ARTICLE IN PRESS

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Journal of Electrostatics 64 (2006) 690–698

www.elsevier.com/locate/elstat

Temporal linear instability analysis of an electrified coaxial jet withinner driving liquid inside a coaxial electrode

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 2 January 2005; received in revised form 13 December 2005; accepted 8 January 2006

Available online 3 February 2006

Abstract

In this paper, a temporal linear stability analysis is performed of a coaxial jet composed of two immiscible liquids inside a coaxial

electrode. This analysis is carried out to investigate the case of an inner driving coaxial electrospray system. The assumption is made that

the inner liquid has high electric conductivity, and the outer liquid is an insulating dielectric. The dimensionless dispersion equation for

both the axisymmetric and non-axisymmetric modes is derived and solved numerically for the axisymmetric case. The effects of the

relevant dimensionless parameters on the instability of the jet are discussed in detail. These parameters include the dimensionless

electrostatic force E, the dielectric constant ratio e, the diameter ratios a and b, the velocity ratio L, the density ratio S, the Weber

number, and the interface tension ratio z. Two independent unstable modes, modes 1 and 2, are found and analyzed. Among the various

parameters, the dimensionless electrostatic force and the dielectric constant have a similar and remarkable influence on modes 1 and 2,

altering drastically the regime of the jet as they vary. The interface tension on the outer interface promotes the instability of both modes 1

and 2 in the region of long wavelengths while suppressing the growth rate in the region characterized by short wavelengths. The interface

tension on the inner interface, however, promotes instability of only mode 2 in the same way. The diameter ratio a has a great effect on

mode 2 while a negligible influence on mode 1. And the diameter ratio b has a slight effect on both the unstable modes.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Electrospray; Coaxial jet; Dispersion equation; Temporal Instability

1. Introduction

Coaxial electrospray is a new, effective method offorming micro and nano capsules. It can be used in thedrug industry, for injecting food additives, in papermanufacturing, and other industries as well. Recently,much experimental research has been performed to find anelectrospray method for generating compound droplets [1].Other research has focused on finding appropriate scalinglaws between electric current and drop size for the cases ofboth outer driving and inner driving [2], as well as thedifferent modes of the coaxial jet electrospray obtained forthe outer driving case [3]. Until now, however, there hasbeen little research dealing with theoretical and numericalmodeling of this phenomenon. Particularly lacking has

e front matter r 2006 Elsevier B.V. All rights reserved.

stat.2006.01.003

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

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

been any instability analyses of electrospray in the coaxialcase.As to the stability analysis of single-liquid electrospray,

various results have been reported, including the effects ofa wide range of parameters on the instability of the jet [4],the temporal linear stability of conductive and dielectricjets in both radial and axial electric fields [5], [6], thetemporal linear stability analysis of a cylindrical electrifiedjet flowing at high velocity inside a coaxial electrode [7], theabsolute and convective instabilities of a cylindricalelectrified jet in a radial electric field [8], a nonlinearelectro-hydrodynamic stability of a finite conducting jet inan axial electric field [9], and the stability of conductingviscosity jets in radial ac electric fields [10].In a previous paper [11], we investigated the linear

stability of an electrified coaxial jet with the outer liquiddriven. In this paper, we attempt to explain the breakup ofthe electrified coaxial jet for the case where the inner liquidis driven. Specifically we utilize the method of temporal

ARTICLE IN PRESSF. Li et al. / Journal of Electrostatics 64 (2006) 690–698 691

linear instability analysis. For the purpose of analysis, thetwo liquids and the ambient gas are assumed to beincompressible Newtonian fluids. The viscosities of allfluids are neglected, and the motion is assumed to beirrotational. The effects of gravity and magnetic fields arealso neglected. The outer liquid and the gaseous phase areassumed to be perfect dielectrics; while inner liquid isconducting and obeys Ohm’s law. Both conductivity anddielectric properties of all liquids are assumed constantover time. The dielectric relaxation time of the system isassumed so small that all free charges distribute on theliquid–liquid interface, achieving an equilibrium state,essentially instantaneously for both unperturbed andperturbed cases.

