Linear Stability of Axisymmetric Perturbations in Imperfectly Conducting Liquid Jets

Linear stability analysis of axisymmetric perturbations in imperfectlyconducting liquid jets

J. M. Lpez-Herrera,a! P. Riesco-Chueca,b! and A. M. Gan-Calvoc!Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n,Sevilla 41092, Spain

sReceived 7 June 2004; accepted 7 January 2005; published online 1 March 2005d

A discussion is presented on the role of limited conductivity and permittivity on the behavior ofelectrified jets. Under certain conditions, significant departures with respect to the perfect-conductorlimit are to be expected. In addition, an exploration is undertaken concerning the validity ofone-dimensional average models in the description of charged jets. To that end, a temporal linearmodal stability analysis is carried out of poor-conductor viscous liquid jets flowing relatively to asteady radial electric field. Only axisymmetric perturbations, leading to highest quality aerosols, areconsidered. A grounded coaxial electrode is located at variable distance. Most available studies inthe literature are restricted to the perfect-conductor limit, while the present contribution is anextension to moderate and low electrical conductivity and permittivity jets, in an effort to describea situation increasingly prevalent in the sector of small-scale free-surface flows. The influence of theelectrode distance b, a parameter a defined as the ratio of the electric relaxation time scale to thecapillary time scale, and the relative permittivity b on the growth rate has been explored yieldingresults on the stability spectrum. In addition, arbitrary viscosity and electrification parameters arecontemplated. In a wide variety of situations, the perfect-conductor limit provides a goodapproximation; however, the influence of a and b on the growth rate and most unstable wavelengthcannot be neglected in the general case. An interfacial boundary layer in the axial velocity profileoccurs in the low-viscosity limit, but this boundary layer tends to disappear when a or b are largeenough. The use of a one-dimensional s1Dd averaged model as an alternative to the 3D approachprovides a helpful shortcut and a complementary insight on the nature of the jets perturbativebehavior. Lowest-order 1D approximations saverage modeld, of widespread application in theliterature of electrified jets, are shown to be inaccurate in low-viscosity imperfect-conductor jets. 2005 American Institute of Physics. fDOI: 10.1063/1.1863285g

I. INTRODUCTION

Electrification of free-surface flows is guided by threeprincipal aims: first, increasing the surface-to-volume ratioby means of a reduction of the liquids apparent surface ten-sion; second, controlling the stability range of the flow; third,directing the flow towards a target. Multiple applications de-rive from these achievements: atomization, particle sorting,ink jet printing, fuel injection, and fiber spinning. For in-stance, jet stimulation by electric fields is increasingly usedin a variety of ink jet printing technologies as a method forcharging and steering the ink drops resulting from jet disin-tegration. Alternatively, continuous deviated ink jet technol-ogy is based on direct deflection of electrified jets. Electriccharges modulate the jets response to electric fields, randomnoise, or specific excitation wavelengths. In many applica-tions, a grounded electrode located at the vicinity of the jetsurface establishes a radial field; the quasielectrostatic pres-sure jump competes with convective, capillary, and viscouseffects to determine the selected wavelength. Another re-search field involving electrified jets is electrospinning, aprocess in which solid fibers are produced from a polymeric

fluid stream delivered through a millimeter-scale nozzle. Ad-ditional achievements are presently reported in pharmaceuti-cal and material sciences. These applications are leading to agrowing interest of researchers in electrified jets. In thesearch for diverse applications, poor conductivity liquids arebecoming increasingly attractive, because food and phar-macy technologies, among others, depend on their efficienthandling.

An important motivation source for the study of electro-hydrodynamics is the search for high quality sprays fromliquid microjets, eagerly demanded in technological applica-tions requiring small and homogeneous droplet size. The axi-symmetric capillary breakup sRayleigh breakupd of smalland steady liquid ligaments into droplets gives rise to con-trollable monodisperse aerosols, unattainable through otherhigh-yield processes.

The foundations for the study of conducting jets in aradial electric field were set by Melcher,1 assuming inviscidflow and constant electric potential. Saville2 generalized theanalysis for arbitrary viscosity liquids. Huebner and Chu3studied the stability of a charged jet in the inviscid limitassuming arbitrary electrode distance b. Setiawan andHeister4 extended this description to the nonlinear growthphase. The influence of the surrounding atmosphere on thestability of the jet was considered by Baudry et al.5 An el-

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egant overview of the linear stability problem was presentedby Garca,6 who explored the perturbative spectrum for avariety of parameters. This work has been recently extendedto ac electric fields by Gonzlez et al.;7 the authors providean attractive formulation of the dc problems, whose notationwe follow here.

The electrical behavior of a moving liquid follows fromthe ratio of the characteristic hydrodynamic time versus theelectrical relaxation time te,i /K, where i and K are theelectrical permittivity and conductivity of the liquid, respec-tively. In the absence of other forcing agents, the time scaleon which perturbations grow and cause the jets breakup isgiven by a balance of surface tension and inertia, summed upby the capillary time tc,srA3 /gd1/2, where r ,g, and A arethe liquid density, surface tension, and the jet radius, respec-tively. If the ratio te / tc is strictly zero the liquid can be con-sidered as a perfect conductor even in the last stages ofpinch-off. In the perfect-conductor limit the calculation ofelectrical stresses on the jet surface is simplified since allcharges are interfacial, the external electric field is normal toit, and the electric field in the jet bulk is zero. In this limit,provided the breakup is axisymmetric, electrification stabi-lizes long-wavelength perturbations while destabilizing shortones.

