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Journal of Electrostatics 66 (2008) 58–70

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

Fang Li, Xie-Yuan Yin, Xie-Zhen Yin�

Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, PR China

Received 21 March 2006; received in revised form 21 June 2007; accepted 29 August 2007

Available online 24 September 2007

Abstract

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.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Coaxial jet; Instability analysis; Non-equipotential; Coaxial electrode

1. Introduction

When two immiscible liquids are injected from twohomocentric capillary tubes, respectively, under appropri-ate conditions of flow rates and electric field, a Taylor conewith a steady coaxial micro-jet at the tip of it will beformed. Then the coaxial jet perturbed breaks up intomicro-compound droplets at some distance downstream.This process is called the cone-jet coaxial electrospray [1].Coaxial electrospray can be simply divided into two cases,i.e. the outer driving and the inner driving [1,2]. The outer-driving case means that the outer liquid of the coaxial jet

e front matter r 2007 Elsevier B.V. All rights reserved.

stat.2007.08.001

ing author. Tel.: +86551 3607645; fax: +86 551 3606459.

ess: xzyin@ustc.edu.cn (X.-Z. Yin).

has sufficiently high conductivity so that all free charges aretransported on the outer gas–liquid interface where electricfield stress must be included in dynamic boundaryconditions. Conversely, the inner-driving case indicatesthat the outer liquid is a perfectly dielectric and the innerliquid is a conductor, so free charge is located on the innerliquid–liquid interface where the electric filed stress needsto be taken into account. Recently, many experimentalresearches have been performed to find the electrospraymethod of generating compound droplets, the scaling lawsbetween electric current and drop size as well as differentmodes of coaxial jet electrospray [1–3].The instability analysis is an important aspect in the

study of jets. For coaxial elelctrospray, the analysis is parti-cularly complicated. Theoretically, according to electrical

ARTICLE IN PRESS

V0

R1

R2

R3

U1

U2

Fig. 1. Sketch of the coaxial jet inside a coaxial electrode.

F. Li et al. / Journal of Electrostatics 66 (2008) 58–70 59

properties of driving liquid, two limits, i.e. equipotentialand non-equipotential, are considered individually [4].It is well known that the electrical relaxation timete��0�=K (�0: the vacuum permittivity; e: the relativedielectric constant of liquid; K: the conductivity of liquid)and the characteristic hydrodynamic time, for example, the

capillary time tc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffirR3=g

q(r: the density of liquid; R: the

radius of jet; g: the surface tension coefficient) are twotypical characteristic parameters involved. In the equipo-tential limit (EP), the driving liquid is considered to havesufficiently high conductivity so that the electrical relaxa-tion time is much smaller compared to the hydrodynamictime. Hence, free charge on the surface can be resetinstantaneously and the surface of driving liquid is keptequipotential over time. Contrarily, in the non-equipoten-tial limit (NEP), the conductivity of driving liquid is sopoor and the velocity of jet is so high that the electricalrelaxation time is much larger than the hydrodynamic time.In this limit, the current due to conduction is negligiblecompared to that due to convection, and free charge seemsto be ‘‘frozen’’ on the surface. As a result, the surface ofdriving liquid is non-equipotential. In the instabilityanalysis of a viscous liquid jet based on the leaky dielectricmodel [5–7], the authors also reduced their dispersionrelation to these two limits and discussed the instabilitybehavior of jet in two limits, respectively. But it should bestressed that the equipotential and the non-equipotentialare two extreme cases and actually coaxial electrosprayusually lies in an intermediate situation.

In the previous study, we discussed the stability of thecoaxial jet subject to a radial electric field. Case EP of theelectrified coaxial jet with outer and inner-driving liquidwas primarily included in [8,9], respectively, and case NEPof the outer-driving coaxial jet was researched in [10]. Inaddition, we studied the case under an axial electric field[11], where both the electric field intensity and the electricalproperties of liquids are proved to play an important rolein the jet instability.

In the current paper, we aim to find out some keys toexplain the breakup of the electrified coaxial jet with inner-driving liquid for case NEP, using the method of temporallinear stability analysis. The paper is organized as follows.In Section 2, the model of the electrified coaxial jet isdescribed and the analytical dispersion relation is derivedwith the dimensionless parameters defined. In Section 3,the dispersion relation is solved numerically and the effectsof main physical parameters, including the electric fieldintensity, dielectric constants of the inner and outer liquids,surface tension as well as velocity difference, on the jetinstability are studied in detail. As a summary to thestability problem of the inviscid coaxial jet under the radialelectric field, in Section 4 the dispersion relations for thefour relevant models studied are written in a generalexpression and the comparisons between these models arecarried out. Finally the main conclusions of the paper aredrawn in Section 5.