2. Theoretical Analysis

The coaxial jet considered in this paper consists of acylindrical inner liquid jet of radius R1, velocity U1, anddensity r, and a coflowing outer liquid jet of radius R2,velocity U2, and density r2. The background gas in theunperturbed case is stationary. An electric potential V0,applied between the central axis anode and the earthedouter cylindrical cathode of radius R3, is kept constant(Fig. 1). Hereafter in this paper, the subscripts 1, 2 and 3shall denote the inner liquid, outer liquid, and backgroundgaseous phase, respectively, when they are used to describebulk physical quantities. These same subscripts will denotethe inner liquid–liquid interface, the outer gas–liquidinterface, and the cylindrical electrode, respectively, whenused to describe interface or boundary physical quantities.

The viscosities of fluids are not considered, hence allshear forces at the liquid–liquid and gas-liquid interfacesdisappear from the equations, and the basic flow velocitiesare allowed to sustain discontinuities at the variousinterfaces. In cylindrical coordinates (r,y,z) , the basicvelocity profiles are assumed to be

~U1ðr; y; zÞ ¼ U1ð0; 0; 1Þ,

~U2ðr; y; zÞ ¼ U2ð0; 0; 1Þ,

~U3ðr; y; zÞ ¼ 0.

V0U2

R1

R2

R3

U1

Fig. 1. Diagram of the model showing relevant coordinates and

dimensions.

Because the entire inner jet of conducting liquid isequipotential, the potential and the electric-field intensityof the inner liquid become

V1ðr; y; zÞ ¼ V0; ~E1 ¼ �rV 1 ¼ 0.

The outer annular liquid jet and the background gaseousphase are both considered as dielectrics, and hence theelectric fields inside them cannot be neglected. Thepotential and electric field intensity in the unperturbedcase can be obtained via the usual electrostatic laws:

V2 ¼ V0lnðr=R1Þ

lnAþ 1

� �; ~E2 ¼ �

V0~r0r lnA

; R1 � r � R2,

V3 ¼e2e3�

V0lnðr=R3Þ

lnA; ~E3 ¼ �

e2e3�

V 0~r0r lnA

; R2 � r � R3,

where ~r0 is the unit vector of the r- direction, and A ¼

ðR2=R3Þe2=e3=ðR2=R1Þ.

During the process of linear stability analysis, we shallmaintain the small amplitude disturbance assumptionthroughout. The interfaces being perturbed consist of thefollowing:

rsi ¼ Ri þ Zi; i ¼ 1; 2,

where Zi is the displacement of the interface from theunperturbed case.The perturbed pressure field can be expressed as

p ¼ p0+p0. When the electric field is perturbed, it will stillbe assumed that V 1ðr; y; zÞ ¼ V0; and ~E1 ¼ �rV 1 ¼ 0.The potential and the electric-field intensity in the outerliquid and in the gaseous phase will be written asVi ¼ V 0i þ V 0i; ~Ei ¼ ~E0i þ ~E

0

ii, where subscript 0 denotesthe unperturbed properties, and the ‘prime’ superscriptdenotes the perturbation of the corresponding quantity. Inthe normal mode method of temporal linear instabilityanalysis, the wave-number k is real, and the frequency o iscomplex function of k: oðkÞ ¼ orðkÞ þ ioiðkÞ: Hence theperturbation can be decomposed into the form of a Fourierexponential:

ðZi;~u0i; p0i;V0iÞ ¼ ðZiðrÞ; ~uiðrÞ; piðrÞ; V iðrÞÞe

otþiðkzþnyÞ, (1)

where Zi,~ui,pi,V i are the perturbation amplitudes of theinterface, velocity, pressure and electrical potential, respec-tively, and n is the azimuthal wave number.Substituting (1) into the linearized, small perturbation

equations for an inviscid fluid, we obtain a modified Besselequation of order n for the amplitude function piðrÞ:

d2pi

dr2þ

1

r

dpi

dr� k2

þn2

r2

� �pi ¼ 0. (2)