1,8

In the present work we aim to study the stability of elec-trified liquid jets when the ratio te / tc is not strictly zero. Ourmodel is directly borrowed from the general formulationSaville provided in 1997 by expanding previous results fromTaylor and Melcher.911 The analysis is applicable to situa-tions where the relaxation time for free charges is short com-pared with the time scale for fluid motion. In the surveypublished in 1997 by Saville,9 the stability of fluid cylindersor free jets and pinned cylinders or liquid bridges is studied.These examples are given as an illustration of the powerfulset of equations for the leaky dielectric model, which buildsthe core of the above survey. The discussion is mostly quali-tative, and the results disclosed are restricted to some par-ticular cases where a solution can be obtained with relativeease. Additional contributions in this field by Saville andco-workers focused on the effect of axial electricforcing.2,1215

The starting point is provided by the linearized NavierStokes equations. A careful description is then made of theparametrical behavior of the resulting dispersion equation.Our study attempts to complete a systematic consideration ofall the input variables swavelength, viscosity, surface tension,electrical conductivity, permittivity and electrification levels,surrounding gas dynamicsd. Previous works do not includeall these terms or do not weigh their relative influence. Theambient influence is explored by means of a sketchy model,which does however allow a preliminary glimpse into thestability of electrified coflowing liquid-gas streams. The si-multaneous forcing of a jet by hydrodynamic focusing sflow-focusing technique16d and by electrification is a differentfield of research, with a variety of presumed applications,e.g., ultrafine atomization; the present contribution may pro-vide a stepping stone for the joint analysis of electrificationand flow focusing.

In addition, under sufficiently general conditions, a one-

dimensional model following Lpez-Herrera et al.17 yieldsexcellent results ffor a comparison on the suitability of eithera three-dimensional s3Dd axisymmetric or a 1D model seefor example Yildirim and Basaran18g, which allows the de-tailed and quite inexpensive exploration of extense para-metrical ranges of interest. It is our aim to determine underwhich conditions the lowest-order 1D approximation sslice-average modeld, widely used in the literature, is to be trusted.It can be anticipated that the assumption of a flat axial ve-locity profile is inaccurate when either of two situations oc-cur: sad the velocity profile is markedly convex; sbd an axialvelocity boundary layer develops, due to the action of shear,at the interface. In both cases, the flat-profile approximationfails to provide a trustworthy description. It will be shownthat jets with moderate conductivity and permittivity oftendisplay sharply convex or boundary-layer velocity profiles.Indeed, nonuniform radial profiles are favored by the combi-nation of small viscosity, poor conductivity and permittivity,and a shear agent. Tangential stress may be generated by avariety of mechanisms, such as surfactant diffusion or, in ourcase, an electric field. Axially oriented electric fields are themost efficient shear agents and they lead to the conversion ofelectric potential into kinetic energy; but even radial fields asstudied here can give rise to significant shear provided a andb are small or moderate. It is important, therefore, to becautious when modeling electrified jets: slice-average mod-els may turn out to be inaccurate, so that a 3D model orhigher-order expansions sparabolicd become unavoidable.

Moderate liquid conductivity and permittivity is readilyobserved in many technological fields, in particular when-ever oils or other organic compounds are manipulated. Otherpractical applications, such as electrospinning or jet printing,may involve imperfect conductors. In the following section,once the main dimensionless groups are introduced, refer-ence will be made to experimental and technological appli-cations involving low conductivity/permittivity liquids.

II. FORMULATION OF THE PROBLEMA. Context and assumptions

The results presented here arise from some previouswork on the conditions leading to jet breakup, under a diver-sity of geometrical and electrification contexts. Lpez-Herrera and Gan-Calvo19 presented a detailed experimen-tal procedure concerning charged capillary jet breakup, in theassumption of perfect-conductor behavior and radial electricforcing. In the course of experimentation, an extended explo-ration was attempted of the poor-conductor range. As statedabove, food, pharmacy, carburation, and other applications athand are increasingly pushing researchers toward the studyof poor-conductors stability and breakup.

Preliminary experimental results provided a wide diver-sity of data leading to considerable trouble in the effort tocalibrate and sort out raw data from a theoretical point ofview. Therefore, a detailed theoretical description of poor-conductor jet behavior appeared to be desirable, both to pro-vide a solid basis for the experimental program, and to ori-entate the production of an efficient numerical code able toinclude breakup. As an intermediate goal, the stability analy-

034106-2 Lpez-Herrera, Riesco-Chueca, and Gan-Calvo Phys. Fluids 17, 034106 ~2005!


sis of a liquid jet is hereby described: it yields inference onsad the size of drops after breakup and the stimulation re-quired therefore; sbd the conditions required in the oppositecase, i.e., when breakup is to be avoided and the integrityand longevity of the jet is to be preserved. Both objectivesare technologically relevant.

We limit the scope of our study to the axisymmetricstability of capillary cylindrical jets of a liquid with densityr, permittivity i, viscosity m, conductivity K, and interfacialsurface tension g. Our interest will be focused on the axi-symmetric capillary jet instability. This choice is dictated bytwo main reasons: first, the prevalence of axisymmetric in-stability in most applications where a high quality spray isrequired se.g., when electrospraying small liquid flow ratesd.Second, axisymmetric modes are dominant in most dc es-timulation regimes provided the Weber number is moderate.Accordingly, we restrict our study to moderate We numbersand ignore sinuous or asymmetric perturbation modes.