2. Theoretical model

Consider a cylindrical coaxial jet consisting of an innercore liquid jet of radius R1 and an outer annular liquid jetof external radius R2, in the cylindrical coordinates ðr; y; zÞas shown in Fig. 1. An electric potential V 0, arising fromthe anode at the inner interface between two liquids(r ¼ R1) and the earthed annular cathode (r ¼ R3), ismaintained constant. The densities of the inner and outerliquids are r1 andr2, respectively. It is assumed that theunperturbed inner and outer jet layers have uniform axialvelocities U1 and U2, respectively. The ambient medium isgas at rest. Both the liquids and gas are assumed to beincompressible and inviscid Newtonian fluids.As is well known, viscosity of liquid plays a significant

role in jet instability. For jets having considerably smalldiameters, such as coaxial jets generated by electrospray(the typical diameter is approximately of the order ofmicrometers), effect of liquid viscosity is of particularimportance. On the one hand, liquid viscosity providesviscous shearing force to balance tangential electric stresson the surface; on the other hand, liquid viscosityinfluences basic velocity profile of liquid jet [5,6,11].However, the study of liquid viscosity can be implementedonly with the help of numerical approaches. In this work,we aim to get some primary understanding about thebehaviors of jet under the radial electric field. From thispoint, an inviscid model is employable. Moreover, in orderto get an analytical dispersion relation, velocity profile isassumed to be uniform. In [6], a parabolic velocity profile isemployed in the instability analysis of a single liquid jet.For the discussion of more realistic profile, Lopez-Herreraet al. [5] provided some numerical results in their study.Furthermore, effect of gravity and magnetic field is

neglected. Mass transfer across interface is also negligible.The electrical conductivity and permittivity of liquids areassumed to be uniform and constant.Before perturbed, the inner liquid keeps equipotential

and accordingly a steady radial electric field is formedbetween the inner liquid–liquid interface and annularelectrode. The electric field intensity can be obtainedreadily according to electrostatics:

E1 ¼ 0; rpR1,

E2 ¼ �V0er

r lnððR2=R3Þ�2=�3=ðR2=R1ÞÞ

; R1prpR2,

ARTICLE IN PRESSF. Li et al. / Journal of Electrostatics 66 (2008) 58–7060

E3 ¼ ��2�3

V 0er

r lnððR2=R3Þ�2=�3Þ=ðR2=R1Þ

; R2prpR3, ð1Þ

where er is the unit vector along the r-axis, and �2, �3 are thedielectric constants of the outer liquid and ambient gas,respectively.

When an arbitrary disturbance is imposed on the coaxialjet, the inner and the outer interfaces depart from theiroriginal equilibrium positions together, resulting in thevelocity, pressure and electric fields being perturbedsimultaneously. According to the modal analysis methodfor infinitesimal perturbations, the locations of theperturbed interfaces can be written as rsi ¼ Ri þ Zi, wheresubscript i ¼ 1, 2 refers to the inner and outer interfaces,respectively, and the infinitesimal radial displacement ofinterface can be decomposed into the exponential form:Ziðy; z; tÞ ¼ Zie

otþiðkzþnyÞ, where Zi is the initial perturbationamplitude, o is the complex frequency, k is the real axialwave number and integer n is the azimuthal wave number.Note that three-dimensional perturbations are consideredhere. The other perturbed physical quantities can also bedecomposed into a basic part and a perturbation part, andfurther these perturbation parts have the same exponentialform as the small displacements of the interfaces.

Following the temporal linear stability analysis straight-forwardly, the dispersion relation can be obtained. Here weomit the derivation process and directly represent the resultin order to focus on the instability analysis. The detaileddescription for the derivation of dispersion relation can befound in [8,9]. On the other hand, in order to non-dimensionalize the dispersion relation, the followingcharacteristic scales are chosen: r2 for density, R2 forlength, U2 for velocity, g2 for surface tension coefficient, �3for dielectric constant and V 0=ðR2 lnðR3=R2ÞÞ for electricfield intensity. Ultimately, the dimensionless dispersionrelation can be expressed as

Dða; bÞ ¼H1D5 þ D4

H2D5 þ D1þ

H4D5 � D6

H3D5 � D3¼ 0, (2)

where

H1 ¼

�Sðbþ iLaÞ2InðaaÞ

I 0nðaaÞþ

GaWea2

ð1� n2 � ðaaÞ2Þ ��p2Ea

a31

�

ðbþ iaÞ2

H2 ¼ ��p2a2E

aðbþ iaÞ2ð�p2 � 1ÞK 0nðaÞD1

T,

H3 ¼

Qb2KnðaÞK 0nðaÞ

þa

Weð1� n2 � a2Þ � �p2Eað�p2 � 1Þ þ �p2a2Eð

ðbþ iaÞ2

H4 ¼�p2a2E

aðbþ iaÞ2ð�p2 � 1ÞK 0nðaÞD6

T,

with

z ¼ D4 ��p2InðaaÞ

�p1I0nðaaÞ

D5,

and

T ¼ �

½InðaÞK 0nðaÞ � �p2I0nðaÞKnðaÞ�½I 0nðaaÞKnðaaÞ �

�p2

�p1InðaaÞK 0nðaaÞ�

I 0nðaaÞ

þ InðaaÞKnðaÞK 0nðaÞ�p2

�p1� 1

� �ð�p2 � 1Þ.

