Numerical modeling of an electrostatically driven liquid meniscus in the cone–jet mode

Aerosol Science 34 (2003) 99–116www.elsevier.com/locate/jaerosci

Numerical modeling of an electrostatically driven liquidmeniscus in the cone–jet mode

Fang Yana, Bakhtier Farouka ;∗, Frank Kob

aDepartment of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, USAbDepartment of Materials Engineering, Drexel University, Philadelphia, PA 19104, USA

Received 30 September 2002; accepted 1 October 2002

Abstract

A numerical model has been developed for an electrostatically driven liquid meniscus for a dielectric 0uid.The model is able to calculate the shape of the liquid cone and the resulting jet, the velocity 2elds insidethe liquid cone–jet, the electric 2elds in and outside the cone–jet, and the surface charge density at the liquidsurface. The mathematical formulas with proper boundary conditions for the relevant physical processes aredescribed in detail. The equations of continuity, momentum and electric potential are solved numerically withan iterative procedure developed for the model. The results of the present model 2t well with experimentalobservations of the cone shape and jet formation.? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Jet break up; Taylor cones; Electrohydrodynamics

1. Introduction

Electrostatically driven jets have been studied for a long time (Zeleny, 1914). When a conductingliquid is supplied to a capillary nozzle at a low 0ow rate and when the interface between air andthe liquid is charged to a su:ciently high electrical potential, the liquid meniscus takes the form ofa stable cone, whose apex emits a microscopic jet. This is the so-called cone–jet mode (Cloupeau& Prunet-Foch, 1989). For Newtonian 0uids, the jet eventually breaks up into small droplets furtherdownstream.

Many researchers have contributed to the understanding of this phenomenon after the pioneer-ing scienti2c description by Zeleny (1914, 1915, 1917). Taylor proposed an electrohydrostatic

∗ Corresponding author. Tel.: +1-215-895-2287; fax: +1-215-895-1478.E-mail address: [email protected] (B. Farouk).

0021-8502/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S0021-8502(02)00146-5

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100 F. Yan et al. / Aerosol Science 34 (2003) 99–116

Nomenclatured diameterE electric 2eldg gravityH distance between the electrodesI currentIk conduction currentIv charge convection currentK electrical conductivityL length of the nozzleP pressureQ liquid 0ow rater radial coordinateR radial distance of the outer boundary for electric 2eldu velocity component in the z directionv velocity component in the r directionz axial coordinate

Greek letters� relative permittivity �=�0�0 absolute permittivity� potential� surface tension� liquid molecular viscosity� liquid kinematic viscosity� liquid density� surface charge density� tangential stress along the interface

Subscriptsn normalt tangentials gas–liquid interface

Superscriptsi inside the liquid meniscuso outside the liquid meniscus′ derivative

explanation of these so-called ‘Taylor cones’ for perfectly conducting liquids (Taylor, 1964). In histheory, he gave an expression for the charged density � (C=m2) at the free surface, which wasobtained from the balance between the surface tension and electrical normal stress. According to histheory, the Taylor cone must have a 2xed angle, which is not supported by experiments (Hayati,

F. Yan et al. / Aerosol Science 34 (2003) 99–116 101

Bailey, & Tadros, 1987) and in Taylor’s analysis, no liquid jet was assumed to be emitted from thecone apex. Melcher and Warren (1971) developed a quasi-one-dimensional non-linear axisymmetricmodel to analyze the electrohydrodynamics of a steady, semi-insulating liquid jet pulled down underthe action of a tangential electric 2eld and gravity. In their model, they applied the appropriateaxial momentum and normal force balances. However, in order to close their model, Melcher andWarren arti2cially imposed an external linear electric 2eld, which was unrealistic. Furthermore, theydid not retain the convective terms in the total electric current carried by the jet and assumed theradial pro2le of axial velocity at each cross section to be almost 0at. Hayati investigated the velocitypro2le inside the liquid cone at the base of an electrically driven jet both numerically (Hayati, 1992)and experimentally (Hayati et al., 1987). Due to interfacial electrical shear stresses an axisymmetriccirculation near the cone apex was observed which showed that the 0at velocity pro2le assumptionwas not accurate enough. However, in solving for the velocity 2eld, they considered a creep 0owof non-viscous liquid with the cone shape pre-determined and 2xed.