The surrounding atmosphere is assumed inviscid and itsinfluence on the dynamics of the jet is modeled with the helpof a simplified approach, which will be justified later byinvoking some restrictions on the Weber numbers. A and Uoare the jet radius, assumed constant, and the average axialvelocity. The equilibrium velocity profile is assumed flat; itsvalue Uo is usually scaled with the capillary wave velocityvc= sg /rAd1/2 by means of the Weber number, We= srAUo

2d /g.In the gas-at-rest limit, where U=0, and assuming

We@1, Keller et al.20 show that the perturbation is rapidlyconvected downstream with the jet velocity, thus avoidingthe simultaneous propagation of disturbances in the upstreamand downstream directions sabsolute instabilityd. See alsoArtana et al.21 for a discussion on the role of absolute andconvective instabilities in the evolution of electrified jets, aswell as ODonnell et al.22 and Chauhan et al.23 for an experi-mental investigation of the convective instability in un-charged jets. Kalaaji et al.24 presented an experimental inves-tigation of the breakup length of a jet and showed thatprovided the Weber number is large enough, temporal andspatial linear perturbation theories coincide. We therefore re-strict our study to moderate or large Weber numbers fap-proximately above 4, see Eq. s5d in Lin and Reitz25g: thisrestriction ensures that the jet velocity is sufficiently higherthan the group velocity of perturbations so that a temporalevolution of the perturbation is observed from a Galileanreference frame moving with the jet. However, when theWeber number exceeds a critical value, either nonsymmetricperturbations grow faster than axisymmetric perturbations orviscous effects in the surrounding atmosphere can no longerbe neglected.

The above references do not consider the coflowing re-gime, where both the jet and the surrounding gas flow swithdifferent mean velocityd in the axial direction. This flow con-figuration is increasingly attractive as a source of potentialtechnical applications. Unfortunately, the parametric limits ofaxisymmetric breakup in coflowing jets have not been ex-plored in detail in terms of the liquid and gas Weber num-bers. A mapping of the breakup regimes as sketched by Linand Reitz25 in the gas-at-rest limit is not available in the

coflowing case. It is known that interfacial shear promotesTaylors instability, where droplets much smaller than the jetdiameter are produced,26 but a systematic exploration of thebreakup regimes prevailing under different gas and liquidWeber numbers is not available.

In the following, however, we narrow the scope of ourattention to flow configurations located in the Rayleigh andfirst wind-induced regimes, where shear effects from the gasside can be ignored. Accordingly, aerodynamic effects sgasliquid interactiond are modeled with a simplified theory; theinfluence of the gas is accounted for in terms of its pressurefluctuation, but gas viscosity is neglected. The surroundinggas is assumed to flow coaxially with an uniform speed Uwhich, depending on the envisaged application, may rangefrom 0 sambient gas at restd to an amount one or two ordersof magnitude above the jet velocity Uo. With this choice, weleave the door open to the study of electrified flow focusingconfigurations sEF3d, a field of considerable interest. Theflow focusing technology is based on the acceleration of aliquid jet surrounded by a coflowing gas stream, whereby afavorable pressure gradient is created, allowing an enhancedcontrol of the problems hydrodynamics.16

Three main simplifications are introduced in our gasliquid interaction model.

sad The gas is incompressible: results can be extrapolatedto the case where the surrounding stream is also liquid,under the stringent condition that it be perfectly insu-lating.

sbd The gas is inviscid, and a uniform flow is assumed sflataxial velocity profile Ud.

scd The surrounding stream is an insulator with vacuumpermittivity: it removes all space charges and is per-fectly unelectrified.

Therefore, the ensuing description will not undertake theanalysis of the velocity boundary layer at the ambient side. Adetailed investigation of the gas velocity profile is performedby Gordillo et al.,27 but their results sthe liquid jet dynamicsis assumed inviscid; no electric effects are consideredd can-not be extrapolated here.

Additional assumptions in our model9 are the following:magnetic fields generated by moving charges can be ne-glected selectrostatic assumptiond; gravitational accelerationis ignored.

Figure 1 shows the geometric outline of the problem.Initially, a potential difference Vo is applied between the jetand a coaxial cylindrical electrode of radius R=bA sur-rounding it. In this geometrical configuration, the axial elec-tric field is negligible compared to the normal electrical field,as is the case in jets emitted by electrospray.28,29 The radialelectric field creates, in the unperturbed state, a surfacecharge density located at the interface, the liquid bulk beingelectrically neutral.