At the same time, from Maxwell’s equations applied toelectro-hydrodynamics, we can obtain a modified Besselequation of order n for the electrical potential perturbationamplitude V iðrÞ:

d2V i

dr2þ

1

r

dV i

dr� k2

þn2

r2

� �V i ¼ 0. (3)

ARTICLE IN PRESSF. Li et al. / Journal of Electrostatics 64 (2006) 690–698692

The solutions of Eqs. (2) and (3) consist of linearcombinations of two nth-order modified Bessel functions ofthe first and the second kinds, InðkrÞ and KnðkrÞ. Todetermine all coefficients in the solutions, and, ultimately,to obtain the dispersion equation, the boundary conditionsfor both hydromechanics and electromagnetics must beused.

The kinematic interface condition applicable to both theinner liquid–liquid and outer gas–liquid interfaces takes thefollowing linearized form:

v0 ¼@Z@tþUs

@Z@z

Here v0 is the interfacial radial velocity in the perturbedcase, Us is the basic flow velocity, and the subscript s

denotes an interfacial property.The dynamic interface conditions for the inner and outer

interfaces are expressed by

p1 ¼ p2 þ g1r �~n1 �~n1 �~~T

el

2 �~n1,

p2 ¼ p3 þ g2r �~n2 �~n2 �~~T

el

3 �~~T

el

2

� ��~n2,

where gi is the interface tension, and ~ni is the normal vectorat each respective interface.

At the inner and outer interfaces, we must consider thecontinuity of the tangential electric field, and Gauss’ lawfor the normal field:

n� ~Ei

�� �� ¼ 0,

~n � ei~Ei

�� �� ¼ qs.

In these equations, the notation || || denotes a jump inmagnitude across the interface, and i ¼ 1, 2 or 3 denotesthe corresponding physical quantity.

Using the characteristic quantities R2, r2, U2, g2 and e3,we obtain the following dimensionless parameters:

a ¼ R2k dimensionless wave numberb ¼ oR2=U2 dimensionless frequency

S ¼r1r2

density ratio of the inner liquid to the outer liquid

Q ¼r3r2

density ratio of the gaseous phase to the outerliquid

L ¼ U1

U2velocity ratio of the inner liquid to the outerliquid

a ¼ R1

R2diameter ratio of the inner liquid jet to the outerliquid jet

b ¼ R3

R2diameter ratio of the coaxial cathode to the outerliquid jet

z ¼ g1g2

interface tension ratio of the inner interface to theouter interface

e ¼ e2e3

dielectric constant ratio of the outer liquid to thegaseous phase

We ¼r2U2

2R2

g2the Weber number of the outer interface

E ¼e3V2

0

r2U22R2

2ln2A

the ratio of electrostatic and inertial forces

Finally the dispersion equation can be written as thefollowing dimensionless form:

Dða;bÞ ¼ ðD�1 þH�2D�5ÞðD

�6 �H�4D

�5Þ þ ðD

�3 �H�3D

�5Þ

� ðD�4 �H�1D�5Þ ¼ 0, ð4Þ

where the symbols in Eq. (4) are listed in Appendix A.Note that in the above dispersion relation (4), the

dielectric constant ratio between two liquids is notinvolved. It is because that the inner liquid is assumed tobe a conductor and there is no electric field in its bulk.Therefore the dielectric constant of the inner liquid doesnot influence on the instability of our model. Actually onlythe tangential electric field condition on the innerliquid–liquid interface is used in the derivation process ofthe dispersion relation.