FernLandez De La Mora and Loscertales (1994) made a major contribution by developing thescaling laws controlling the current emitted by Taylor cones, which bears directly on the amount ofcharge carried by the resulting spray droplets. Using experimental values and dimensional analysisthey reported the relations, which are used to estimate the droplet radius and the current throughthe liquid cone. Their scaling laws are only valid for liquid cones with a 0at radial pro2le of theaxial liquid velocity in the jet. GanLan-Calvo, Davila, and Barrero (1997) revised these relationsand proposed a new scaling of current and droplet size. GanLan-Calvo (1997) also established aquasi-one-dimensional electrohydrodynamic physical model comprising temporal balance equationsof mass, momentum, charge, the capillary balance across the surface and the inner and outer electric2eld equations and applied the model to an experimental analysis. The measured current and cone–jet shape served as input parameters in his model. The application of this model is also limitedbecause without the knowledge from the experimental data, the electric 2eld, axial velocity andstresses inside the liquid cone jet cannot be investigated. Hartman, Brunner, Camelot, Marijnissen,and Scarlett (1999b) developed a physical model to calculate the shape of the liquid cone and jet,the electric 2elds in and outside the cone, and the surface charge density at the liquid surface. In themodel, a one-dimensional momentum equation was used for the 0ow 2eld simulation. A jet break-upmodel and a spray evolution model were also subsequently developed by Hartman, Borra, Brunner,Marijnissen, and Scarlett (1999a) and Hartman, Brunner, Camelot, Marijnissen, and Scarlett (2000).

In the present paper, we describe a two-dimensional axi-symmetric formulation of the 0ow andelectric 2elds of the electrostatically driven meniscus through a capillary nozzle. Based on the gov-erning equations, a numerical model has been developed to calculate the shape of the liquid coneand the resulting jet, the electric 2elds, and the surface charge density along the liquid surface. Theliquid properties, liquid 0ow rate and electrode con2guration are needed as input parameters forthe model. Two-dimensional axi-symmetric equations of continuity, momentum and Gauss’ law aresolved numerically. The simulation results are compared with experimental data.

2. Problem geometry

The con2guration for an electrostatically driven meniscus is depicted in Fig. 1. A dielectric liquidwith electrical conductivity K and relative permittivity � issues in the form of a steady capillary jet


Φ0

rz

liquid

H

Ldnozzle

ground D

A B'

C

R

B

Fig. 1. Con2guration of the physical model.

from an injection nozzle of length L and diameter dnozzle. A high potential (voltage) �0 is applied onthe top electrode and the nozzle so that an external electric 2eld is formed between the nozzle andthe ground along the jet direction. The distance between the nozzle exit and the ground is (H − L).The liquid 0ow rate Q is an input parameter to the model. Other physically relevant parametersof the problem are the liquid–gas surface tension �, liquid density �, and liquid viscosity �. As aresult of the electric charge carried by the jet, there will be a net electric current I 0owing from theinjecting nozzle towards the ground. There are two possibilities depending on the 0ow rate and thestrength of the electrostatic 2eld: (i) the jet is continuous and strikes the ground before breaking upinto droplets (Melcher & Warren, 1971); (ii) the jet breaks up into charged droplets which traveltowards the ground under the external electrostatic 2eld force. In the present paper, only the 2rstsituation (the cone–jet mode) is considered.