In most of the applications, charged jets issue from aliquid source, where a voltage Vo is applied, and end up atthe breakup region. A transition from conduction to convec-tion, as the relevant charge transport mechanism, takes placein the vicinity of the needle. Charges relax from the bulk to

034106-3 Linear stability analysis Phys. Fluids 17, 034106 ~2005!


the surface in a characteristic time te; this process takingplace in a region of length Lr,Uote much shorter than thetotal jet length Lj. The jet length is generally of the order ofUotc, but in many practical situations, the longevity of the jetcan be extended by avoiding all instability factors. Most ofthe voltage drop along the jet DV is located in the chargerelaxation region, which can be modeled as a resistor of or-der VR,Lr / sKpA2d; therefore, the characteristic relativevoltage drop is

DVVo

,VRIVo

, VRoU ,WebS te

tcD2 ! 1, s1d

where I is the current transported by the jet. Thus, the axialvoltage drop in the relaxation stretch is negligible, and thepotential at the jet surface can be estimated, in first approxi-mation, as the potential applied at the needle Vo. Furtherinsight on the structure of the potential drop along the axialcoordinate of a jet is provided by Gamero-Castao andHruby30 and Higuera.31

The equations of the problem are made dimensionlesswith the radius of the unperturbed jet A, the density r, sur-face tension g, and the characteristic surface charge densityoEo , o being the vacuum permittivity and Eo=Vo / fA lnsbdg the characteristic radial electric field, whileb=R /A. Our choice of Eo rather than Vo as a scaling factorhas the following implication: at the interface, the normal-ized electrostatic pressure is of Os1d. This is unrealistic whenb, because an infinite potential difference Vo is requiredin order to preserve the scaling; however, the results are con-sistent, and can be applied safely for any b@1. The advan-tage in choosing Eo rather than Vo is that the normalizedelectrostatic pressure oEo

2 does not become vanishinglysmall in the b limit. Setiawan and Heister,4 who used theVo scaling, alluded to the shortcomings of this choice: as theground location is moved far from the jet, the effect of elec-trostatic contributions vanishes. Note, however, that Eomust be bounded in our problem; otherwise, dielectric rup-ture would lead to corona discharge effects.

The above scaling leads to a set of nondimensional pa-rameters which characterize each particular jet and perturba-tion.

s1d The half wavelength of the perturbation l=L /A or thedimensionless wavenumber k=p /l.

s2d The radial position of the ground electrode b=R /A.s3d The Weber number We=rAUo

2 /g= sUo /vcd2, where vc= sg /rAd1/2 is the capillary velocity, measuring the rela-tive importance of the jets absolute inertia with respectto the capillary forces. Some authors prefer to use itsinverse, which is referred to as a Euler number.

s4d An external flow Weber number We=rAU2 /g

= sU /vcd2, measuring the relative importance of the am-bient fluids absolute inertia with respect to the capillaryforces. Note that the liquids density sand not the gasd isused. In many applications, the ambient gas is at rest, sothat We=0. An additional parameter of interest in de-scribing the liquid-gas interaction is the ratio betweenthe gas and liquid densities r=rg /r.

s5d The Ohnesorge number C=m / srgAd1/2=ntc /A2 weigh-ing viscous forces against capillary forces. It is the in-verse of a Reynolds number based on the capillary ve-locity; it can also be understood as the ratio between thecapillary time tc and the viscous radial diffusion timetv,A2 /n. The usual or convective Reynolds number,Re=UoA /n, can be expressed as Re=We1/2 /C.

s6d The electric number x=AoEo2 /g, also known as Taylor

number or electrical Bond number, comparing the elec-tric pressure with the capillary pressure. It can be con-ceived as the ratio between the capillary time scale and atime scale obtained by geometric averaging of the vis-cous diffusion time tv and the shear or electrohydrody-namic time scale ts=m / soEo

2d, i.e., x, tc2 / stvtsd

,Ctc / ts. Saville9 introduced a Reynolds number basedon the electrohydrodynamic velocity us=A / ts; it can bewritten as Res=roA2Eo

2 /m2=x /C2.s7d The liquids relative permittivity b=i /o. The ambient

fluid is supposed to have vacuum permittivity.s8d The relaxation parameter a= frA3K2 / sgi

2dg1/2= tc / te. Itcoincides with the ratio of the electrical relaxation to thecapillary time sgenerally, the shortest hydrodynamictime of the processd. The combination ab will be shownlater to describe the relative importance of conductiveversus convective charge transport. The perfect-conductor limit is recovered when ab@1 sgenerallysimplified to a@1 by taking into account customary bvaluesd: this limit implies that relaxation is quicker thandeformation,32 so that the jet remains isopotential atany time, and its internal electric field is zero. In theopposite situation ab!1, charges are glued to the inter-face shoney-bubble limitd, and are therefore passivelyconvected and stretched with it.

Each perturbative situation is therefore characterized by a setof eight free parameters: a , b , C , x , b, We, We, and l,whose influence will be explored next. It is important to notethat our hypothesis about the electric charge being relaxed tothe jet surface does not necessarily imply a@1. Indeed, pro-vided that the jet is long enough, the relaxation process canbe considered to have taken place at an earlier location.Therefore, the influence of a when it is of Os1d or evensmaller can be explored under the assumption that all

FIG. 1. Geometrical sketch of the problem.



charges are superficial. This is a realistic situation, frequentlyobserved in experimental work.

Electrospray can be used to illustrate the above. This isan apt technique for the generation of electrified capillarymicrojets nanojets and sprays. It gives rise to a great varietyof jet configurations and atomization modes; among them,the cone-jet mode is preferred because of its steadiness andspray characteristics.33 Cone-jet operation can be attainedwith a wide range of liquids34 exhibiting enormous diversityin the relevant physical constants sconductivity K, surfacetension s, density r, relative permittivity b, and viscosity md.Conductivity and, to a lesser extent, viscosity coefficients,may exhibit radically different orders of magnitude. Conejets have been obtained with toluene sK=1011 S/md, in-tensely acid water solutions sK=1 S/md, hexane sm=33104 Pa sd or glycerol sm=1.3 Pa sd.