The triangle symbols appearing in the dispersionequation are listed below:

D1 ¼ InðaaÞK 0nðaaÞ � I 0nðaaÞKnðaaÞ,

D2 ¼ InðaaÞKnðaÞ � InðaÞKnðaaÞ,

D3 ¼ I 0nðaaÞKnðaÞ � InðaÞK 0nðaaÞ,

D4 ¼ InðaaÞK 0nðaÞ � I 0nðaÞKnðaaÞ,

D5 ¼ I 0nðaaÞK 0nðaÞ � I 0nðaÞK0nðaaÞ,

D6 ¼ InðaÞK 0nðaÞ � I 0nðaÞKnðaÞ,

where InðxÞ, KnðxÞ are the nth-order modified Besselfunctions of the first and second kinds, respectively, andthe prime denotes derivation with respect to variable x.The involved dimensionless parameters are as follows:

the density ratios S ¼ r1=r2 and Q ¼ r3=r2, the diameterratios a ¼ R1=R2 and b ¼ R3=R2, the velocity ratioL ¼ U1=U2, the surface tension coefficient ratioG ¼ g1=g2, the relative dielectric constants �p1 ¼ �1=�3 and�p2 ¼ �2=�3, the Weber number We ¼ r2U

22R2=g2 and the

dimensionless electrostatic force E ¼ ð�3V20Þ=ðr2U

22R2

2ln2

ðb�p2=aÞÞ. In Eq. (2), a ¼ R2k and b ¼ oR2=U2 are the non-dimensional axial wave number and complex frequency,respectively. In the following numerical analysis, thecomplex frequency b will be regarded as the function ofthe axial wave number a and other non-dimensionalparameters. Note that the diameter ratio b is assumed to

þ aaK 0nðaÞD3 � �p2KnðaÞD5

T

�,

�p2 � 1Þ2K 0nðaÞz

T,

ARTICLE IN PRESSF. Li et al. / Journal of Electrostatics 66 (2008) 58–70 61

be much larger than unit (i.e. R3bR2) so that thedispersion relation is simplified reasonably. Furthermore,if the density of gas is considerably smaller than that of theliquids, the dynamic effect of gas phase is negligible. But asmentioned in our previous papers [8,9], for relatively densergas, its effect on jet instability is significant. Though we donot mean to study this issue especially in the current paper,the term related to gas phase in the dispersion relation isretained for a thorough view.

When the outer liquid is assumed to be the same mediumas the ambient gas, the outer interface disappears and thecoaxial jet is changed into a single liquid jet with finiteconductivity. As a result, the dispersion relation (2) can bereduced to the following equation which is written in thedimensional form:

� r1ðoþ iU1kÞ2InðkR1Þ

I 0nðkR1Þþ

g1kR2

1

ð1� n2 � ðkR1Þ2Þ

þ r3o2 KnðkR1Þ

K 0nðkR1Þ�

�3V20

R31 ln

2ððR2=R3Þ

�2=�3=ðR2=R1ÞÞkð1þ kR1BÞ ¼ 0,

with B ¼ ððKnðkR1ÞÞ=ðK0nðkR1ÞÞ � ð�3InðkR1Þ=�2I

0nðkR1ÞÞÞ

�1.The relevant linear instability analysis of the single liquidjet in the NEP was firstly investigated by Artana et al. [4]and the same dispersion relation was derived there.

3. Results and discussion

Apparently the dispersion relation (2) is a transcendentalquartic equation which must be solved numerically. Thereare four roots (eigenvalues) of the complex frequency b ¼br þ ibi (br: the growth rate of perturbation; bi: the phasefrequency; i: the imaginary unit) for an axial wave numbera. But only two of the roots have positive real parts (i.e. thegrowth rate br40), and correspond to the unstable modesthat induce the instability of jet ultimately. In this section,these two eigenvalues will be studied. The effects of severalimportant parameters, including the radial electric fieldintensity, electrical permittivity of liquid, surface tensionand velocity difference, on the jet instability will beinvestigated individually. These important physical quan-tities are mainly contained in the following dimensionlessparameters: the dimensionless electrostatic force E, therelative dielectric constants �p1, �p2, the Weber number,the interface tension coefficient ratio G and the velocityratio L.

Before calculating, a group of reference parametersshould be selected. For the sake of approaching theexperimental situations, water and sunflower oil are chosenas the inner and outer liquids, respectively. Their physicalproperties can be found in [2]. The typical dimensionlessvalues are obtained after simple calculation: S ¼ 1.19,Q ¼ 0.001, G ¼ 0:51, �p1 ¼ 80, �p2 ¼ 3:4, a ¼ 0.8, L ¼ 1:25,We ¼ 10 and E ¼ 0:01. It should be pointed out that incoaxial electrospray experiments the configuration ofelectrode is usually needle-plane, where the direction ofelectric field is neither radial nor axial, actually both

involved. Moreover, the magnitude of electric field is not aconstant. Therefore, the determination of the value ofparameter E according to experiments seems an intractablecase. In our model, we use a cylindrical electrode instead ofthe needle-plane one for the convenience of analysis,following the other authors [5,12–15]. The choice of acylindrical electrode configuration is based on the fact thatfor the jet part of coaxial electrospray the tangentialcomponent of electric field is much smaller compared to thenormal component. Certainly, the electric potential V0

assumed here cannot be equal to that imposed by theneedle in experiments. As to the electric field in coaxialelectrospray, both electric potential imposed on needle andsurface charge carried by jet (especially, the latter) must betaken into account [16–21]. In the calculation, thedimensionless parameters will be fixed at the above valuesexcept clarified otherwise. Considering that the breakupprocess of the liquid jet undergoing in coaxial electrosprayis axisymmetric where the azimuthal wave number n=0,only the axisymmetric instability is solved and analyzed inthis section, though Eq. (2) is derived for all theaxisymmetric and non-axisymmetric modes.

