Instability of a charged drop in Coulomb and dipole fields

Abstract

The instability of the nth oscillation mode of a drop oscillating in a nonuniform electrostatic field produced by a point charge or a point dipole is studied analytically by expanding in two small parameters: the dimensionless equilibrium deformation of the drop and the ratio of the oscillation amplitude to the radius of the drop. When the charge is smaller than the Rayleigh critical value, the instability mechanism for a drop in a nonuniform electrostatic field remains the same as for an uncharged drop or for a charged drop in a uniform field. The critical value of the field parameter saturates with increasing mode number, but the asymptote that is parallel to the abscissa axis and to which the field parameter tends with increasing mode number declines as the drop’s charge grows. In the nonuniform field, the mode coupling becomes stronger, generating a set of modes even if initially only one mode is excited. As the nonuniformity of the field grows, so does the mode coupling.