3. Mathematical formulation

For a dielectric liquid with low electric conductivity, if the electrical relaxation time te = ��0=Kis small compared to the hydrodynamic time th ∼ L=U (where U ∼ Q=R2 is the characteristic jetvelocity and L and R are the axial and radial characteristic lengths of the jet), the liquid bulk isquasi-neutral and free charges are con2ned to a very thin layer underneath the liquid–air interface(GanLan-Calvo, 1997). Therefore, we can use the assumption that all free charges accumulate onlyat the liquid–air surface. In the liquid bulk, the velocity 2eld is governed by the Navier–Stokesequations for Newtonian 0uids. A laminar, axisymmetric, and steady jet 0ow, as shown in Fig. 1,is described by the following equations:

Equation of continuity:

@@r

(rv) +@@z

(ru) = 0: (1)


Equation of momentum in the radial and axial directions:

v@v@r

+ u@v@z

=−1�@p@r

+ �(@2v@r2

+1r@v@r

− vr2

+@2v@z2

); (2)

v@u@r

+ u@u@z

=−1�@p@z

+ �(@2u@r2

+1r@u@r

+@2u@z2

)− g: (3)

In these equations, u and v denote the axial and radial velocity components, p the pressure, g thegravitational acceleration, � the density and � the kinematic viscosity.

For both inside and outside the liquid bulk (except the interface), the potential 2eld is given bythe equation:

1r

@@r

(r@�@r

)+

@@z

(@�@z

)= 0: (4)

3.1. Simpli2cations and boundary conditions

The terms in the radial momentum equation (2) are considered to be small compared with thosein axial momentum equation (3). Therefore, Eq. (2) can be omitted. If the derivative @2u=@z2 inEq. (3) is also neglected, Eqs. (1) and (3) simplify to the well-known boundary layer equations andare used for solving v and u velocity, respectively.The boundary conditions for the hydrodynamic equations are enunciated as follows:At the tip of the nozzle:

z = H − L; 06 r6 r0: v= 0; u is assumed to be parabolic

along axis (r = 0):

r = 0; (H − L)¿ z¿ 0: @u=@r = 0; @v=@r = 0; v= 0

along the liquid–air interface:

r = rs; (H − L)¿ z¿ 0: �= Et; ps = pair + Qp;

where rs and � are the cone jet radius and charge density along the surface, respectively, and � andEt are the tangential shear stress and tangential electric 2eld on the surface, respectively.

The conditions on the free surface allow a pressure jump ‘Qp’ over the interface from air toliquid. The pressure ps inside the liquid 0ow at the surface is given by

Qp= �∇ · n− �02[Eo2

n − �Ei2n + (� − 1)E2

t ] (5)

and

∇ · n=(1r2

− 1r1

); (6)


where r1 and r2 are the main (outer and inner) radii of curvature of the free surface. The followingequations apply for the radii r1 and r2:

1r1

=d2rs=dz2

(1 + (drs=dz)2)3=2=

r′′s(1 + r′2

s )3=2(7)

and

1r2

=1

rs(1 + (drs=dz)2)1=2=

1rs(1 + r′2

s )1=2: (8)

The electric 2eld equation (Eq. (4)) is solved for the computational domain ABB′CD (seeFig. 1). The computational domain includes both liquid and gas regions and also the nozzle sur-face. For the surfaces AB; B′C and DA, zero gradient boundary conditions were used. For thesurface BB′ and the nozzle surfaces, ! = �0 was used, while for the grounded region, we use!= 0.

Along the liquid–air interface, charge density must also satisfy:

� = �0(Eon − �Ei

n); (9)

while the inner and outer tangential electric 2elds are continuous along the interface.

Eit = Eo

t (10)

there is a jump between the inner and outer normal electric 2elds, Ein and Eo

n , due to the presenceof surface charge density. To evaluate the charge density along the free surface, two charge transfermechanisms must be considered. One is the conduction of the charge because of the electric 2eldinside the liquid. The other is the free charge convection along the surface. If ‘s’ is the distancealong the surface, the conservation equation for charge (FernLandez De La Mora and Loscertales,1994) is given as

d(2"rsus�)ds

= 2"rsKEin: (11)

Eqs. (1), (3) and (4) and the boundary conditions described above close the system. A numericalscheme is then developed to obtain the solution in terms of cone/jet radius, the velocity 2eld withinthe jet and the electric 2eld. It is noted that the nozzle length L also has some eTects on the electric2eld and therefore on the characteristics of the cone jet. The potential 2eld is solved for the wholegeometry including the nozzle (region ABB′CD in Fig. 1) and the 0ow 2eld is solved only forliquid cone jet starting from the nozzle tip.