3. Numerical Results

In this section, the effects of parameters involved in thedispersion equation on the instability of the electrifiedcoaxial jet are discussed in detail. The dimensionlessdispersion equation (4) is quartic for the complexfrequency b and must be solved numerically. Here we useMATLAB which can implement symbolic computationand solve the transcendental equation easily. An exampleof the procedure is given in Appendix B. There exist foursolutions of the complex frequency b, corresponding tofour different modes. The real parts of two solutions arepositive, and the real parts of the other two are negative.Because only positive solutions induce jet instability, thenegative solutions will be omitted in the analysis. The tworemaining modes to be analyzed will be labeled modes 1

and 2.The dimensionless number E ¼ e3V 2

0=r2U22R

22ln

2 A

which describes the ratio of the electrostatic force to theinertial force, has a prominent effect on both modes 1 and2(Fig. 2a and b). As the dimensionless number E increases,the growth rate br increases significantly for a fixed wavenumber a. For each plot, there exists a maximum growthrate, brmax, with a corresponding wave number armax andcut-off wave number ac above which the growth ratedecreases to zero. These results show that the existence of amaximum growth rate definitely plays a role in the breakupof the jet. Those wave numbers corresponding to themaximum value, or those close to it in value, have a strongconnection to the size of the droplets that break awayfrom the coaxial jet. With increasing E, the parametersbrmax, armax, and ac also increase quickly, suggesting thatthe coaxial jet is more unstable, the droplet sizes aresmaller, and the breakup of the jet probably movesfrom the Rayleigh regime to the wind-induced andatomization regimes. Comparing mode 1 with mode 2,the maximum growth rate brmax of the former is smallerthan the maximum growth rate of the latter. And thewave number armax corresponding to brmax for mode 1 isclearly smaller than that for mode 2 all the way. At the

ARTICLE IN PRESS

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

β r

mode 1

(a)

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

β r

mode 2

(b)

E=0E=0.0001E=0.0002E=0.0003E=0.0005

α

α

E=0E=0.0001E=0.0002E=0.0003E=0.0005

Fig. 2. Effect of the dimensionless number E on the growth ratebr for (a)mode 1 ; (b) mode 2. a ¼ 0.4, b ¼ 2.0, S ¼ 0.8, Q ¼ 1� 10�3, L ¼ 0:8,z ¼ 0:2, e ¼ 20, We ¼ 100, n ¼ 0.

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

α

β r

mode 1

(a)

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

β r

mode 2

(b)

ε=10ε=15ε=20ε=25ε=30

α

ε=10ε=15ε=20ε=25ε=30

Fig. 3. Effect of the dielectric constant ratio e on the growth rate br for (a)mode 1; (b) mode 2. a ¼ 0.4, b ¼ 2.0, S ¼ 0.8, Q ¼ 1� 10�3, L ¼ 0:8,z ¼ 0:2, We ¼ 100, E ¼ 0:0003, n ¼ 0.

F. Li et al. / Journal of Electrostatics 64 (2006) 690–698 693

region of relatively long wavelengths, the growth rate ofmode 1 is generally larger than that of mode 2, whereas atthe region of relatively short wavelengths, the growth rateof mode 2 exceeds that of mode 1 for a fixed wave number.And mode 2 has a much larger cut-off wave number ac thanmode 1.

The dimensionless number e ¼ e2=e3, which representsthe dielectric constant ratio of the outer fluid to the gaseousphase, has a similar effect on both modes 1 and 2 withchanging dimensionless number E (Fig. 3a and b). But itdestabilizes mode 1 more violently than E does, whereas itsinfluence on mode 2 is comparatively small. The distinctinstability effect of the dimensionless parameters E and eon the two unstable modes can be explained by noting thatthe most essential influencing factor affecting the instabilityof the jet is the electric field intensity within the liquid jetand at the interfaces.

