium configurations for the same value of x , only the middle one is unstable.