While the present model is similar to that presented by Hartman et al. (1999b), we point out thefollowing diTerences between the present model and that reported by Hartman et al. (1999b):

(a) This is the 2rst cone–jet model coupled with an axi-symmetric 0ow 2eld simulation whileHartman et al. (1999b) considered only one-dimensional 0ow 2eld.


(b) We provide a full description of the boundary conditions used in our model, both for thehydrodynamic and the electrostatic equations.

(c) We use an adaptive grid generation scheme which is essential to update the cone–jet shape.

4. Numerical procedure

In the present model, the velocity and electric 2elds are tightly coupled through complex interac-tions. The 0uid motions are in0uenced by electrical 2eld and surface charge density. In turn, thesemotions modify the charge density distribution along the interface through the convective term of thecurrent density in the corresponding charge conservation equation. The electric potential 2elds aredetermined by the surface charge distributions, and the motion may substantially change the valueof the electric 2elds.

Fig. 2 shows a schematic representation of the present cone–jet numerical model developed. Theiteration procedure is similar to what has been described by Hartman et al. (1999b). First, a puregravity-driven meniscus is considered with the 0ow rate used for electrostatically driven jet andthe meniscus shape and velocity 2eld are calculated accordingly. Simulations are then obtained forthe electrostatically driven jet by sequentially increasing the applied potential. The 0ow rate is keptconstant in each simulation. The resulting cone–jet shape and the associated velocity and electric2elds are solved for the prescribed electric potential. The electric 2eld is solved using Gauss’ Law(except for the air–liquid interface) and the charge distribution along the surface is determined bycurrent balance. For the grid points along the air–liquid interface, the electric 2eld is computed byusing Eq. (9). The calculation of the electric 2eld and the current balance are repeated until thechanges in surface charge prediction is very small. After the convergence of the calculations of theelectric 2eld and the surface charge, the velocity 2eld is simulated under the eTect of the electricinterfacial stress, gravity, and surface tension. The cone–jet shape is updated with the condition thatthe mass 0ow rate along each axial section is constant. If the new cone shape is equal to the formerone, the iteration process is stopped. Otherwise the new shape is used as input for a new cycle ofcomputation.

The system of diTerential equations (1), (2) and (4) is solved numerically using a 2nite diTer-ence method. Second-order central diTerence scheme is applied to discretize the diTerential equa-tions. The numerical solution of the hydrodynamic equations is similar to that described by Mitrovicand Ricoeur (1995). A variable-sized mesh in both axial and radial directions is used so that nodesmay be concentrated in areas with large gradients. Speci2cally, grids are arranged on the interface asshown in Fig. 3. During the calculation, the axial grid spacing is 2xed while the radial grid stretches.At each iteration step, the grids are re-generated with the updating of the cone–jet shape. Very 2negrid size is required near the liquid/gas interface where all the parameters experience signi2cantvariation. Along the developed jet, rather coarse grids can be arranged due to very small gradients.

The computation of the electric potential is the most time consuming part in this simulationbecause the potential 2eld is tightly coupled with the 0ow 2eld and tens of thousands of iterationsare required to reach a converged solution. In order to accelerate the convergence, a well-establishedmultigrid scheme (Hackbush, 1985) is applied to solve Eq. (4). A standard ‘V’ cycle is employedwith three grid levels. As a consequence, the multigrid scheme is seen to be about 3.5 times fasterthan the regular method in this application.


no

yes

start from gravity

driven jet

impose potential

calculate the potential

field

current balance at the

interface

Electric field converged ?

calculate the velocity

field

update the cone jet

shape

cone jet shape

converged ?

no

yes

stop

Fig. 2. Flow chart of the numerical scheme.