Experimental and theoretical investigations28,31,3537have led to the introduction of the following scaling magni-tudes:

Qo =gorK

, Io = Sog2r

D1/2, do = S o2gp2rK2

D1/3. s2dThe above magnitudes set the scale for the injected flow rateQ, the electric current transported by the jet I, and the jetdiameter A. With their help, a universal ratio is established:I / Io,sQ /Qod1/2 and A /do,sQ /Qod1/2. Provided second-order effects such as the needle geometry are ignored, theabove scaling is accurate in a wide range of situations in-volving diverse liquids and geometries.

On the basis of electrospray scaling, a and C can beestimated in typical liquid jets issuing from a cone-jetsource. Table I shows a and C values drawing on experimen-tal data from Gan-Calvo;36 different organic liquids areused. Some additional data from Gan-Calvo et al.38 wereobtained by electrospraying low-viscosity liquids sheptane,dioxane, as well as dodecanold. As can be gathered from thenew data, Ohnesorge numbers well below unity can be foundby electrospraying nonpolar liquids such as toluene, cyclo-hexane, or heptane, characterized by their low conductivity

and viscosity. Such low Ohnesorge numbers are called toplay an increasing role in multiple technological applicationsinvolving organic liquids.

These results indicate that low a and b are met in real-istic experimental explorations. Electrical polarization, thekey to dielectric energy storage, is the result of a wide vari-ety of processes, including distortion or reorientation of mol-ecules and orbitals as well as electrochemical reactions. Lowvalues of b, implying weak polarizability, are frequently ob-served in connection with organic or long-chain molecules,particularly when water is absent from the composition ofthe liquid. Ionic charge transport is reduced in low-b liquidsbecause of their reluctance to dissolve ions, so that low con-ductivities are frequent in such cases. Accordingly, the rou-tine assumption that jets are perfect conductors cannot bemade without risk. In addition, very low Ohnesorge numbersare frequently observed in low-viscosity jets.

In particular, Lpez-Herrera and Gan-Calvo19 carriedout a set of experiments with precisely the same electric fieldand configuration proposed here. In the above paper, theelectrification of the jet tries to achieve the condition of zero-tangential stress. Experiments leading to breakup were car-ried out and compared with a theoretical model assumingperfect-conductor behavior. Three mixtures of water andglycerine were used in the study. Agreement between experi-ments and theory was in general good, a logical consequenceof the fact that all the jets tested were water solutions; and asmall amount of water is all that is needed for the permittiv-ity to become high sso that ab@1, close to the perfect-conductor limitd. Should the experimental range be expandedto include purely organic liquids, the perfect-conductor hy-pothesis would no longer hold. The need for a stabilitytheory of imperfect conductors follows from such consider-ations.

B. Equations

Under the conditions and assumptions described above,the nondimensional, cylindrical coordinates equations in areference frame moving with the jet velocity Uo are

TABLE I. Estimated values in electrospray jets: diameter A, relaxation parameter a, and Ohnesorge number C;experimental data and scaling law from Gan-Calvo sRef. 36d and Gan-Calvo et al. sRef. 38d.

Liquid dosmmd Qosml/mind b Q /Qo Asmmd a C

Dodecanola 1.67 2.34 6.52 2.36 0.08 1.70

30 9.16 0.62 0.86

1-Octanola 1.52 1.88 102 2.15 0.05 1.25

30 8.33 0.41 0.63

Propylenglycola 1.24 1.54 31.22 1.76 0.02 5.18

30 6.82 0.13 2.63

Dioxaneb 5.82 14.5 2.52 4.62 0.09 0.11

30 20.72 0.86 0.05

Heptaneb 7.43 21.2 1.92 5.89 0.12 0.04

30 26.46 1.13 0.02aReference 36.bReference 38.



srUdrr

+ Wz = 0, s3d

Ut + UUr + WUz = Pr + CsUzz Wzrd , s4d

and

Wt + UWr + WWz = Pz + CSWrr + Wrr

+ WzzD , s5dwhere r and z are the radial and the axial coordinates, t is thetime, U and W are the radial and axial velocity, and P is thepressure. Subscripts denote partial derivatives.

In the chosen frame, neglecting the gas viscosity, we canapproximate the axial speed of the gas as UUo, which inour scaling amounts to We

1/2We1/2. Therefore, the equa-

tions for the gas phase are

srUgdrr

+ sWgdz = 0, s6d

rfsUgdt + sWe1/2

We1/2dsUgdzg = sPgdr, s7d

and

rfsWgdt + sWe1/2

We1/2dsWgdzg = sPgdz. s8d

The hydrodynamic variables are subject to periodicity con-ditions in the axial direction

Fzuz=0 = Fzuz=l = 0, s9d

Wuz=0 = Wuz=l = 0, s10d

Uzuz=0 = Uzuz=l = 0, s11d

where r=Fsz , td is the interface equation. In addition, theregularity conditions at the axis demand

Uur=0 = Wrur=0 = Prur=0 = 0. s12d

The electric problem is modeled by the Laplace equationfor both the inner potential fi and the outer potential fo