3.1. Effect of the radial electric field intensity on the jet

instability

Fig. 2 illustrates the effect of the dimensionless electro-static force E on the instability of the electrified coaxial jetfor case NEP (solid lines). For the convenience ofcomparison, the corresponding EP case under the samecondition is also plotted in the figure (dashed lines).Fig. 2(a) and (b) represent two different groups of unstableeigenvalues with positive growth rate br, respectively,which correspond to different unstable modes. Thecalculation results show that the mode in Fig. 2(a) has anearly out-of-phase interface appearance and thereforebelongs to the para-varicose mode (Fig. 3(b)). The growthrate in Fig. 2(b) has two branches: the left one is of thepara-sinuous mode with a nearly in-phase interfaceappearance (Fig. 3(a)) and the right one is of thetransitional mode having a continuously changing phasedifference from 01 to 1801 (the description about the para-varicose, para-sinuous and transitional modes can befound in [10,11,22,23]). Note that in Fig. 2 all the unstablemodes generally lie in the Rayleigh regime, that is, most ofthe unstable dimensionless axial wave number a related tothe disturbance wavelength l ¼ 2p=a is comparable to orsmaller than unit. The result is in well agreement with theexperiments where the Rayleigh breakup took place [1–3].From the figure it can be seen that the growth rate of the

para-varicose mode is diminished greatly with the electricfield intensity increasing. Meanwhile, the para-sinuous andtransitional modes are slowly combined, with the formerbeing destabilized and the latter being stabilized for mostwave numbers. Apparently, the para-sinuous mode is themost unstable and therefore has the most possibility to beobserved in experiments [1–3]. Moreover, the dominant

ARTICLE IN PRESS

z

�1 �2

R1 R2

zR1 R2

�1

�2

Fig. 3. Schematic description of the unstable modes: (a) the para-sinuous

mode and (b) the para-varicose mode.

0 0.4 0.8 1.2 1.6 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.005

0.01

0.02, 0.03

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.005

0

0.01

0.02

B

A

E=0.03

E=0

� r� r

�

�

Fig. 2. Effect of the dimensionless electrostatic force E on the growth rate

of (a) the para-varicose mode and (b) the para-sinuous (left peaks) and

transitional (right peaks) modes.

F. Li et al. / Journal of Electrostatics 66 (2008) 58–7062

axial wave number amax corresponding to the maximumgrowth rate bmax for the para-sinuous mode movesdistinctly towards short wavelengths, indicating that theelectric field makes compound droplets smaller. The gapbetween cases NEP and EP is amplified by the electric field,but NEP is always less unstable than the EP. It comes tothe conclusion that well-conducting media can acceleratethe breakup of jet compared with those having relativelypoor conductivity.Comparing the inner-driving case considered in the

present paper with the outer-driving case studied in [9](both are NEP), the radial electric field destabilizes thepara-sinuous mode and stabilizes the transitional mode inboth cases significantly. However, the electric field exhibitsa greatly destabilizing effect on the para-varicose mode ofthe outer-driving case [8,10] while the converse behavior isfound in the inner-driving case. In practice, it is notdifficult to explain this phenomenon, noting that for theinner-driving case the basic electric field intensity in Eq. (1)exists not only in the ambient gas but also in the outerliquid. And so do the electric field perturbations. That it,the electric fields in both the gas and outer liquid contributeto the stress balance on the outer interface. Though theelectric stress due to the gas phase enhances the perturba-tion on the outer interface, the greater electric stress due tothe outer liquid suppresses the instability. Therefore, theouter interface is generally stabilized by the electric fieldand the corresponding para-varicose mode that mainlyresults from the outer interface has a decreasing growthrate. However, for the outer-driving case there is no basicelectric field in the liquids, in spite of the existence of theperturbation of electric field in them in case NEP. So themagnitude of electric stress from the liquids is a smalldegree quantity compared to that from the gas phase.Ultimately, the outer interface is destabilized by the electricfield and the corresponding para-varicose mode is desta-bilized. But the phenomenon is not invariable. In thefollowing analysis, it can be seen that at relatively highWeber numbers the electric field plays a remarkablydestabilizing effect on the para-varicose mode in relativelyshort wavelength region.It is well worth noting that in Fig. 2(b) all the curves

intersect at points A (0.7729, 0.1266) and B (0.6784, 0.1219)for cases NEP and EP, respectively. At these intersectionpoints the jet instability are not influenced by the electricfield; in the left region of point A or B, the para-sinuousmode is suppressed slightly, and in the right it is promotedgreatly. It comes to the conclusion that the radial electricfield plays a dual role in the Rayleigh instability. Inpractice, the dual role of radial electric field has been wellacknowledged in the study of single liquid jets [5,6,24].