5. Results and discussion

The cone–jet shape depends on the liquid 0ow rate, the applied electric potential and the liquidproperties such as density, viscosity, conductivity, permittivity and surface tension. Also, the electrodecon2guration has some in0uence on the process. Table 1 lists the cases studied here. The casesare characterized by the type of liquid and the 0ow rate. Table lists the physical properties of the0uids considered. The cases were simulated with diTerent values of applied voltage. Fig. 4 shows thecone shape predicted by the model for case 1 and measurements reported under similar conditions(Hartman et al., 1999b). The diameter of the nozzle is 8 mm and the length of the nozzle, L is6 mm. The distance from the nozzle tip to the ground is 34 mm. A liquid (ethylene glycol) 0ow


free surface

Fig. 3. Schematic of the mesh arrangement for 0ow 2eld calculation at the free surface.

Table 1Cases considered for simulation

Case Liquid Nozzle diameter Distance between Flow rate Applied potential Nozzle length(mm) electrodes (mm) (10−10 m3=s) (kV) (mm)

1 Ethylene glycol 8.0 34.0 14 20 6.02 1-octanol 1.2 38.5 44.5 4.65 4.53a Ethylene glycol 1.0 9.0 1.1 1.0 2.03b Ethylene glycol 1.0 9.0 1.1 0.1 2.03c Ethylene glycol 1.0 9.0 1.1 0.05 2.03d Ethylene glycol 1.0 9.0 1.1 0.01 2.03e Ethylene glycol 1.0 9.0 1.1 0.0 2.04 Ethylene glycol 1.0 9.0 5.5 1.0 2.05 1-octanol 1.0 9.0 1.1 1.0 2.0

rate of 1:4× 10−9 m3=s is considered for this case. The applied potential diTerence is 20 kV. It canbe seen that the numerical solution agrees very well with the experimental measurement.

The computations were carried out and compared with another experimental measurement (seeFig. 5) by GanLan-Calvo (1997). The liquid used is 1-octanol (see Table 2).The nozzle diameter is set to be 1:20 mm. The liquid 0ow rate is 4:45×10−9 m3=s and the applied

voltage to the nozzle is 4:65 kV. The distance from the nozzle bottom to the ground is 38:50 mm.The nozzle length is reported in GanLan-Calvo (1997), however, in our simulation it is chosen as4:5 mm. The agreement between the numerical results and the experimental data again is found tobe good in Fig. 5.

To explore the physical characteristics of electrically driven jets, we simulated seven additionalcases (3a, 3b, 3c, 3d, 3e, 4, and 5) under the same electrode con2guration. The nozzle diameteris chosen to be 1 mm. The nozzle length is 2 mm and the distance between the nozzle tip and theground is 9 mm. The outer radius of the computational domain was 20 mm. The applied potential


0 1 2 3 4

r (mm)

0

1

2

3

4

dis

tan

ce f

rom

th

e n

ozz

le (

mm

)

experimental data

numerical results

Fig. 4. Comparison of numerical results of the cone–jet radius with experimental data (Hartman et al., 1999b) for ethyleneglycol (case 1).

0 0.1 0.2 0.3 0.4 0.5 0.6

r (mm)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

dis

tan

ce f

rom

th

e n

ozz

le (

mm

)

numerical results

experimental data

Fig. 5. Comparison of numerical results of the cone–jet radius with experimental data (GanLan-Calvo, 1997) for 1-octonal(case 2).

is 1 kV for all three cases. For case 3a–3e, results are reported for the applied potential varyingbetween zero and 1 kV. Fig. 6 shows the development of the cone–jet shape with the variationof potential 2eld from zero to 1 kV for cases 3a–3e. It can be seen from the 2gure that a nearlyvertical jet with radius about 50 �m is formed initially at 50 V. When the potential diTerence is