For mode 1, when the diameter ratio b ¼ R3=R2 issmaller than 2, the growth rate br decreases slowly withincreasing b (Fig. 4a). At the same time, brmax, armax and acdecrease also. However, when the diameter ratio b exceeds2, the values of br, brmax, armax and ac change little as b

increases. This means that the influence of b on theinstability of the jet approaches a limiting state as b

approaches the value of 2. For mode 2, the growth rate br

together with brmax, armax and ac, increases obviously withdiminishing diameter ratio a ¼ R1=R2 (Fig. 4b). This resultsuggests that the diameter ratio b is not an importantinfluencing factor affecting the instability of the electrifiedcoaxial jet in the inner-driven case, while the diameter ratioa affects the instability of the electrified coaxial jet verymuch. On the other hand, b has a negligible influence onmode 2 and while a has a negligible influence on mode 1.Hence mode 1 is most likely associated with the instability

ARTICLE IN PRESS

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

β r

mode 1b=1.2b=1.5b=2b=3b=10

(a)

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

β r

mode 2a=0.15a=0.25a=0.4a=0.8a=0.85

(b)

α

α

Fig. 4. (a) Effect of the diameter ratio b on the growth rate br for mode 1.

a ¼ 0.4, S ¼ 0.8, Q ¼ 1� 10�3, L ¼ 0:8, z ¼ 0:2, We ¼ 100, e ¼ 20,

E ¼ 0:0003, n ¼ 0. (b) Effect of the diameter ratio a on the growth rate

br for mode 2. b ¼ 2.0, S ¼ 0.8, Q ¼ 1� 10�3, L ¼ 0:8, z ¼ 0:2, We ¼ 100,

e ¼ 20, E ¼ 0:0003, n ¼ 0.

0 2 4 6 8 10 120

0.5

1

1.5

β r

mode 1S=0.001S=0.01S=0.1S=0.8S=1.0S=2.0S=4.0S=10S=20

(a)

0 2 4 6 8 10 120

1

2

3

4

5

6

β r

mode 2

(b)

α

α

Λ=0Λ=0.4Λ=0.8Λ=1.0Λ=1.3Λ=1.4Λ=1.8

Fig. 5. (a) Effect of the density ratio S on the growth rate br for mode 2.

a ¼ 0.4, b ¼ 2.0, Q ¼ 1� 10�3, L ¼ 0:8, z ¼ 0:2, We ¼ 100, e ¼ 20,

E ¼ 0:0003, n ¼ 0. (b) Effect of the velocity ratio L on the growth rate

br for mode 2. a ¼ 0.4, b ¼ 2.0, S ¼ 0.8, Q ¼ 1� 10�3, z ¼ 0:2, We ¼ 100,

e ¼ 20, E ¼ 0:0003, n ¼ 0.

F. Li et al. / Journal of Electrostatics 64 (2006) 690–698694

of the outer gas-liquid interface, while mode 2 is associatedwith the inner liquid-liquid interface.

The density ratio S ¼ r1=r2 and the velocity ratio L ¼U1=U2 exert negligible effects on mode 1, but they haveremarkable effects on mode 2 (Fig. 5a and b). Therefore itis likely that modes 1 and 2 are associated with the outerand inner interfaces, respectively. In the calculationwindow of the wave number a from 0 to 12, the growthrate br increases with decreasing the density ratio S exceptfor very small density ratios and relatively large wavenumbers. When the velocity ratio L is equal to unity (i.e.,the outer and inner liquids have the same velocity), thegrowth rate br, together with brmax, armax and ac, attainsthe smallest values. As the velocity ratio L deviates fromunity, all these quantities quickly increase. Specifically,when the velocity difference across the interface increases,

the jet becomes more unstable. The origin of instability inthis case is the Kelvin-Helmholtz instability.The Weber number We ¼ r2U2

2R2=g2, which representsthe ratio of the inertial force of the outer fluid to theinterface tension of the outer interface, has distinct effectson both modes 1 and 2 (Fig. 6a and Fig. 6b). When We issmall, the growth rate br and the maximum growth ratebrmax increase markedly with decreasing We, but armax andac change little. When We surpasses a critical value, allthese quantities increase with We. It is coincident with theformer conclusion that the interface tension on the outerinterface promotes instability of the jet in the region of longwavelength, and it suppresses instability in the region ofshort wavelength [12]. The Weber number We has almostthe same influence on mode 2 as that on mode 1. The onlydifference is that the influence is a little slight.

