## Abstract

We discover novel types of stationary cone-jet steams emitting from a nozzle of a syringe loaded with a conductive liquid. The predicted cone-jet-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 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.

## 1. Introduction

When an electrically conductive liquid meniscus or a conductive drop dwell under high voltage, their free surface deforms and may adopt a stable conical shape whose apex emits a thin straight jet if the electric potential exceeds some critical value [1–6]. Far away from the apex, the jet can become whipping or disintegrates in droplets. The conical feature is known as the Taylor cone: the cone angle for an ideal conducting liquid had been calculated by Taylor in 1964 [2] using the balance between the capillary and the electrostatic forces and assuming equipotentiality of the cone surface. He found that the angle at the cone apex is 2*θ*_{T0}=98.6°. The Taylor approach is essentially based on the hydrostatic arguments: the motions of charges and the liquid inside the cone as well as their leakage from the apex do not occur in his model.

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 [11–13]. 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 [22–24]. Significant progress has been achieved in the study of non-stationary behaviour of EHD flows [25–28]. Recent computer simulations of the EHD equations in three dimensions enable tracing the evolution of the meniscus [26] 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 [26], 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 [26–28]. 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 [11–13]. The conventional point of view is that the diffusion current is relatively small [26–29]. 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 [16–24]. 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 [15]. It is noteworthy that the electrically driven surface current was taken into account in [28]; 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 [30–32]. 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 [16]. 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.

## 2. Model and basic equations

The following model is considered: the electrically conductive liquid is pumped at the flow rate *Q* through a nozzle (of diameter *D*) by an extra pressure *P*_{0} inside it (figure 1). Obviously, the flow rate is a function of the pressure, *Q*=*Q*(*P*_{0}). The electrodes (top, located near the nozzle, and bottom, located below the jet) generate a high voltage *Φ*_{0} and a homogeneous (external) electric field *E*_{0} along the *z*-axis. The liquid is characterized by its density *ρ*, electrical conductivity *K*, viscosity *η*, dielectric constant *ε* and surface tension *γ*. For the surrounding gas medium *ρ*=*K*=0 and *ε*=1. The electric conductivity *K* depends on the volume concentrations of positive *n*_{+} and negative *n*_{−} ions, their mobilities *μ*_{+}, *μ*_{−}; the ions are monovalent (with charge *e*), so *K*=*e*^{2}(*n*_{+}*μ*_{+}+*n*_{−}*μ*_{−}). For the sake of convenience, let us specify that the nozzle is positively charged. The negatively charged ions move to the positive electrode where they are neutralized, whereas the positively charged ions move downwards and then travel to the collector. For simplicity, we consider incompressible liquid in a steady flow state. The volume of the liquid is electrically neutral, i.e. *en*_{+}−*en*_{−}=0 or *n*_{+}=*n*_{−}=*n*. The neutrality follows from the Ohm law, the general charge conservation and mass conservation laws, and the Maxwell equations: the current density inside the fluid is **j**=*K***E**+*υ***v**, where **v**(**x**,*t*) is the fluid velocity, **E** is the electric field and *υ* is the volumetric charge density; ∂*υ*/∂*t*+**∇**⋅**j**=0, **∇**⋅**v**=0 and **∇**⋅*ε***E**=*υ*/*ε*_{0}. Here *ε*_{0} is the permittivity of vacuum.

The behaviour of a conducting Newtonian liquid can be described by a set of EHD equations including the momentum equation (the Navier–Stokes equation of motion), the electrostatic equations and the force and charge balance equations on the free boundary of the liquid (the surface charge transport and traction conditions).

### (a) Momentum equation

The momentum equation involves the velocity field **v**(**x**,*t*) inside the liquid
** Σ**=

*η*(

**∇**

**v**+

**∇**

**v**

^{T}),

**∇**is the gradient operator,

**I**is the unit tensor and

*p*is the excess pressure (relative to the pressure outside). The velocity field

**v**(

**x**,

*t*) is additionally restricted by the incompressibility condition

**∇**⋅

**v**=0. The differential equation (2.1) should be supplemented by the boundary conditions on the free surface implying the balance of the viscous, capillary and electric forces (note that the normal component of the velocity is always zero as we focus on the stationary flows in this paper)