Table 2Liquid properties

Liquid � (kg=m3) � (kg=m s) � (N/m) � K (�S=m)

Ethylene glycol 1109 0.02 0.048 37 731-octanol 827 0.0081 0.0235 10 8.05

0 0.1 0.2 0.3 0.4 0.5

r (mm)

0

1

2

3

4

5

6

7

8

9

dis

tan

ce f

rom

th

e n

ozz

le (

mm

)

0 V

10 V

50 V

100 V

1000 V

Fig. 6. Development of the cone–jet shape with change of applied potentials for cases 3a–3e.

increased, the jet thins substantially and the cone apex moves upward. At 1 kV, a jet with radius2:3 �m is emitted from the cone base.

Fig. 7 shows the potential contours for case 3a with 1 kV potential. The location of nozzle is alsoindicated in the 2gure. Due to the accumulation of charge along the interface, the potential contoursshow a ‘kink’ along the air–liquid interface. Fig. 8(a) depicts cone–jet radius and the distributionof the charge density along the free surface for case 3a with 1 kV potential. It can be observed thatthe charge density slowly increases 2rst along the cone surface and reaches a maximum value veryrapidly near the cone apex. Then it drops along the jet surface gradually. This can be explained bythe predicted variation of normal electric 2eld inside the liquid Ei

n shown in Fig. 8(b). The chargedensity at the surface of a stationary conductor in equilibrium is such that the internal electric 2eldis null. When the conductor is a 0uid and its surface is set to motion, there is charge transportationdue to convection. As a result, the electric 2eld inside the liquid is no longer zero. We can see inFig. 8(b) that Ei

n is extremely small (for case 3a) near the nozzle. Near the cone apex, Ein increases

signi2cantly to a peak value and drops abruptly to approach zero as the jet develops. The tangential


0 5 10 15 20r (mm)

0

1

2

3

4

5

6

7

8

9

10

11

12z

(mm

)

937.5

812.5

687.5

562.5

437.5

312.5

187.5

62.5

Fig. 7. Electric potential contours (applied voltage 1 kV) for case 3a (values in V).

electric 2eld Et along the interface is shown in Fig. 8(c) for case 3a with 1 kV potential and thenormal electric 2eld outside the liquid Eo

n is shown in Fig. 8(d). The interior normal electric 2eld Ein

is rather small compared to the tangential electric 2eld Et (shown in Fig. 8(c) and the normal electric2eld outside Eo

n (shown in Fig. 8(d)). For case 3a, Eon is about 103 times larger than Ei

n along theinterface. The outside normal electric 2eld Eo

n is approximately equal to �=�0 since �Ein is negligible

compared to �=�0 (see Eq. (9)). However, Ein determines the distribution of the charge density along

the interface. As shown in Fig. 8(c), Et changes slowly along the cone base but increases rapidlynear the cone apex and decreases to a certain value when a stable jet is formed. It is clear from thesimulations that the charge density, the normal electric 2eld (Ei

n and Eon) and the tangential electric

2eld, all experience signi2cant variation within a very short distance near the cone apex and reachesmaximum values. In this case (case 3a), the cone apex is 2:4 mm from the nozzle. Fig. 8(e) showsthe variation of the surface liquid tangential velocity along the cone jet. Due to the tangential electricstress, the surface tangential velocity is accelerated rapidly from almost zero at the nozzle tip toabout 3 m=s at the cone apex for this case. The tangential velocity keeps on increasing graduallyalong the jet due to the reduction of the jet radius.