3.2. Effect of the electrical permittivities of the liquids on the

jet instability

The liquid electrical permittivity plays a significant rolein the instability of the electrified coaxial jet. In Eq. (2), the

ARTICLE IN PRESSF. Li et al. / Journal of Electrostatics 66 (2008) 58–70 63

dimensionless parameters �p1 and �p2 representing therelative dielectric constants of the inner and outer liquids,respectively, are recognized. The effect of them is studied inthis subsection. Fig. 4(a) and (b) illustrate the margincurves under the reference state for both cases and onlyNEP when �p1 is fixed to 10, respectively, where symbol Odenotes the oscillatory region (i.e. br ¼ 0). In Figs. 2 and4(a), it is evident that case NEP is very close to EP.However, in Fig. 4(b), the domains of the unstabletransitional and neutral oscillatory modes are bothextended, making the para-sinuous mode dominant regionnarrower considerably. From this point, inner liquidhaving small electrical permittivity does not favor therealization of coaxial electrospray.

0 0.005 0.01 0.015 0.02 0.025 0.030

1

2

3

4

para-sinuous

O

transitional

0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

2.5

3

transitional

para-sinuous

O

O

transitionalO

O

transitional

O

O

O

O

��

E

E

Fig. 4. The marginal lines versus the dimensionless electrostatic force E

for (a) the reference state and (b) the non-equipotential case when

�p1 ¼ 10.

Note that the larger the dielectric constant �p1, the closercase NEP to EP. In practice, the difference between casesNEP and EP in the analytical dispersion relations lie in twoitems z and T. Supposing �p1 !1 and keeping �p2 finite, zand T for NEP are both reduced to the correspondingexpressions for EP. In his study of the instability of singleliquid jet, Lopez-Herrera et al. [5] discovered that increas-ing the electrical permittivity of liquid has the sameinfluence on the growth rate with increasing the conduc-tivity of liquid. From this point, the permittivity andconductivity of the inner liquid in coaxial jet are just likethose of the single liquid jet, that is, the inner liquid jet canbe considered as a single liquid jet to some extent, with twodifferent fluids (the outer liquid and the ambient gas)surrounding it.Besides, the normalized eigenfunctions of the electric

physical quantities for NEP when �p1 is 10, i.e. theperturbations of the electric potential, as well as the axialand radial components of the electric field intensity, arerepresented in Fig. 5. The calculation point is fixed at thedominant wave number amax and maximum growth ratebmax of the para-sinuous mode. (The values of amax andbmax are 0.8596 and 0.0902, respectively.) It can be seenthat when the permittivity of inner liquid is relatively smallthe electric field perturbations in the inner liquid are veryevident. And the discontinuities on both the interfaces aregreat too.Though the relative dielectric constant of the outer

insulating liquid �p2 is assumed to be much smaller thanthat of the inner conducting liquid �p1, its influence on theinstability of both NEP and EP is much greater than �p1

(see Fig. 6). It is shown that both the para-varicose mode(Fig. 6(a)) and the transitional mode (Fig. 6(b), the rightpeaks) are suppressed distinctly, whereas the para-sinuousmode (Fig. 6(b), the left peaks) is enhanced as �p2 increases.However, another phenomenon is observed in Fig. 7,where �p1 is fixed to 10. The effect of �p2 on the growth rateof the para-sinuous mode is quite different from that inFig. 6, with the para-sinuous mode (Fig. 7(b), the leftpeaks) being suppressed by �p2 and its dominant regionbeing reduced significantly. Hence the permittivity of theouter insulating liquid behaves differently as the permittiv-ity of the inner conducting liquid varies. On the other hand,it can be seen from the dispersion relation (2) that therelative dielectric constants of liquids are apparently non-monotonical and it seems difficult to find general behaviorcharacteristics of them.

3.3. Effect of the surface tensions and velocity discontinuity

on the jet instability

The Weber number (We) representing the ratio of theinertial force to surface tension is one of the mostimportant parameters influencing jet instability. Fig. 8illustrates its effect on the growth rates of the unstablemodes. In the figure the para-varicose mode is inhibited bythe Weber number remarkably. For the para-sinuous and

ARTICLE IN PRESS

5

0 0.4 0.8 1.2 1.6 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

3.4

�p2 = 1

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

3.4

1

�p2 = 5� r

� r

�

�

Fig. 6. Effect of the relative dielectric constant �p2 on the growth rate of

(a) the para-varicose mode and (b) the para-sinuous (left peaks) and

transitional (right peaks) modes.

0 0.5 1 1.5 2 2.5 3 3.5 4-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Ez

r

0 0.5 1 1.5 2 2.5 3 3.5 4-1

-0.8

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1ˆ

r

0 0.5 1.5 2 2.5 3 3.5 4-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Er

r

2

-0.6

ˆ

1

�

ˆ

ˆ

Fig. 5. The normalized dimensionless eigenfunctions of the electric

potential perturbation j, the radial electric field intensity perturbation

Er and the axial electric field intensity perturbation Ez for the non-

equipotential case when �p1 is fixed to 10. The triangle, square and circle

markers denote the absolute magnitude, the real and imaginary parts of

corresponding eigenfunctions, respectively.