**n**is the unit vector normal to the surface and directed outside the liquid phase,

*C*=div

**n**is twice the mean surface curvature and

**F**is the electric force per unit area. Furthermore, we use the cylindrical system of coordinates (

*z*,

*r*,

*ϕ*) and assume that the cone-jet surface is axially symmetric (with the axis

*z*), so it can be described by the function

*r*=

*h*(

*z*,

*t*). The vectors normal and tangential to the surface are, respectively,

**e**

_{z}is the unit vector along

*z*-axis and

**e**

_{r}is the radial unit vector. The electric force reads [35]

**E**

_{i}and

**E**

_{o}are the electric fields inside and outside the liquid, respectively. The subscripts ‘

*n*’ and ‘

*τ*’ are used correspondingly for the normal and tangential components of the variables in question.

### (b) Electrostatic equations

The electric fields **E**_{i} and **E**_{o} can be expressed through the corresponding potentials *Φ*_{i} and *Φ*_{o} obeying the Laplace equations, i.e. **E**_{i}=−**∇***Φ*_{i}, **E**_{o}=−**∇***Φ*_{o} and
*σ*_{c} is the surface density of conductive charge. Far away from the boundary (outside the liquid phase), the electric field is *E*_{0}**e**_{z}.

The above equations for the electric field can be reformulated in the integral form. To this end, we introduce the surface density of the polarization charge *σ*_{p}. The potential created by the conductive and polarization charges and by the electrodes is written as
*Φ*=*Φ*(**r**) coincides with *Φ*_{i} when **r** is inside the liquid and equals to *Φ*_{o} when **r** is on the outside. Here **r**−**r**_{1} is the distance between the point **r** and a point **r**_{1}=*z*_{1}**e**_{z}+*h*(*z*_{1},*t*)**e**_{r} on the surface. The integration **r**_{1} over the free surface *A* where the conductive and polarization charges are localized.

Let us consider a small surface area d*A*. The electric field created by the surface charges arranged outside of this area element is given by
*Φ* is defined by equation (2.6). The normal components of the electric field inside and outside of the selected area correspondingly read

The first term in the above equations is the electric field created by the surface area d*A* and **E** is given by (2.7). Substitution of equation (2.8) in the boundary condition (2.5) results in the equation for the surface charge densities *σ*_{c}, *σ*_{p}
*σ*_{c} and *σ*_{p}
** τ** is the tangential vector.

The electric force **F** acting on the free surface, see equation (2.3), can be expressed via the total surface charge density *σ*=*σ*_{c}+*σ*_{p} and the fraction of the conductive charge *w*=*σ*_{c}/*σ* (below we consider the case 0≤*w*≤1):

### (c) Charge balance equation

The electric field inside the liquid **E**_{i} induces current with density **j**(**E**_{i}) which obeys the linear Ohm law **j**=*K***E**_{i}. The normal component of this current *j*_{n}=*KE*_{i,n} is responsible for charging the free surface. The charge balance equation on the surface reads
*v*_{τ} and *v*_{n} are the tangential and normal components of the flow velocity at the surface, **∇**_{S} is the divergence of the tangential vector field on the surface and *eμ*_{+}*E*_{τ} is the drift velocity of the surface charges due to the action of the tangential electric field *E*_{τ}. Note that the surface is positively charged by virtue of the cone geometry (figure 1), hence it is the mobility *μ*_{+} that is involved in the above equation. It is shown below that the surface current induced by the tangential electric field is crucial for the cone formation. In the stationary regime, equation (2.13) can be written as
*h*(*z*) and using the condition **∇**⋅**j**=0 gives the total electric current at the cross-section (*z*):
*E*_{i,z} varies slower than *h*^{−2}, the conductive volume current decreases as *h*(*z*) and tends to zero at the cone apex where *h*→0. Therefore, there is some length scale *h*=*h*_{0}, so that the total current is mainly determined by the surface charges for *h*<*h*_{0} (the latter condition is always satisfied if *h*_{0}<*D*). In the next section, we consider the situation where the surface current due to the tangential electric field is the dominating part of the current. The conditions of validity of this approximation are considered at the end of §3 (the length *h*_{0} is also specified there).