The present simulations also show that the axial velocity distribution along each axial cross sectionis not uniform. Fig. 9 illustrates the radial pro2le of the axial velocities at two axial sections (0.8and 2:0 mm from the nozzle exit) for case 3a with 1 kV potential. Because of the tangential stress atthe surface, surface velocity is always larger than the velocity at the centerline. The velocity pro2lesat the two sections are, however, of similar shape. At 0:8 mm from the nozzle exit the axial velocityat the interface is only slightly larger than that at the center as shown in Fig. 9(a). At 2:0 mmfrom the nozzle, Fig. 9(b), the axial velocity diTerence between the interface and the center line


0 2 4 6 8

distance from the nozzle (mm)

0

0.1

0.2

0.3

0.4

0.5r

(mm

)

0 2 4 6 8

distance from the nozzle (mm)0 2 4 6 8


0 2 4 6 8


0

0.1

0.2

0.3

0.4

0.5

r (m

m)

0

1E-05

2E-05

3E-05

4E-05

5E-05

char

ge

den

sity

(C

/m2 )

cone jet shapecharge density

0 2 4 6 8


0

0.1

0.2

0.3

0.4

0.5

r (m

m)

0

1

2

3

4

5

6

7

8

9

10

Eni

(103

V/m

)

cone jet shape

normal electric field at thefree surface inside theliquid En

i

0

1

2

3

4

5

Et(

105

V/m

)

cone jet shapetangential electric fieldat the free surface Et

0

0.1

0.2

0.3

0.4

0.5

r (m

m)

0

1

2

3

4

5

6

7

Eno

(10

6V

/m)

cone jet shape

normal electric field at the freesurface outside the liquid En

o

0

0.1

0.2

0.3

0.4

0.5

r (m

m)

0

1

2

3

4

5

6

7

surf

ace

velo

city

(m

/s)

cone jet shape

surface velocity

(a) (b)

(c) (d)

(e)

Fig. 8. (a) Variation of charge density along the free surface for case 3a (applied voltage 1 kV); (b) Variation of normalelectric 2eld Ei

n along the free surface for case 3a (applied voltage 1 kV); (c) Variation of tangential electric 2eld alongthe free surface for case 3a (applied voltage 1 kV); (d) Variation of normal electric 2eld Eo

n along the free surfacefor case 3a (applied voltage 1 kV); (e) Variation of tangential velocity along the free surface for case 3a (appliedvoltage 1 kV).


0 0.1 0.2

r (mm)

r (mm)

4.27

4.28

4.29

axia

l vel

oci

ty (

10-4

m/s

)

0 0.01 0.02 0.033.73

3.74

3.75

3.76

axia

l vel

oci

ty (

10-2

m/s

)

(a)

(b)

Fig. 9. (a) Radial pro2le of axial velocity at section 0:8 mm from the nozzle or case 3a (applied voltage 1 kV); (b)Radial pro2le of axial velocity at section 2 mm from the nozzle case 3a (applied voltage 1 kV).

is enlarged due to the increased electrical shear stress at the free surface. As the jet develops, thevelocity pro2le inside the jet becomes almost uniform.

We found from the simulation results the total current, i.e., the surface convection plus the bulkconduction current across a transversal section of the jet, for case 3a is 3:69 nA. It is in agreement


0 2 4 6 8


0

0.5

1

1.5

2

2.5

3

3.5

curr

ent

(nA

) convection current

conduction current

total current

Fig. 10. Variation of conduction current and convection current along the cone jet for case 3a (applied voltage 1 kV).

with the current scaling law by GanLan-Calvo et al. (1997), I ∼ (QK�)1=2=(� − 1)1=4. Fig. 10 showsthe variation of convection and conduction current along the cone jet. At the base of the liquid cone,the conduction current is dominant and the convection current is very small. At the position wherethe jet emerges from the cone, convection current increases signi2cantly and becomes dominant. Itis noted that the convection current decreases slowly to approach a constant value after the coneapex due to the decrease of charge density along the free surface of the developed jet.

Fig. 11(a) presents the distribution of the surface tension stress and normal electric stress alongthe cone jet for case 3a. Compared to normal electric stress, the surface tension stress is much moredominant and increases all the way along the free surface. The normal electric stress is negligiblealong the liquid cone base and reaches a peak value at the cone apex. The normal electric stressdrops slowly along the developed jet. Fig. 11(b) shows the tangential electric stress along the conejet. The negative sign indicates that the tangential stress is in the opposite direction to z-axis. Thetangential electric stress is also very small along the cone base and increases signi2cantly to amaximum value near the cone apex. The large tangential electric stress rapidly accelerates the liquidat the free surface. This eTect at the cone apex is essential for high speed jet emitted from the conebase. Along the jet, the tangential electric stress drops slowly and the velocity at the free surfaceincreases gradually.

