Small-amplitude shape oscillation and linear instability of an electrically charged viscoelastic liquid droplet

Abstract

In this paper the small-amplitude oscillation and linear instability of an electrically charged liquid droplet suspended in a vacuum is studied. The liquid is assumed to be viscoelastic and leaky dielectric. The problem is linearized and a characteristic equation is derived. The complex frequency is solved numerically as a function of the relevant dimensionless parameters. The effects of surface charge and the electrical properties of the liquid on the oscillation and instability characteristics of the droplet are investigated for several cases of viscoelasticity and the quadrupole mode n=2. In the case of large or small elasticities, increasing the charge level leads to a decrease in the critical Ohnesorge number at which a supercritical bifurcation occurs and narrows down the interval of the Ohnesorge number for oscillation. For large viscosities, oscillation can be induced by elasticity when the relative stress relaxation time is between 0.004 and 10 approximately, and the electric field may diminish this upper limit to a certain extent. The existence of the electric field also gives rise to new aperiodic branches first appearing at large viscosities or elasticities with smaller damping rates. The effect of finite conductivity on the oscillation behavior of the droplet is rather complicated. Most significantly, when the electrical relaxation time is of the order of the capillary time or a little smaller, new supercritical and subcritical bifurcations with new intervals that favor the appearance of oscillation occur at moderate/relatively large viscosities or elasticities. In addition, it is found that the Rayleigh limit expressed in the form χ=n+2 where χ the electrical Bond number and n is the order of the mode is identical with that for a perfectly conducting, inviscid droplet. Above the Rayleigh limit, the viscoelastic droplet becomes unstable, and high-order modes may be dominant over the quadrupole one when the charge level is high enough.