On the stability of a highly charged droplet suspended in the superposition of gravitational and electrostatic fields

For a highly charged droplet suspended at rest in the superposition of gravitational and electrostatic fields, the critical conditions of realization of its instability due to intrinsic and induced charges are determined. All computations are performed in the fourth order of smallness in the stationary deformation of the spherical droplet and in the first order of smallness in the dimensionless amplitude of its capillary oscillations. The dependences of Rayleigh $W_{cr}$ and Taylor wcr critical parameters on the radius of an initially spherical droplet, density, coefficient of surface tension, gravity acceleration, and number of oscillation mode are found that are different from those for a free droplet. As the number of modes increases, the critical value of the Rayleigh parameter increases and approaches asymptotic value $W_{cr}\approx$ 0.95 whereas critical parameter $w_{cr}$ decreases and approaches asymptotic value $w_{cr}\approx$ 1/25 $\times$ 10$^{-4}$. These variations in $W_{cr}$ and $w_{cr}$ are explained by the condition of immobility for the mass center in the suspension that couples $W_{cr}$, $w_{cr}$, and gravity acceleration.