The simulations are repeated with the same con2guration (as in cases 3a–3e) but for ethyleneglycol at a larger 0ow rate 5:5× 10−10 m3=s (case 4) and for 1-octanol at the same 0ow rate (case5). The comparison of the cone–jet shapes is shown in Fig. 12(a). When the 0ow rate is increased(cases 3a and 4), the cone base is much longer and the radius is thicker. Under the same conditions(cases 3a and 5), the cone base for 1-octanol case is smaller than ethylene glycol case. This is


0 2 4 6 8

distance from the nozzle (mm) distance from the nozzle (mm)

0

50

100

150

200

250

no

rmal

ele

ctri

c st

ress

(N

/m2 )

0

5

10

15

20

25

surf

ace

ten

sio

n s

tres

s (k

N/m

2 )

surface tension stress

normal electric stress at the free surface

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

r (m

m)

-30

-25

-20

-15

-10

-5

0

tan

gen

tial

ele

ctri

c st

ress

(N

/m2 )

cone jet shape

tangential electric stressalong the free surface

(a) (b)

Fig. 11. (a) Variation of surface tension stress and normal electric stress along the free surface for case 3a (appliedvoltage 1 kV); (b) Variation of tangential electric stress along the free surface for case 3a (applied voltage 1 kV).

because of the smaller surface tension of 1-octanol. The variations of charge density for diTerentcases (3a, 4 and 5) are depicted in Fig. 12(b). For ethylene glycol, the charge density reacheshigher value at the cone apex for low 0ow rate (case 3a compared to case 4). The charge densityfor ethylene glycol is larger than that for 1-octonal under the same condition (cases 3a and 5) dueto larger electric conductivity of the ethylene glycol.

6. Conclusions

In this work, we present a comprehensive model for the description of electrostatically drivenmeniscus in the cone–jet mode. The axi-symmetric equations of continuity, momentum and electricpotential are solved numerically with speci2ed boundary conditions. With this physical model, thecone–jet mode can be predicted without any 2tting parameters, given the liquid properties, liquid0ow rate and the electrode con2gurations. This model is able to calculate the cone–jet shape andthe velocity 2eld within the meniscus. The numerical results are compared with experimental dataand good agreement is obtained between the predictions and the measurements. This model can alsoestimate the electric 2eld inside and outside the liquid meniscus. The charge density, the tangen-tial electric 2eld along the interface and the normal electrical 2eld inside the liquid are found tochange sharply near the cone apex and all reach the maximum values at the cone apex where thestable jet emerges. This phenomenon is important in understanding the mechanism of electricallydriven jets. Due to the tangential electric stress, the axial velocity at the surface is larger than thatat the centerline near the cone base. However, it becomes more uniform in the downstream jetsection.


0 0.1 0.2 0.3 0.4 0.5r (mm)

0

1

2

3

4

5

6

7

8

9

dis

tan

ce f

rom

th

e n

ozz

le (

mm

)

Ethylene glycol at 1.1 X 10-10 m3/s


1-octanol at 1.1 X 10-10 m3/s



1-octanol at 1.1 X 10-10 m3/s

0 1 2 3 4 5 6 7 8 9distance from the nozzle (mm)

0

1

2

3

4

5

6

char

ge

den

sity

(10

-5 C

/m2 )

(a)

(b)

Fig. 12. (a) Cone–jet shape for cases 3a, 4 and 5 listed in Table 1; (b) Surface charge densities for cases 3a, 4 and 5listed in Table 1.


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Numerical modeling of an electrostatically driven liquid meniscus in the cone–jet mode

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