Three-dimensional analysis of the steady-state shape and small-amplitude oscillation of a bubble in uniform and non-uniform electric fields

Abstract

A three-dimensional analysis is performed to investigate the effects of an electric field on the steady deformation and small-amplitude oscillation of a bubble in dielectric liquid. To deal with a general class of electric fields, an electric field near the bubble is approximately represented by the sum of a uniform field and a linear field. Analytical results have been obtained for steady deformation and modification of oscillation frequency by using the domain perturbation method with the angular momentum operator approach.It has been found that, to the first order, the steady shape of a bubble in an arbitrary electric field can be represented by a linear combination of a finite number of spherical harmonics Yml, where 0[les ]l[les ]4 and [mid ]m[mid ][les ]l. For the oscillation about the deformed steady shape, the overall frequency modification from the value of free oscillation about a spherical shape is obtained by considering two contributions separately: (i) that due to the deformed steady shape (indirect effect), and (ii) that due to the direct effect of an electric field. Both the direct and indirect effects of an electric field split the (2l+1)-fold degenerate frequency of Yml modes, in the case of free oscillation about a spherical shape, into different frequencies that depend on m. However, when the average is taken over the (2l+1) values of m, the frequency splitting due to the indirect effect via the deformed steady shape preserves the average value, while the splitting due to the direct effect of an electric field does not.The oscillation characteristics of a bubble in a uniform electric field under the negligible compressibility assumption are compared with those of a conducting drop in a uniform electric field. For axisymmetric oscillation modes, deforming the steady shape into a prolate spheroid has the same effect of decreasing the oscillation frequency in both the drop and the bubble. However, the electric field has different effects on the oscillation about a spherical shape. The oscillation frequency increases with the increase of electric field in the case of a bubble, while it decreases in the case of a drop. This fundamental difference comes from the fact that the electric field outside the bubble exerts a suppressive surface force while the electric field outside the conducting drop exerts a pulling force on the surface.

Deformation and motion of a charged conducting drop in a dielectric liquid under a nonuniform electric field

“…The applied electric field near the drop is assumed to be approximated by the sum of a uniform field and a linear field as (for the properties of G, see Ref. [6])…”

Transport and deformation of droplets in a microdevice using dielectrophoresis

“…Lee and Kang [23] and Lee et al [24] extended the previous studies to the case where the electric field is no longer uniform, but instead varies linearly in space in the vicinity of the drop. They analytically studied the influence of the spatial variation of the electric field on the steady drop shape and on the circulating flow inside the drop.…”

Rotational regimes of freely suspended liquid crystal films under electric current in presence of an external electric field

“…Electric field can produce instabilities in non-polar (Atten 1996;Saville 1997;Lee et al 2000;Lee and Kang 1999;Collins et al 2008) and polar (Ramos et al 1998(Ramos et al , 1999 liquids. The effect of electric field on liquid crystal (LC) films have been studied widely.…”