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

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 fi elds, an electric field near the bubble is approximately represented by the sum of a uniform field and a linear field. Analytical results have been ob tained for steady deformation and modification of oscillation frequency by using the domain perturbation method with the angular momentum operator app roach. 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 Y-l(m), where 0 less than or equal t o l less than or equal to 4 and \m\ less than or equal to I. For the oscill ation about the deformed steady shape, the overall frequency modification f rom the value of free oscillation about a spherical shape is obtained by co nsidering two contributions separately: (i) that due to the deformed steady shape (indirect effect), and (ii) that due to the direct effect of an elec tric field. Both the direct and indirect effects of an electric field split the (2l+1)-fold degenerate frequency of Y-l(m) 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 shap e 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 und er the negligible compressibility assumption are compared with those of a c onducting drop in a uniform electric field. For axisymmetric oscillation mo des, deforming the steady shape into a prolate spheroid has the same effect of decreasing the oscillation frequency in both the drop and the bubble. H owever, 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 fi eld outside the bubble exerts a suppressive surface force while the electri c field outside the conducting drop exerts a pulling force on the surface.