Electrohydrodynamics of stationary cone-jet streaming
Proceedings of the Royal Society A
We discover novel types of stationary cone-jet steams emitting from a nozzle of a syringe loaded with a conductive liquid. The predicted conejet-flow geometries are based on the analysis of the electrohydrodynamic equations including the surface current. The electric field and the flow velocity field inside the cone are calculated. It is shown that the electric current along the conical stream depends on the cone angle. The stable values of this angle are obtained based on the Onsager's
... e Onsager's principle of maximum entropy production. The characteristics of the jet that emits from the conical tip are also studied. The obtained results are relevant both for the electrospraying and electrospinning processes. on July 20, 2018 http://rspa.royalsocietypublishing.org/ Downloaded from 2 rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20150290 The conical surfaces have also been predicted for the ideal dielectric liquids whose dielectric constant exceeds some critical value ε > ε c ≈ 17.6: in this case the conical angle depends on ε and varies in the range 0 < θ T < 49.3 • [7-10]. However, these theories assume that the liquid does not contain mobile charges; therefore, the electro-induced jet emanation from the corresponding cones is virtually impossible. On the other hand, the presence of free ions in the dielectric liquid must change its flow behaviour. This behaviour could be elucidated based on the system of electrohydrodynamic (EHD) equations which involve the Navier-Stokes equation, the electrostatic equations and the force and the charge balance equations on the free interface    . Together these equations are complicated for analysis; therefore, different approximations and simplifications are usually employed. Widespread employment of a combination of analytical and scaling methods [14-21] allowed identification of several stationary regimes with different scaling dependences for the radius of the jet and the electric current when the external flow is imposed. The shape of the jet has also been investigated numerically using the slender body approximation which reduces the three-dimensional problem to one dimension    . Significant progress has been achieved in the study of non-stationary behaviour of EHD flows     . Recent computer simulations of the EHD equations in three dimensions enable tracing the evolution of the meniscus  and the droplet [27, 28] up to formation of progeny drops, and to find their size as a function of the bulk and the surface conductivities. In particular, it was predicted that the drop size scales linearly with viscosity  , and that the droplet charge at the formation instant is below (but comparable to) the Rayleigh stability limit [27, 28] . Larger progeny drops were obtained in the case of lower conductivity    . These results are in agreement with the corresponding scaling formulae obtained by the authors based on the comparison of capillary, viscous and conduction relaxation times. One of the most important parameters defining the jet properties is the electric current related to the charge balance equation. There are four main contributions to the current, namely the conductive current of mobile ions in the bulk of the liquid which charges the surface, the drift current of the surface ions in the electric field, the current of the surface ions governed by the flow and finally the surface diffusion current of the ions    . The conventional point of view is that the diffusion current is relatively small     . Most theoretical approaches assume that the total current has two main contributions, namely the conductive current and the convective current of the surface charges governed by the flow          . The surface current caused by the tangential electric field is usually neglected [26, 27] . This approximation is related to the widely used assumption that the tangential field is inversely proportional to the square of the distance to the apex  . It is noteworthy that the electrically driven surface current was taken into account in  ; however, it is only inviscid flow inside charged droplets that was considered there. Recent experiments show that evaluation of the current carried by the jet is a delicate problem due to the secondary spraying, ionization of the surrounding gas media and appearance of the corona discharge   . Several scaling formulae for the current have been proposed based on the experimental data [30, 33] , but they have not been matched theoretically yet. Despite the recent magnificent advances in our understanding of the electrodispersion processes the fundamental theoretical problem concerning the existence of self-similar solutions of the EHD equations differing from the Taylor cone remains actual [5, 6] . The current theories do not explain one of the basic features: the variation of the conical angles observed in experiments [6, 34] . In this paper, we show that the electrical drift of the surface ions induced by the tangential electric field is a crucial factor defining the cone geometry of the emitting liquid flow. Based on the system of the EHD equations, we discovered a novel type of self-similar conical solutions. In the next section, we define the model and the basic hydrodynamic and electrostatic equations. The self-similar solutions to these equations are established in §3, where we also predict the stable cone angles as a function of the dielectric constant of the liquid, and specify the region of validity of the model. The jet flow emerging from the cone apex region is analysed in §4. It is noteworthy that a cone-jet flow has been considered for perfectly conducting liquids  . In this case, the electric field inside the liquid vanishes, and the cone semi-angle is restricted to θ T0 . These approximations are lifted in this study.