ARTICLE IN PRESS

0 2 4 6 8 10 120

1

2

3

4

5

6

β r

mode 1

We=0.01We=0.1We=1We=10We=100We=200We=1000

(a)

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

β r

mode 2

We=0.01We=0.1We=1We=10We=100We=200We=1000

(b)

α

α

Fig. 6. Effect of the Weber number We on the growth rate br for (a) mode

1 ; (b) mode 2. a ¼ 0.4, b ¼ 2.0, S ¼ 0.8, Q ¼ 1� 10�3, L ¼ 0:8, z ¼ 0:2,e ¼ 20, E ¼ 0:0003, n ¼ 0.

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

β r

mode 1

(a)

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

β r

mode 2

(b)

α

α

ζ=0.01

ζ=100ζ=0.2

ζ=0.01ζ=0.1ζ=0.2ζ=0.4ζ=1ζ=10ζ=100

Fig. 7. Effect of the interface tension ratio z on the growth rate br for (a)mode 1 ; (b) mode 2. a ¼ 0.4, b ¼ 2.0, S ¼ 0.8, Q ¼ 1� 10�3, L ¼ 0:8,We ¼ 100, e ¼ 20, E ¼ 0:0003, n ¼ 0.

F. Li et al. / Journal of Electrostatics 64 (2006) 690–698 695

The dimensionless number z ¼ g1=g2, which representsthe tension ratio across of the inner-to-outer surfaceinterface, has no marked effect on mode 1, as has beenassumed (Fig. 7a). For mode 2, we see that when z isgreater than unity, the growth rate br and the maxi-mum growth rate brmax decrease with decreasing z(Notethat z ¼ 1 means the outer interface and the inner inter-face have the same interface tensions.) When the dimen-sionless number z is smaller than unity, br and brmax

increase with decreasing z. However, armax and ac de-crease all the time with increasing the interface tension.The variation of the growth rate indicates that the inter-face tension on the inner interface promotes instabilityof the jet at first for long wavelengths, and then suppre-sses the growth rate in the region of the short wave-lengths if the interface tension of the outer interface isfixed.

4. Conclusions and discussions

The instability of the inner driven coaxial electrospray hasbeen investigated in this paper. The system has been modeledas a coaxial cylindrical jet inside a coaxial cylindricalelectrode. The dimensionless dispersion equation is derivedto investigate the effects of the parameters involved on thebreakup of the electrified coaxial jet. Following the methodof temporal instability analysis, two independent modes,modes 1 and 2, are discovered. As the axisymmetric mode isthe most unstable [13], the axisymmetric case of thedispersion equation is solved numerically while assumingthat the azimuthal wave number n is equal to zero.The dimensionless parameter E ¼ e3V 2

0=r2U22R

22ln

2A andthe dielectric constant e ¼ e2=e3 have a similar effect onmodes 1 and 2. These two parameters affect both modes 1

and 2 significantly, which is quite different from the case of

ARTICLE IN PRESSF. Li et al. / Journal of Electrostatics 64 (2006) 690–698696

the outer-driving coaxial electrospray where mode 1 is muchmore sensitive to the electric field than is mode 2 [11]. Boththe modes can be changed from the Rayleigh regime to thewind-induced and atomization regimes while E or e increasecontinuously. The diameter ratio b ¼ R3=R2 has a slighteffect on the growth rate of mode 1. The opposite is knownto be true for the outer driven jet [11]. And the diameterratio a ¼ R1=R2 has a great effect on the growth rate ofmode 2, the same as the out-driving case. The effect of a onmode 1 and the effect of b on mode 2 are shown to benegligible, indicating that modes 1 and 2 are associated withthe outer and inner interfaces, respectively. This conclusionis supported by the results of the velocity ratio L ¼ U1=U2