2fi = 2fo = 0, s13d

subject to the radial boundary conditionsfour=b = 0 s14d

and

fri ur=0 = 0, s15d

and periodicity in the axial direction

fzisr,zduz=0,l = fz

osr,zduz=0,l = 0 s16d

for both the inner and outer potentials.At the interface r=Fsz , td, the electrical boundary condi-

tions are

sEno

bEni dur=Fsz,td = se s17d

and

four=Fsz,td = fiur=Fsz,td, s18d

where se is the surface charge density and Eno and En

i are thenormal component of the electric field for the outer and innerdomain, respectively. The normal and tangential componentsof the electric field at the interface can be written in terms ofthe potential as

Eno,i

=

1s1 + Fz

2d1/2s fr

o,i + Fzfzo,idur=Fsz,td s19d

and

Et =1

s1 + Fz2d1/2

s Fzfro

fzodur=Fsz,td. s20d

In addition, the kinematic conditions for the liquid andfor the gas, the continuity equation of the surface chargedensity, and the stress balance in the normal and tangentialdirection read

sFt U + FzWdur=Fsz,td = 0, s21d

hUg U + sWe1/2 We1/2dFzjur=Fsz,td = 0, s22d

ssedt +FzU + W1 + Fz

2 ssedz abEni

se

1 + Fz2 hUr + Fz

2Wz FzsUz + Wrdjur=Fsz,td = 0, s23d

sP Pgdur=Fsz,td pce

2C1 + Fz

2 hUr + Fz2Wz FzsWr + Uzdjur=Fsz,td = 0, s24d

and

C1 + Fz

2 h2FzsUr Wzd + s1 Fz2dsWr + Uzdjur=Fsz,td

xseEt = 0. s25d

The capillary-electric pressure term is defined as

pce =1

s1 + Fz2d1/2S 1F Fzz1 + Fz2D

x

2fsEn

od2 bsEni d2 + sb 1dsEtd2g . s26d

Note that the electrification number x weighs the relativeimportance of the capillary pressure jump sdestabilizingfor long wavelengthsd against the normal component of



Maxwells stress, thereby creating the appearance of an al-tered surface tension. It will be shown later that provided x.1 and b.2.718 28, a wavelength range emerges in thevicinity of k=0 where axisymmetric disturbances are stable.

In our linear analysis, initial conditions are only used todefine the unperturbed solution, as will be done in Eq. s27d.

III. LINEAR ANALYSIS: 3D APPROACHA. Perturbed equations and boundary conditions

In this section we describe the linear analysis of theproblem considered. A similar approach has been adopted byMestel,39,40 who analyzed the linear dynamics of charged jetsforced by an axial electric field in the quasi-inviscid andhighly viscous limits. In both works the jet is charged andsubject to an external uniform axial electric field, the electri-cal relaxation time being a free parameter. Gamero-Castaoand Hruby30 studied filaments emitted by cone jets assumingsmall wave axisymmetric disturbances. Their results arebased on the perfect-conductor simplification sEi=0, Et=0d,an approximation implicitly requiring that ab; there-fore, the application of their results to the honey-bubble limitscharge-convection dominant, i.e., a=0d demands specialcaution. Hartman et al.41 studied small wave disturbancessboth axisymmetric and helical modesd in a jet issuing froma nozzle cone with the help of a lowest-order 1D model forthe slice velocity. Their analysis is restricted to the perfect-conductor limit. On the other hand, Gonzlez et al.7 provideda thorough stability analysis of conducting jets under ac ra-dial electric fields. A brief description of the dc case is madedrawing on previous work by Garca.6 In their paper,Gonzlez et al.7 take into account surrounding gas dynamicseffects using a semiempirical model. Azimuthal perturbationmodes snonaxisymmetric fluctuationsd are included in theirdescription, and modal competition is shown to take placeunder the stimulus of the imposed ac frequency. However,their equations are restricted to the perfect-conductor limit.In the present paper, as indicated in the Introduction, onlyaxisymmetric modes will be considered, a hypothesis gener-ally satisfactory when only dc forcing is considered. Mostresearch on the subject agrees in considering nonaxisymmet-ric modes as evanescent, i.e., either damped or displayingslower growth as axisymmetric disturbances ssee for instanceLin and Webb42d. Our study concentrates on the influence ofarbitrary viscosity, permittivity b, and conductivity K in thepresence of a dc radial electric field. The interplay of theseparameters is shown to be a key factor in the emergence ofan interfacial boundary layer. In addition, the effect of elec-trification x and electrode distance b is discussed.

The temporal linear analysis looks out for solutions inthe form of a small sinusoidal spatial perturbation over thestatic or basic solution. We take the perfect cylinder to be ourstatic reference; therefore, solutions as follows are examined:

1Usz,r,tdWsz,r,tdPsz,r,tdPgsz,r,tdfosz,r,tdfisz,r,tdsesz,tdFsz,td

2 =100

1 x/2P

lnsb/rdlnsbd

11

2 +1usr,z,tdwsr,z,tdpsr,z,tdpgsr,z,td

f osr,z,td

f isr,z,tdsesz,td

fsz,td

2 , s27dand the perturbation is written as

1usr,z,tdwsr,z,tdpsr,z,tdpgsr,z,td

f osr,z,td

f isr,z,tdsesz,td

fsz,td

2 = Re31usrdwsrdpsrdpgsrd

f osrd

f isrdse

f2z4 , s28d

where z is equal to eVt+ikz , k being the nondimensionalwavenumber p /l sl is the half wavelengthd, and V=s+viis the complex eigenvalue involving the growth rate s andthe oscillation frequency v. Hatted variables denote smallamplitudes. Neglecting nonlinear terms of the NavierStokesequations we obtain

