Instability of a charged droplet in an inhomogeneous electrostatic field of a thin rod

The problem of stability of oscillations of a charged droplet in an inhomogeneous electrostatic field of a thin charged rod is investigated in the nonlinear formulation using the asymptotic expansion in two small parameters, namely, the dimensionless equilibrium droplet strain and the ratio of the droplet oscillation amplitude to the droplet radius. It is shown that, when the droplet charge is less than the Rayleigh critical charge, in the inhomogeneous electrostatic field the droplet instability implementation mechanism remains the same as for the charged droplet in the field of a point charge. As the oscillation mode number increases, the critical field parameter reaches saturation tending to the horizontal asymptotics. The longer the rod, the higher the level of the asymptotics. As the rod length increases, the amplitudes of the related droplet oscillations and the increments of the unstable droplet oscillations in the electrostatic field of the rod decrease.