F. Li et al. / Journal of Electrostatics 66 (2008) 58–7064

transitional modes (Fig. 8(b)), there exists a critical axialwave number ðacr � 1:2Þ. When aoacr, the para-sinuousmode is stabilized sharply by We, and otherwise it isdestabilized. That is, the surface tension on the outerinterface increases the instability of jet in the relatively longwavelength region ðao1:2Þ while suppresses it in the

relatively short wavelength region ða41:2Þ, which is inagreement with the case without electric field [22]. Ingeneral, the electric field does not change the behaviors ofsurface tension in quality. On the other hand, as the surfacetension decreases, the transitional mode is enhanced andmoves towards short wavelengths distinctly, predicting thatsmall surface tensions are not suitable for the formation ofcompound droplet. But we need not worry about thisbecause in electrospray experiments surface tensions aregenerally much larger than usual considering the micro-scale of jet diameter. Note that case NEP and EP are notdiscernible when the Weber number is sufficiently small. Itis probably because that the surface tension rather than theelectric field is predominant in this situation.In addition, the effect of the dimensionless electrostatic

force E on the instability of the electrified coaxial jet at

ARTICLE IN PRESS

1

1020

30

0 0.4 0.8 1.2 1.6 20

0.05

0.1

0.15

0.2

0.25

We = 0.5

1

10

20

30

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

� r� r

�

�

We = 0.5

Fig. 8. Effect of the Weber number We on the growth rate of (a) the para-

varicose mode and (b) the para-sinuous and transitional modes.

0 0.4 0.8 1.2 1.6 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

3.4

5

0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

3.4

3.4

5

5

5

1

�p2 = 1

�p2 = 1

�

� r� r

�

Fig. 7. Effect of the relative dielectric constant �p2 on the growth rate of

(a) the para-varicose mode and (b) the para-sinuous (left peaks) and

transitional (right peaks) modes for the non-equipotential case when �p1 isfixed to 10.

F. Li et al. / Journal of Electrostatics 66 (2008) 58–70 65

higher Weber numbers is calculated. As an example, Fig. 9illustrates the unstable modes when the Weber number isup to 100. The phenomenon is quite different from Fig. 2where We is 10. The growth rate of the para-varicose mode(Fig. 9(a)) is decreased to null when the dimensionlesselectrostatic force is small but is increased by relativelylarge dimensionless electrostatic forces. The stabilizingeffect of the electric field happens the Rayleigh regime(nearly ao1), which is in agreement with the previousobservation (Fig. 2(a)). It is of interest to mention that atlarge dimensionless electrostatic forces, the region ofunstable wave numbers moves towards short wavelengthdirection and all the unstable modes lie in not the Rayleighregime but the wind-induced regime (in the wind-inducedregime the right cut-off wave number is much larger thanunit [9]) where the electric field has a remarkable

destabilizing effect on the jet instability. Moreover, thepara-sinuous and transitional modes (Fig. 9(b)) are mergedindiscernibly, with growth rates much larger than thoseshown in Fig. 2(b). As a result, larger Weber number isapparently disadvantageous to electrospray. In addition,case NEP is very close to EP. It is probably because theinstability of the coaxial jet is dominated by the inertialforce rather than the electric field force in this situation. Inpractice, in the wind-induced regime the para-varicose,para-sinuous and transitional modes may be changed intothe other two unstable modes [9]. The large Weber numbercase as shown in Fig. 9 should be dealt with caution inexperiments.The surface tension coefficient of a liquid–liquid inter-

face is generally smaller than that of a gas–liquid interface.As a result, the parameter G ¼ g1=g2 standing for the ratioof surface tension coefficient of the inner to outer interface

ARTICLE IN PRESS

0.7

0.9

Γ = 0.9

0.10.51

0.51

Γ = 0.1

0.3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

� r� r

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

�

�

Fig. 10. Effect of the interface tension coefficient ratio G on the growth

rate of (a) the para-varicose mode and (b) the para-sinuous and

transitional modes for the non-equipotential case.

� r� r

�0 2 4 6 8 10

0

�0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

1.2

0

0.005, 0.010.02

Ε = 0.03

Ε = 0.03

0.5

1

1.5

2

2.5

3

0.005

0

0.01

0.02

Fig. 9. Effect of the dimensionless electrostatic force E on the growth rate

of (a) the para-varicose mode and (b) the para-sinuous and transitional

modes when the Weber number is fixed to 100.

F. Li et al. / Journal of Electrostatics 66 (2008) 58–7066

is smaller than unit. In Fig. 10, the effect of G on thegrowth rates of unstable modes is illuminated, where onlycase NEP is plotted, for case EP is very similar to NEP. Inthe figure it can be seen that the para-varicose and para-sinuous modes are stabilized slightly as G decreases,predicting that the surface tension of the inner interfacehas the same but relatively greater influence on the jetinstability, compared to the surface tension of the outerinterface. Moreover, the transitional mode is destabilizedby the surface tension with the unstable region movingtowards short wavelengths. Surely, this situation does notfavor electrospray.

Velocity discontinuity on interface is another basicinfluencing factor inducing the well-known Kelvin–Helm-holtz instability. In the dispersion relation (2), the velocityratio of the inner and outer liquids L ¼ U1=U2 is involved.But in essence, it is the relative velocity difference j1� Lj

that is the essential influencing factor in the mechanism ofjet instability. Therefore in Fig. 11 the effect of j1� Lj onthe growth rates of unstable modes is plotted. Note thatwhen j1� Lj is 0 (i.e. the case that the inner and outerliquids have the same velocity), the growth rates are theleast and only the para-varicose and para-sinuous modesare unstable. As j1� Lj increases, the growth rates of allthe unstable modes are augmented significantly. The rightcut-off wave number aco moves into the wind-inducedregime ultimately. Within the range of the relative velocitydifference under consideration, case NEP is close to EP tosome extent.