## 3. Electrohydrodynamics of the cone

Below we analyse the conic part of the liquid flow involving both conductive and polarization surface charges in the limit of vanishing flow rate, *Q*=0. The cone occupies the region *z*≤0 with its apex located at *z*=0 (figure 1). The cone surface is determined by the revolution of the line *h*(*z*)=−*Bz* around the axis *z*, where *θ*_{T} is a half of the apex angle. The curvature of the cone surface is

As discussed above (and verified at the end of this section), the electric current *I*_{0} is mainly determined by the drift of the surface charges under the tangential electric field

The balance between different forces involved in equation (2.2) implies validity of the following scaling relations
*Σ*_{nn}=*Σ*_{τn}=0. Taking into account that *F*_{n} is a quadratic function of the surface charge density and the tangential electric field (see equation (2.12)) and assuming that the fraction of the conductive charges *w*=const., we arrive at the following scaling relations for the charge densities and the electric field: *I*_{0}∼*hσ*_{c}*E*_{τ}=const., and with equation (2.9) provided that *E*_{0} is negligible compared to the last term in equation (2.9). The latter condition is justified as follows: it is natural to assume that *E*∼*E*_{0} near the orifice (*h*(*z*)∼*D*/2), hence (by virtue of the scaling relation *E*∼*h*^{−1/2}) *E*≫*E*_{0} in the cone region of interest, *h*(*z*)≪*D*/2. Note that the electric current *I*_{0}=0 in the limiting cases of the perfectly conductive liquid (*E*_{τ}=0) and the dielectric liquid (*σ*_{c}=0).

Let us turn to equation (2.9), where we now neglect the contribution of the external field formally setting *E*_{0}=0. Substituting the total charge density *A**=const. in equation (2.9) after some algebra we get a transcendental equation defining *ε* and the fraction of the conductive charges *w*=*σ*_{c}/*σ*:
*θ*_{T}=30°.

Using equation (3.3) we find the cone angle as a function of the dielectric constant for different values of *w* (figure 3). It is interesting to note that at *w*=1, we arrive at the classical Taylor cone value *θ*_{T0}=49.3°, and that the case *w*=0 corresponding to the dielectric fluids (no conductivity) completely agrees with the results of references [8,9]. At *ε*<17.59, the half angle is in the region 0°≤*θ*_{T}≤49.3° (0≤*B*≤1.16) depending on the value of *w*.

After substitution of the total surface charge density *I*_{2}(*B*=1.16)=0 as it should be for the classical Taylor cone. The plot of the function **E**, which is given by equation (2.7), depend on the cone angle.

Let us turn to the Laplace equations (2.4) which can be solved using the spherical system of coordinates (*R*,*θ*,*φ*). The resulting axially symmetric potential fields with the proper *R*-dependence can be expressed in terms of the Legendre functions [8,9]
*E*_{R}=−*E*_{τ}. Hereby we find
*w*=0, equation (3.7) coincides with the corresponding equation defining the cone angle for dielectric liquids [8,9].

Numerical calculations show that roots of equation (3.7) coincide with the roots of equation (3.3). This fact follows from the equivalence between the differential and integral forms of the electrostatic problem. Elimination of (1−*w*)/(*ε*−1) from equation (*a*) using equation (3.3) yields the following mathematical identity
*A**

Let us now analyse the liquid flow inside the cone. Introducing the stream function *ψ* and omitting the inertial term (as the present study is restricted to the regime of low Reynolds numbers, see equation (3.21)) we rewrite the momentum equation (2.1) as [36]
*θ*=*θ*_{T} are given by
**e**_{R}. Substituting *E*_{τ} from equation (3.4) in equation (3.12) and using *A**,
*θ*_{T} and *w* is given by equation (3.3), i.e.

Equations (*a*), (*b*) and (3.11) are solved analytically assuming that *ψ*=*R*^{2}*Ψ*(*θ*). The latter assumption naturally comes from the idea that viscous stress at the surface must be comparable with Laplace pressure (and with the dielectric force), in agreement with the scaling relations (3.2). After some algebra we find the stream function, the pressure and the velocity components

The flow inside the cone shows significant vorticity. Note that the vortex flows in the cone regions have been observed experimentally [37,38] and obtained theoretically [16,39,40]. In particular, a qualitatively similar recirculating flow pattern was predicted [40] in the case of low viscosity and low bulk conductivity. However, the obtained flow fields still differ from our prediction, equation (*b*), due to approximations on the electric field inside the cone involved in the previous studies. Interestingly, the velocity components depend on the angle *θ* but not on *R*, so the radial stress component is zero, *Σ*_{RR}=0. The pressure increases in the direction towards the apex: *p*∼1/*R*.

