Instability analysis of an inner-driving coaxial jet inside a coaxial electrode for the non-equipotential case

In this paper, the temporal linear stability analysis of a coaxial jet with two coflowing immiscible liquids inside a coaxial electrode is carried out to investigate the non-equipotential case of the inner-driving coaxial electrospray. Accordingly the outer liquid is assumed to be insulating and the inner liquid as the driving medium with finite conductivity. However, the current due to conduction is assumed to be much smaller than that due to convection so that charge is transported mainly by means of convection. In this case, free charge seems to be “frozen” on the interface between two liquids. The analytical dimensionless dispersion relation is derived straightforwardly through theoretical analysis. In order to study the jet instability, the complex frequencies are solved using numerical approach. According to the calculation results, three different unstable modes, i.e. the para-varicose mode, the para-sinuous mode and the transitional mode, are found in the Rayleigh regime. It is also found that the non-dimensional electrostatic force and the Weber number have the similar effects on the unstable modes. In particular, these two parameters both stabilize the para-sinuous mode for relatively long wavelengths and destabilize it for relatively short wavelengths. Moreover, the unstable regions are extended into the wind-induced regime as the electric field intensity or liquid velocity increases, or surface tension decreases. It is predicted that the non-equipotential case closes to the equipotential one when the inner-driving liquid has sufficiently large electrical permittivity. Conversely, if the permittivity of the inner liquid is considerably small, the non-equipotential case is much less unstable than the equipotential case. In addition, the permittivity of the outer insulating liquid shows an unexplainable non-monotonical effect on the jet instability. As a summary, a general dispersion relation expression is given for all the four involved models, according to equipotential and non-equipotential, as well as inner and outer driving. And the essence of the effect of the electric field on the jet instability is outlined for these cases.