4. Comparison between different models

Now we have established four different models in theresearch of the instability and breakup of the coaxial jet

ARTICLE IN PRESSF. Li et al. / Journal of Electrostatics 66 (2008) 58–70 67

under the radial electric field on the assumption that theinner and outer liquids are inviscid and the basic state hasuniform velocity profiles. As outlined before, according toelectrical properties of the inner core and outer annularliquids, coaxial electrospray is roughly divided into innerdriving and outer driving. For each type, two limitingcases, i.e. the equipotential and the non-equipotential, aretaken into account. Hence, four models are conceived, i.e.the equipotential outer-driving case (abbreviated as OEP),the non-equipotential outer-driving case (ONEP), theequipotential inner-driving case (IEP) and the non-equipotential inner-driving case (INEP). In [8–10] and thecurrent paper, we simply analyzed these four cases,respectively. Since the four models are representative, itseems particularly necessary to summarize and comparethem, in order that readers can get a comprehensive view ofvarious situations in coaxial electrospray. On the otherhand, the comparisons between different models provide ussome significant general laws which will help us easilymanipulate the experiments to get ideal results.

For the convenience of comparison, it is reasonable towrite the dispersion equations of the four models in auniform form. The following formulation like Eq. (2) ischosen:

Dða; bÞ ¼H1D5 þ D4

H2D5 þ D1þ

H4D5 � D6

H3D5 � D3¼ 0,

with

H1 ¼

�Sðbþ iLaÞ2InðaaÞ

I 0nðaaÞþ

GaWea2

ð1� n2 � ðaaÞ2Þ ��p2Ea

a31þ aa

K 0nðaÞD3 � �p2KnðaÞD5

T

� �h1

ðbþ iaÞ2;

H2 ¼ ��p2a2E

aðbþ iaÞ2ð�p2 � 1ÞK 0nðaÞD1

Th2,

H3 ¼

Qb2KnðaÞK 0nðaÞ

þa

Weð1� n2 � a2Þ � Eað1þ azÞh3

ðbþ iaÞ2,

H4 ¼�p2a2E

aðbþ iaÞ2ð�p2 � 1ÞK 0nðaÞD6

Th4:

The symbols h1, h2, h3, h4, z, T and x are listed in Table 1for the four cases. All the other symbols such as D1�D6 andthe involved non-dimensional parameters are coincidentwith those defined in this paper.

The three terms in the numerator of H1 represent theeffect of the inertial force of the inner liquid jet, the surfacetension of the liquid–liquid interface and the electrostaticforce generated on the inner interface on the instability ofjet, respectively. The three terms in the numerator of H3

denote the effect of the inertial force of the gas phase, thesurface tension of the gas–liquid interface and theelectrostatic force on the outer interface, respectively.Symbols H2 and H4 are considered to represent the effect

of the electrostatic force due to the coupling of the electricfield on the inner and outer interfaces, respectively.In Table 1, for cases OEP and ONEP symbols h1, h2, and

h4 are null, indicating that there is no electric field withinthe whole coaxial jet. The only electrostatic force arisesfrom the ambient gas phase (symbol h3). Moreover, caseONEP has a rather more complex expression of z than caseOEP, but z can be reduced to the corresponding formula-tion of OEP provided that the dielectric constant ratio �p2 isto approach infinite. For cases IEP and INEP, thesituations are more complicated. The electrostatic forcesappear in every term. And the difference between IEPand INEP lies in symbols T and x. Letting the dielectricconstant ratio �p1 tends to infinite, INEP will be reducedto IEP.In the following, we abstract the general behaviors of the

electrified coaxial jet and the effects of the radial electricfield as well as the electric properties of the liquids on thejet instability for the four cases.As found in the non-electrified jet, three different

unstable modes, i.e. the para-varicose mode, the para-sinuous mode and the transitional mode, are identified inthe Rayleigh regime for all the four models. And the para-sinuous mode dominates the jet breakup under mostsituations, regardless of the magnitude of the electric field.In all the cases, the radial electric field shows a

destabilizing effect on the liquid jet because it enhancesthe growth rate of the para-sinuous mode distinctly.

As to the para-varicose mode, cases OEP and ONEPexhibit a destabilized feature while cases IEP and INEPshow a stabilized feature. The electric field stabilizes thetransitional mode of all the cases. Although the radialelectric field increases the growth rate of the para-sinuousmode more greatly than the other modes, which benefitsthe realization of coaxial electrospray, it is unsuitable toaugment the electric field excessively, for the para-sinuousmode as well as the para-varicose and transitional modeswill probably be transformed into the other unstable modes[9] if the electric field is sufficiently large. Therefore inexperiments the used voltage must be handled delicatelyand kept moderate. The Weber number has almost the

ARTICLE IN PRESS

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.0450.8

0.6

0.4

0.250

0.4

0.25

� r� r

|1−Λ| = 0

|1−Λ| = 0.8

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

�

�

Fig. 11. Effect of the diameter ratio L on the growth rate of (a) the para-

varicose mode and (b) the para-sinuous and transitional modes. The

values of j1� Lj are: 0, 0.25, 0.4, 0.6, and 0.8.

F. Li et al. / Journal of Electrostatics 66 (2008) 58–7068

same effect on the jet instability as the radial electric field.On the other hand, the non-axisymmetric modes will bedestabilized with the increase of the electrostatic force orWeber number [9]. These situations should be avoided inelectrospray.