and the density ratio S ¼ r1=r2. The Weber number We ¼

r2U22R2=g2 has a significant influence on both modes 1 and

2. It can be seen that the outer interface tension influencesnot only the outer interface but also the inner interface in theinner-driven case through its interaction with the electricfield intensity, while it influences only the outer interface inthe outer-driven case [11]. Conversely, the interface tensionratio z ¼ g1=g2 influences only mode 2.In this paper we do not consider the effect of the liquid

viscosity on the stability of the coaxial jet, which will becarried out in future.

Acknowledgements

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

Appendix A

The symbols appearing in Eq. (4) are:

A� ¼a

be,

D�1 ¼ InðaaÞK 0nðaaÞ � I 0nðaaÞKnðaaÞ; D�2 ¼ InðaaÞKnðaÞ � InðaÞKnðaaÞ,

D�3 ¼ I 0nðaaÞKnðaÞ � InðaÞK 0nðaaÞ; D�4 ¼ InðaaÞK 0nðaÞ � I 0nðaÞKnðaaÞ,

D�5 ¼ I 0nðaaÞK 0nðaÞ � I 0nðaÞK0nðaaÞ; D�6 ¼ InðaÞK 0nðaÞ � I 0nðaÞKnðaÞ,

D�7 ¼ InðaÞKnðabÞ � InðabÞKnðaÞ; D�8 ¼ I 0nðaÞKnðabÞ � InðabÞK 0nðaÞ,

D�9 ¼ InðaÞK 0nðabÞ � I 0nðabÞKnðaÞ; D�10 ¼ I 0nðaÞK0nðabÞ � I 0nðabÞK 0nðaÞ,

H�1 ¼

S bþ iLað Þ2InðaaÞ � za

a2We1� n2 � ðaaÞ2� �

I 0nðaaÞ þeaEI 0nðaaÞ

a31þ

aaI 0nðaaÞ

InðaaÞþ

aaInðaÞInðaaÞ

D�1 D�8�eI 0n ðaÞIn ðaÞ

D�7

� �eD�4D

�7�D

�2D�8

� �

bþ iað Þ2I 0nðaaÞ

,

H�2 ¼ea2E

a bþ iað Þ2�

e� 1ð ÞD�1D�8

eD�4D�7 � D�2D

�8

,

H�3 ¼Qb2 D�9

D�10þ a

We1� n2 � a2� �

� eaE e� 1ð Þ � ea2Ee�1ð Þ

2D�4D�8

eD�4D�7�D

�2D�8

bþ iað Þ2

,

H�4 ¼

ea2Ea�

InðaÞInðaaÞ

I 0nðaÞInðaÞþ

D�4 D�8�eI 0n ðaÞIn ðaaÞ

D�7

� ��

eInðaaÞInðaÞ

D�6D�8

eD�4D�7�D

�2D�8

� �

bþ iað Þ2

,

where I 0nðxÞ;K0nðxÞ are the derivatives of the n-order modified Bessel functions of the first and the second kinds, InðxÞ;KnðxÞ;

respectively.

Appendix B

A MATLAB procedure for solving the complex frequency b in the dispersion relation (4): syms dalpha ddisp betaformat long g

n ¼ 0 %

n is the azimuthal wave number f ¼ diff(besseli(n,dalpha),dalpha) % derivative of the Bessel function In(x) g ¼ diff(besselk(n,dalpha),dalpha) % derivative of the Bessel function Kn(x) a ¼ 0.4 % setting of the parameter a ¼ R1=R2