The electric current *I*_{0} is a function of *θ*_{T}; the corresponding relationship follows from the boundary conditions, equation (3.11) [41]:
*I*_{0} as a function of *θ*_{T} is plotted in figure 5. The dependence has one branch if *ε*<17.5 and two branches if *ε*>17.6. Substitution of *I*_{0} (as defined in equation (3.16)) in equation (*b*) yields the following scaling formulae for the flow velocity in the cone: *v*_{θ}∼*v*_{R}∼*γ*/*η*. Similarly, we get the scaling relations for the electric fields inside and outside the cone:

The present theory is based on several assumptions which give rise to some restrictions on the cone size, as discussed below. Firstly, it was assumed that the surface current dominates over the conductive current. The conductive current towards the side surface of the cone with the base of radius *h*_{0} is given by
*I*≪*I*_{0} is equivalent to *h*≪*h*_{0}, where *h*_{0} is defined by equation Δ*I*≃*I*_{0} leading to
*v*_{τ}<*eμ*_{+}*E*_{τ}. On using equations ()–(), this condition can be written as
*E*_{0} in the cone region we considered (see equation (2.9)). It implies that *E*_{n}≫*E*_{0}, hence
*ρv*_{R}*h*/*η*, which increases with *h*, should be smaller than some critical value Re_{c}. Therefore, we must demand
*h*≪*h*^{#}, where the crossover radius of the cone *h*^{#}, the cone must be conjugated with the meniscus. The region *h*≥*h*^{#} needs a special analysis which is beyond the scope of this paper.

An important question is how to find the electric current and the cone angle. They cannot be defined unambiguously based solely on the asymptotic EHD theory considered above. To this end, an additional analysis of the crossover flow regime in the region between the orifice and the asymptotic cone would be necessary. To bypass this difficulty we invoke Onsager’s principle of maximum entropy production [42] which is well established for linear stationary processes [43,44]. It says that a stable stationary process makes the total entropy production a maximum, or, equivalently, it maximizes the work of external forces per unit time, *Φ*_{0} (the applied electric potential difference between the nozzle and the collector) and the excess pressure *P*_{0} in the nozzle: *Q*. In the next section, we show that *Q* is sufficiently small, so that the second term in *Φ*_{0} the maximum of entropy production coincides with the maximum of the electric current. According to the theory presented above, the latter is a function of the cone angle *θ*_{T}. The angles *θ**_{T} corresponding to the maximum of *I*_{0} are shown in figure 6 as a function of the dielectric constant *ε*.

This function shows two branches. The first branch starts at *ε*=1 and exhibits a monotonic decrease of the angle *θ**_{T} with increasing *ε*. By contrast, the second branch reveals an increasing *θ**_{T} as a function of *ε* starting at *ε*≈12.6. The fraction of the mobile charges on the cone surface as a function of *ε* for both brunches is shown in figure 7.

## 4. Flow dynamics in the cone apex and jet regions

In the previous section, we have explored the conical flow structures with free surface based on the full system of EHD equations for the case *Q*→0. Let us now consider the jet emission from the cone which must take place if some flow rate *Q*>0 is imposed. The flow perturbation due to a small *Q* can be roughly described by an additional radial velocity field inside the cone
*R*^{−3} and the inertial term *v*_{1R}∝*R*^{−2} grows faster than the electrical drift velocity of the surface ions, *eμ*_{+}*E*_{R}∝*R*^{−1/2}. Close enough to the apex the viscous and inertial stress contributions coming from the two sources or the drift velocity of the surface ions and the convective flow velocity become of the same order, therefore, the conical structure may fail. This notion allows to identify three characteristic radii and correspondingly three different flow regimes of how the cone transforms into the jet. In regime I, the conical structure is broken because the surface current starts to be governed by the flow when *v*_{1R}≃*eμ*_{+}*E*_{R}. The electric field in the cone *E*_{R} is given by equation (*a*); therefore, the radius of the cone-jet transition zone is *h*(*z*)≃*a*_{1}=(*Q*^{2}*ε*_{0}/*e*^{2}*μ*^{2}_{+}*γ*)^{1/3}*f*_{1}(*ε*). In regime II, the cone-jet transition is caused by inertia once *p* is the pressure for *Q*=0, equation (*a*), and the radius of the transition zone reads *h*(*z*)≃*a*_{2}=(*ρQ*^{2}/*γ*)^{1/3}*f*_{2}(*ε*). Finally, in regime III, the cone-jet transition occurs when the viscous force becomes comparable with the capillary force, *p*≃*Σ*_{1θθ}, where *Σ*_{1θθ} is the stress perturbation, equation (4.2). From here we find the radius *h*(*z*)≃*a*_{3}=(*Qη*/*γ*)^{1/2}*f*_{3}(*ε*). The functions *f*_{1}(*ε*), *f*_{2}(*ε*), *f*_{3}(*ε*) are found from the above mentioned equations after the substitution *h*∼|*z*|∼*b* whose exact geometry is unknown. The exact description of the proximal apex region should be based on a further analysis of the full system of EHD equations which is beyond the scope of this paper.

