Nonlinear vibrations of a charged drop in a third-order approximation in the amplitude of multimode initial deformation

Electrically induced bubble deformation, translation and collapse

“…The observed form of the electric field is explained by noting that for λ = 0, no charge is induced on the bubble surface. Therefore at leading order A = 1/2 and B = 3/2 which are the first coefficients in the respective electric field Fourier-Legendre expansions (14), (15). Note, for the field outside the bubble, by the form of the expansion, the P 1 contribution is forced towards zero along the symmetry line, on the bubble surface.…”

“…In order to obtain expressions for the dimensionless coefficients A and A k , we substitute the expansions given by (14) and (15) in the dimensionless versions of the interfacial conditions (2) and (3), which, after making use of identities given in the Appendix and equating coefficients in the resultant Legendre polynomials, yields…”

“…In the two latter studies, for the case of a spatially uniform electric field, only the effect on the volume and a single shape mode was accounted for. Approaches that allow for nonlinear coupling between shape modes have been developed extensively for droplets in an electric field (see for example [6][7][8][9][10][11][12][13][14][15][16][17][18]), but appear not yet to have been applied directly to bubbles, and it remains unclear which and how bubble shape modes interact in an electric field. To our knowledge, an analysis that is nonlinear in shape-mode amplitudes and allows for the collapse of a bubble has also not been attempted previously.…”