ARTICLE IN PRESSF. Li et al. / Journal of Electrostatics 64 (2006) 690–698 697

b ¼ 2 %

setting of the parameter b ¼ R3=R2 lamda ¼ 0.8 % setting of the parameter lamda ¼ U1=U2 S ¼ 0.8 % setting of the parameter S ¼ rho1=rho2 Q ¼ 1e-3 % setting of the parameter Q ¼ rho3=rho2 zeta ¼ 0.2 % setting of the parameter zeta ¼ gamma1/gamma2 epsilon ¼ 20 % setting of the parameter epsilon ¼ epsilon2/spsilon3 We ¼ 100 % setting of the parameter We ¼ rho2nU22nR2=gamma2 Epsilon ¼ 0.0003 % setting of the parameter Epsilon ¼ epsilon3*V0^2/(rho2*

%

U2^2*R2^2*(lnA)^2) alfa ¼ 0.01:0.4:12 % setting of the wave number alpha si ¼ length(alfa) one ¼ ones(4,si) % generating a matrix to store the four solutions alfabeta ¼ [alfa; one] % the matrix storing the wave number and solutions ok ¼ sym(‘ddisp ¼ 0’) % formation of the dispersion relation

for p ¼ 1:1:sialpha ¼ alfa(p)% computing the involved Bessel functionsin ¼ besseli(n,alpha)ina ¼ besseli(n,a*alpha)inb ¼ besseli(n,b*alpha)kn ¼ besselk(n,alpha)kna ¼ besselk(n,a*alpha)knb ¼ besselk(n,b*alpha)inalfa ¼ subs(f,dalpha,alpha)inalfaa ¼ subs(f,dalpha,a*alpha)inalfab ¼ subs(f,dalpha,b*alpha)knalfa ¼ subs(g,dalpha,alpha)knalfaa ¼ subs(g,dalpha,a*alpha)knalfab ¼ subs(g,dalpha,b*alpha)delta1 ¼ ina*knalfaa-inalfaa*knadelta2 ¼ ina*kn-in*knadelta3 ¼ inalfaa*kn-in*knalfaadelta4 ¼ ina*knalfa-inalfa*knadelta5 ¼ inalfaa*knalfa-inalfa*knalfaadelta6 ¼ in*knalfa-inalfa*kndelta7 ¼ in*knb-inb*kndelta8 ¼ inalfa*knb-inb*knalfadelta9 ¼ in*knalfab-inalfab*kndelta10 ¼ inalfa*knalfab-inalfab*knalfaB ¼ epsilon*delta4*delta7-delta2*delta8

H1 ¼ (

S*(beta+i*lamda*alpha)^2*ina-zeta*alpha*(1-n^2-(alpha*a)^2)*inalfaa/(a^2*We)+epsilon*alpha*Epsilon*inalfaa/a^3*(1+alpha*a*inalfaa/ina+alpha*a*in*delta1*(delta8-epsilon*inalfa*delta7/in)/(ina*B)))/((beta+i*alpha)^2*inalfaa)

H2 ¼ e

psilon*alpha^2*Epsilon*(epsilon-1)*delta1*delta8/(a*(beta+i*alpha)^2*B) H3 ¼ ( Q*beta^2*delta9/delta10+alpha*(1-n^2-alpha^2)/We-epsilon*alpha*Epsilon*(epsilon-1)-

epsilon*alpha^2*Epsilon*(epsilon-1)^2*delta4*delta8/B)/(beta+i*alpha)^2

H4 ¼ i n/ina*epsilon*alpha^2*Epsilon/a*(inalfa/in+(delta4*(delta8-epsilon*inalfa*delta7/in)-

epsilon*ina*delta6*delta8/in)/B)/(beta+i*alpha)^2

disp ¼ ( delta1+H2*delta5)*(delta6-H4*delta5)+(delta3-H3*delta5)*(delta4-H1*delta5) equ ¼ subs(ok,ddisp,disp) betas ¼ double(solve(equ)) % getting the four complex frequencies alfabeta(2:5,p) ¼ betas

end

ARTICLE IN PRESSF. Li et al. / Journal of Electrostatics 64 (2006) 690–698698

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