We are now in the position to return to the general expression for the work done by external forces, *Φ*_{0} is higher than the electric potential difference Δ*Φ* between the cone section at *Φ*_{0}>Δ*Φ*. The difference Δ*Φ* can be estimated as *P*_{0}, it cannot exceed the pressure near the cone base *p*_{max}∼*I*_{0}/(*eμ*_{+}*h*^{#})∼*γ*/*h*^{#} (see equation (*a*)): *P*_{0}<*p*_{max}. Therefore, *P*_{0}*Q*/(*Φ*_{0}*I*_{0})<*p*_{max}*Q*/(Δ*ΦI*_{0})∼*v*_{1R}/(*eμ*_{+}*E*_{R})≪1 (at *R*∼*z*^{#}) and the term *P*_{0}*Q* can be always neglected as long as the conditions of validity of the theory presented in §3 are satisfied.

Let us turn now to the jet below the apex. The jet is supposed to be thin, so it can be described using the slender body approximation (|*h*′_{z}|≪1). Then, the momentum equation can be simplified as [14,21–24]
**∇**⋅**v**=0 bring about the volume conservation equation ∂*h*^{2}/∂*t*+(∂/∂*z*)(*h*^{2}*v*_{z})=0. Therefore, the velocity and the elongation rate inside the jet are approximately given by (note that the flow inside the jet is basically elongational)

Returning to the stationary regime, one can rewrite equation (4.3) as

The electrical field far below the cone (*z*≫*b*) is mainly created by the charged surface of the cone (and the electrodes): the contribution of the jet is negligible (this can be shown, for example, based on the jet-flow solution described below). The electrical field of the cone, *z*-axis, *z*≫*b*, the result is

This field is of course much stronger than *E*_{0} directly generated by the electrodes (for *z*≪*z*^{#}). Far below the apex, for *z*≫*b*, the electrical field gets weaker with *z*, while the flow velocity increases: it is much faster in the jet than in the cone. Therefore, the convective surface current should become dominant, *v*_{z}≫*eμ*_{+}*E*_{z} for *z*≫*b*. In what follows we focus on this latter regime. (The case when the electrically driven surface current is still dominant also in the jet, *v*_{z}≪*eμ*_{+}*E*_{z}, can be considered in a similar way). Neglecting the electrically driven surface current in the jet, the total current can be written as *I*_{0}≃2*Qσ*_{c}/*h*. Taking into account that *F*_{n} is negligible in the regime under consideration) we get from equation (4.5)
*h*(*z*) at large *z*, we find
*h*≃*h** at *z*≪*z**. The asymptotics *h*∝*z*^{−1/8} has been predicted [16] for an inviscid flow of a perfectly conducting liquid.

Here
*a*), is always valid if *h*_{3}<*h*_{1} (cf. equations (3.19), (3.21)). The crossover length *z** does depend on the viscosity and conductivity (ion mobility) in this regime. The alternative regime, equation (4.7*b*), can occur only if *h*_{3}≫*h*_{1}. In both regimes, *h**≪*b*≪*z** and the jet gets thinner for lower flow rate *Q*.

The condition *z*≫*z** ensures that both the viscous term and capillary term are negligible. For *h*∝*z*^{−1/4} for *z*≫*z*^{#} (this regime was predicted in [14]). The dependence *r*(*z*) can be, therefore, summarized as follows:
*b*), (4.8*c*), while viscous (at low *Q*) or capillary (at higher *Q*) forces also play a role in the proximal regime *z*≤*z**. Note that in all the cases the jet is thin: *h*≪*b* as long as *b*≪*h*^{#}.