In case OEP, the electrical permittivities of the inner andouter liquids have not much effect on the jet instability.However, in case ONEP, the permittivities of the inner andouter liquids are involved. The numerical results showedthat the para-varicose and para-sinuous modes are bothdestabilized with the dielectric constant of the outer liquidincreasing, while the dielectric constant of the inner liquidhas little effect. In cases IEP and INEP, the permittivityof the outer liquid destabilizes the para-varicose andpara-sinuous modes remarkably. In particular, in caseINEP the permittivity of the inner liquid is involved,representing a non-monotonic influence on the unstable

modes. On the other hand, the difference between theequipotential and NEPs depends greatly on the relevantpermittivity. For cases OEP and ONEP, the dielectricconstant of the outer liquid enhances the gap betweenthem. And so does the dielectric constant of the innerliquid for cases IEP and INEP.In general, all the cases well predict the generation of

compound droplets in coaxial electrospray under certainconditions. However, our models are somewhat over-simplified, where the liquids are assumed to be inviscid andthe basic velocity profiles to be uniform. Moreover, thedriving liquid is close to be perfectly conducting. We canonly get some qualitative results based on it. In futureanalysis, these assumptions are expected to be modifiedand the more practical leaky dielectric model taking intoaccount liquid viscosity and real velocity profile should beestablished.

5. Conclusions

The instability of the inner-driving coaxial electrosprayis studied in this paper, utilizing a simplified model of aninviscid coaxial jet having uniform basic velocities inside acoaxial cylindrical electrode for the non-equipotential case.The analytical non-dimensionalized dispersion equation isderived to investigate the effects of relevant parametersinvolved on the instability of the electrified coaxial jet.Following the temporal linear instability analysis, threeindependent unstable modes, i.e. the para-varicose, para-sinuous and transitional modes, are identified in theRayleigh regime. And the axisymmetric case is studiednumerically.The effect of the radial electric field on the

unstable modes is considerably significant. In general, itinhibits the para-varicose and transitional modes distinctlyand enhances the para-sinuous mode in the Rayleighregime in most situations. The dominant region of thepara-sinuous mode moves towards relatively shortwavelengths with the electric field intensity increasing,even into the wind-induced regime. On the other hand, theelectric permittivity of the inner liquid diminishes thegrowth rate of the para-sinuous mode when it decreases.The dielectric constant of the outer liquid is foundto influence the jet instability significantly and complicat-edly. The non-equipotential case is predicted to havenarrower para-sinuous dominant region than the equipo-tential case.It is of interest that the Weber number has a similar

effect on the jet instability with the electrostatic force,stabilizing the jet at relatively long wavelengths anddestabilizing it at relatively short wavelengths. Namely,the surface tensions on both the interfaces enhance the jetinstability at long wavelengths and suppress it at shortwavelengths regardless of the magnitude of the electricfiled. The relative velocity difference also plays animportant role in the jet instability, inducing the Kelvin–Helmholtz instability.

ARTICLE IN PRESS

Table 1

The symbols h1, h2, h3, h4, z, T and x in the uniform dispersion equation expression

Cases h1 h2 h3 h4 z T xOEP 0 0 1 0 K 0nðaÞ

KnðaÞ/ /

ONEP 0 0 1 0 a / /

IEP 1 1 �p2ð�p2 � 1Þ 1�ð�p2 � 1Þ

K 0nðaÞxT

D2K 0nðaÞ � �p2D4KnðaÞ D4

INEP 1 1 �p2ð�p2 � 1Þ 1�ð�p2 � 1Þ

K 0nðaÞxT

b

D4 ��p2InðaaÞ

�p1I 0nðaaÞD5

a

�p2K 0nðaÞ InðaaÞI 0nðaaÞK 0nðaÞ�p2�p1� 1

� �þ I 0nðaÞðI

0nðaaÞKnðaaÞ �

�p2�p1

InðaaÞK 0nðaaÞÞ

� �

½�p2I 0nðaÞKnðaÞ � InðaÞK 0nðaÞ� I 0nðaaÞKnðaaÞ ��p2�p1

InðaaÞK 0nðaaÞ

� �þ InðaaÞI 0nðaaÞKnðaÞK 0nðaÞð�p2 � 1Þ

�p2�p1� 1

� �.

b

�

½InðaÞK 0nðaÞ � �p2I 0nðaÞKnðaÞ� I 0nðaaÞKnðaaÞ ��p2�p1

InðaaÞK 0nðaaÞ

� �

I 0nðaaÞ

þ InðaaÞKnðaÞK 0nðaÞ 1��p2�p1

� �ð�p2 � 1Þ.

F. Li et al. / Journal of Electrostatics 66 (2008) 58–70 69

According to the analyses, the non-equipotential casepredicts less advantage compared to the equipotential casein inner-driving coaxial electrospray. Liquid having rela-tively good conductivity are suggested to serve as inner-driving one. Moreover, moderate electric field and rela-tively small surface tensions together with relatively smallliquid velocity difference are favorable to the generation ofcompound droplets.

Acknowledgments

This work was supported by the National NaturalScience Foundation of China Project no. 10572137 and theGraduate Innovation Project of USTC no. KD2005036.

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