Linear stability analysis of axisymmetric perturbations in imperfectly conducting liquid jets

Abstract

A discussion is presented on the role of limited conductivity and permittivity on the behavior of electrified jets. Under certain conditions, significant departures with respect to the perfect-conductor limit are to be expected. In addition, an exploration is undertaken concerning the validity of one-dimensional average models in the description of charged jets. To that end, a temporal linear modal stability analysis is carried out of poor-conductor viscous liquid jets flowing relatively to a steady radial electric field. Only axisymmetric perturbations, leading to highest quality aerosols, are considered. A grounded coaxial electrode is located at variable distance. Most available studies in the literature are restricted to the perfect-conductor limit, while the present contribution is an extension to moderate and low electrical conductivity and permittivity jets, in an effort to describe a situation increasingly prevalent in the sector of small-scale free-surface flows. The influence of the electrode distance $b$ , a parameter $\alpha$ defined as the ratio of the electric relaxation time scale to the capillary time scale, and the relative permittivity $\beta$ on the growth rate has been explored yielding results on the stability spectrum. In addition, arbitrary viscosity and electrification parameters are contemplated. In a wide variety of situations, the perfect-conductor limit provides a good approximation; however, the influence of $\alpha$ and $\beta$ on the growth rate and most unstable wavelength cannot be neglected in the general case. An interfacial boundary layer in the axial velocity profile occurs in the low-viscosity limit, but this boundary layer tends to disappear when $\alpha$ or $\beta$ are large enough. The use of a one-dimensional (1D) averaged model as an alternative to the 3D approach provides a helpful shortcut and a complementary insight on the nature of the jet’s perturbative behavior. Lowest-order 1D approximations (average model), of widespread application in the literature of electrified jets, are shown to be inaccurate in low-viscosity imperfect-conductor jets.