## 5. Summary and concluding remarks

In this research we studied a novel type of conical features of a stationary conductive liquid flow emitting from a narrow orifice in the presence of a strong electric field. The flow is electrically driven, giving rise to a thin charged jet developed below the cone apex. The conical flow considered here is akin to the well-known Taylor cone predicted for ideal conducting liquids [2]. However, in the present study the cone surface is not assumed to be equipotential, and, moreover, there is a finite electric current along the cone, dominated by the surface current. The driving electric field shows *R*^{−1/2} scaling dependence on the distance *R* to the apex in contrast to the much stronger *R*^{−2} singularity considered before [15]. This new feature arises from the electrically driven surface current, whose effect was disregarded before, and whose importance is demonstrated in this paper.

Schematically the theoretical approach can be summarized as follows: first, we establish the scaling laws for the electric field *E*, the surface charge *σ* and flow velocity *v* as a function of *R* based on the capillary/electric/viscous stress balance. The resultant scaling law, *σ*=const.⋅*R*^{−1/2}, is then postulated (with arbitrary const) and the problem is solved for the electric field (with an arbitrary cone angle *θ*_{T}). This field *E* is used in the EHD problem to obtain the flow field. Finally, the unknown prefactor (const.) and the fraction *w* of conductive surface charge are obtained self-consistently using the force and normal electric field balance on the cone surface. We stress that *θ*_{T} is not necessarily equal to the Taylor semi-angle *θ*_{T0}, rather *θ*_{T} tends to *θ*_{T0} in one asymptotic regime (see below).

Thus, both the fluid flow and electric fields, **v**(**x**) and **E**(**x**), are predicted within the cone. It is shown that the field patterns are self-similar, *v*∼*γ*/*η*, *E*∼(*γ*/*R*)^{1/2}, while the pressure *p*∼*γ*/*R*, and the surface charge density *σ*∼*E*. The self-similar solutions of EHD equations are valid not too close to the cone apex *R*≫*b*, where *b* is the size of the apex region defined by the flow rate *Q*. The total current *I*_{0} along the cone-jet is calculated as a function of the cone angle *θ*_{T} and the dielectric constant of the liquid *ε*. The stable cone angles are predicted based on Onsager’s principle of maximum entropy production: *θ*_{T}≈27° for *ε*=1; it decreases down to *θ*_{T}≈15° at *ε*≈12.6, and it tends to 0 at large *ε*. The second branch of higher stable angles is found for *ε*>12.6 (figure 6); the second angle is *θ*_{T}≈36° at *ε*≈12.6, and tends to the classical Taylor angle *θ*_{T0}=49.3° as *w* of conductive surface charges tends to 50% for both branches at high *ε*. Note that the formation of conical tips on the lemon-shaped inviscid charged drops was predicted numerically in ref. [28]. The cone semi-angle was found equal to the Taylor value *θ*_{T0} in the case of perfectly conducting drops. In all other cases, the simulated angle is lower than *θ*_{T0} in qualitative agreement with the present results.

Our analysis of the thin jet emitting from the cone apex shows that its thickness slowly decreases as *z*^{−1/8}, and then as *z*^{−1/4} with the distance *z* from the apex, the crossover between the two regimes is defined by the total cone length *z*^{#}.

Variations of the Taylor cone angles predicted by our theory are in agreement with the experimental data [6,34]. For example, in reference [34] the angles vary in the interval 32°<*θ*_{T}<46°, which could be attributed to the second branch of the predicted stable angles. Moreover, the correlation between the cone angle and the emission current found in [34] is in accordance with our results (see equation (3.16)). One of the important predictions is that the cone angle is a function of the dielectric permittivity. Experimental study of this dependence appears to be a challenging problem.

## Authors' contributions

Both authors contributed to the conception and design of this study. Both authors drafted the manuscript and gave final approval for publication.

## Competing interests

The authors have declared that no competing interests exist.

## Funding

A.V.S. acknowledges financial support from the Russian Science Foundation, grant no. 14-23-00003. A.N.S. acknowledges a partial support from the International Research Training Group (IRTG) ‘Soft Matter Science: Concepts for the Design of Functional Materials’.

- Received May 5, 2015.
- Accepted September 14, 2015.

- © 2015 The Author(s)