= v = 0 s29d

and

vt = = p + C2v . s30d

A separate equation for the pressure can be obtained by ap-plying the divergence operator to s30d and using s29d; thesame holds for the gas pressure:

2p = 0, 2pg = 0. s31d

On the other hand, taking the Laplacian of s30d leads to

2S2 1C ]]tDv = 0. s32dDue to the linear character of the operators in the latter equa-tion we can split the velocity into two terms v= vv+ vnv,where vnv is the inviscid contribution and vv the viscouscontribution; the following equations are satisfied:

2vnv = 0, S2 1C ]]tDvv = 0, vnvt = = p . s33dThe perturbations comply with the regularity condition at theaxis psr ,z , tdur=0 and wsr ,z , tdur=0 finite and usr ,z , tdur=0=0.By Taylor expansion around r=1 at the interface, the kine-matic conditions and the stress balance yield

uur=1 = ft, s34d



su ugdur=1 = iks We1/2 + We1/2df , s35d

sp pgdur=1 = f fzz + 2Curur=1 + xsf + f rodur=1, s36dand

Csuz + wrdur=1 + xf zour=1 = 0. s37d

The outer and the inner electrical potential perturbations sat-isfy the Laplace equation

2f o,i = 0 s38d

with regularity conditions,

f isr,z,tdur=0 finite, f osr,z,tdur=b = 0. s39d

At the interface the inner and the outer potential are coupledthrough the normal electric field jump and the continuity ofthe potential

s f ro + bf r

idur=1 = se s40d

and

f our=1 = f iur=1 s41d

together with the continuity equation for the charge surfacedensity

ssedt urur=1 = abf ri ur=1. s42d

Substituting the solution s27d in the bulk equations s31ds33dand s38d we obtain the ODE system which governs the am-plitudes of the variables

C9 +C8

r k2C = 0, s43d

wv9 +wv8

r kv

2wv = 0, s44d

unv9 +unv8

r Sk2 + 1

r2Dunv = 0, s45d

uv9 +uv8

r Skv2 + 1

r2Duv = 0, s46d

where C stands for p , pg , wnv, and f o,i. The primes denotedifferentiation with respect to r and kv is defined as

kv2

= k2 +V

C. s47d

The additive term V /C,V A2 /n, tv / td is occasionally iden-tified as a Reynolds number based on the problems unsteadycharacter, V being the physical magnitude of V;39 it can alsobe viewed as the ratio between the viscous diffusion timescale tv and the disturbance time scale td. In addition, thecontinuity and momentum equations read

u8 +u

r+ ikw = 0, p8 = Vunv, ikp + Vwnv = 0. s48d

The gas momentum equation in the radial direction is

rfV + iksWe1/2

We1/2dgug = pg8. s49d

The boundary conditions closing the problem are

us0d = 0, ws0d, ps0d and f is0d finite, pgsd = 0; s50d

us1d = Vf, ugs1d = us1d + iksWe1/2 We1/2df , s51d

ps1d pgs1d = s 1 + k2df + 2Cu8s1d + xff + sf od8s1dg ,s52d

Cfikus1d + w8s1dg + xikf is1d = 0, s53d

f + f os1d = f is1d , s54d

f sf od8s1d + bsf id8s1d = se, s55d

Vse u8s1d = absf id8s1d , s56d

f osbd = 0. s57d

B. The dispersion equation

The general solution of the ODE set s43ds48d consistsof a combination of modified Bessel functions. Recallings50d and s57d, the solution reads

psrd = AVIoskrdkI1skd

, pgsrd = AgKoskrdkK1skd

, s58d

usrd = unvskrd + uvskrd = AI1skrdI1skd

B I1skvrdI1skvd

, s59d

and

wsrd = wnvskrd + wvskrd = AiIoskrdI1skd

Bi kvIoskvrdkI1skvd

s60d

for the hydrodynamic variables, and

f osrd =AefKoskbdIoskrd IoskbdKoskrdg

kfKoskbdI1skd + IoskbdK1skdgs61d

and

f isrd = BeIoskrdkI1skd

s62d

for the outer and inner electrical potential, whereA , Ag , B , Ae, and Be are real constants. Note that, underadequate scaling, the radial profiles of the pressure and theelectric potential are identical. The substitution of the previ-ous solutions in the boundary conditions at the interfaces50ds57d and the elimination of the constantsA , Ag , B , Ae, and Be leads to the dispersion relation



D = V2jskd + 2CVf2k2jskd 1g + 4k2C2fk2jskd

kv2jskvdg +

2JskdCV

k2f2 + Gskdgfk2jskd kv2jskvdg

+ JskdS2k2jskd + k2jskdkv2jskvdGskd + 1GskdD+ S1 + Jskdk2Gskd

V2fkv

2jskvd k2jskdgDTskd = 0, s63dwhere the driving term Tskd sums up all the interfacial forc-ing ssurface tension, aerodynamic effects, electric chargingdand is defined as

Tskd = k2 1 + xS1 + 1GskdD rfV + iksWe

1/2 We1/2dg2Gskd , s64d

while the following auxiliary functions are introduced:

Gskd =KoskbdIoskd IoskbdKoskd

kfKoskbdI1skd + IoskbdK1skdg, jskd =

IoskdkI1skd

,

s65d

Jskd =xjskdEskd

, Eskd = GskdbS1 + aVD jskd , s66d

and

Gskd = KoskdkK1skd

. s67d

Jskd is a combined electrification number, barely sensitive tob, condensing all the influence of a and b; its magnitudeincrease as the conductivity or the permittivity decrease.Jskd becomes zero in the perfect-conductor limit, when ei-ther a , b, or their combination tend to . Eskd is alwaysnegative. In the limit where b@1, GskdGskd. The leadingterm in D is V2jskd, representing the effect of inertia, i.e.,forces caused by the unsteadiness of the disturbance. In low-viscosity jets sC!1d, this term becomes dominant and isbalanced by the driving term alone. In the opposite limitsC@1d, the driving term is balanced by the viscous termsalone ssecond and third terms in Dd. The calculations leadingto the dispersion relation are detailed in the Appendix.

It can be readily verified that ab implies that Be0, so that the inner electric field becomes vanishinglysmall. Assuming the disturbances are small, it follows aswell that the tangential field Et becomes negligible. There-fore, the perfect-conductor case can be recovered by select-ing ab@1.

Our description of the influence of the surrounding gasmirrors the procedure introduced by Gonzlez et al.;7 in theirpaper they suggest a simple semiempirical correction, basedon Sterling and Sleicher,43 allowing the description of the gasviscosity influence. Additionally, concerning the validityrange of the theory, the work of Lin and Reitz25 is cited tojustify the parametric range where the above applies: We.4, rWe,6.5. These limits are empirically consistent forunelectrified jets only, but can be used as orientative land-

marks in our case. The first restriction is aimed at avoidingabsolute instability sdripping moded; it sets a criterion forRayleigh breakup. The second restriction excludes situationsbelonging to the second wind-induced breakup regime,where gas shear becomes important. However, the range ofvalidity is not well studied in the case where the outer fluid isflowing co-axially with a speed U. In spite of the resultinguncertainty, the dispersion equation s63d is to be trustedwhenever dripping or second wind-induced breakup can beexcluded.

Figure 2 shows a characteristic stability map; the growthrate s and the oscillation frequency v are plotted against thewavenumber k. The jet is characterized by an Ohnesorgenumber C=0.1, Taylor number x=0.6, relative permittivityb=10, relaxation parameter a=1.0, and b=. Aerodynamiceffects are ignored. A single unstable branch is observed,corresponding to values of k ranging from 0 to 1.167 03,associated with aperiodic growth of perturbations. The otherbranches are stable, some of them being oscillatory stable sI,II, and IIId. There are four complex solutions associated toeach wavenumber. In Fig. 3 the velocity fields and externalelectric fields svectoriald are depicted for each one of thesolutions; k=0.5, a regime where all behaviors are aperiodicsV is a real numberd. We observe that sid the electric field isalways symmetrical with respect to the wavelength; siid theonly unstable mode is essentially capillary sAd while the oth-ers are stable. The main stabilizing force in mode B is thesurface tension. Modes C and D are characterized by strongrecirculations of the liquid where the viscous term balancesthe inertial term, the capillary term being negligible.

An even number of roots are observed. The problemdealt with by Hohman et al.44 displays a distinctive differ-ence compared to our problem: the presence of an axial,rather than radial, electric field. The dispersion relation ob-tained by them is cubic; accordingly, they find three modes;on the other hand, the radial geometry of our electric forcingleads to a fourth-order dispersion relation sfour modesd.

A salient physical difference between the two problemsis that the first one displays antisymmetric electric charges in

FIG. 2. Solution of the dispersion relation sa=1, b=10, C=0.1, and x=0.6d.



the axial direction along a perturbation wavelength, whileours is symmetric. Their having an odd-order dispersion re-lation rather than an even-order one may plausibly be a re-flection of the antisymmetry inherent to axial forcing. In ad-dition, as illustrated in Fig. 2, the radial electric field inducesan additional stable mode sthree out of fourd while the physi-cal picture drawn by Hohman et al. in the presence of atangential field clearly hints at the destabilizing role of axialelectric forcing. A simplistic but illustrative picture is thefollowing: given a symmetric scosine-liked perturbation, aradial electric field induces additional dynamical actionswhich are symmetrical in the axial direction with respect tothe perturbation sand these may be stabilizing or destabiliz-ingd, while the axial electric field gives rise to antisymmetricforces which are always destabilizing: see Figs. 9 and 11 inHohman et al.44

Aerodynamic effects modify the above map in that non-zero r, We, or We give rise to an imaginary component ofthe dispersion equation D. This has an immediate implica-tion: it is impossible for a real V to satisfy the dispersionequation. Therefore, all growth behaviors are modulated,lmsVd;v0, i.e., aperiodic growth sas shown in Fig. 2 forthe unstable branch Ad cannot exist.

C. Parametrical patterns of the dispersion equation

In the uncharged limit sx=0 in Dd, the dispersion equa-tion coincides with the expression obtained byChandrasekhar45 after Rayleigh.46 Setting the additional con-straint C=0 leads to the original result for the capillary in-stability due to Rayleigh. Setiawan and Heister4 obtained anexpression for D in the electrified inviscid limit, assumingperfect conductivity and b=. Huebner and Chu3 developeda variational approach leading to an expression for D in theperfect-conductor limit, assuming zero viscosity and arbi-trary electrode distance b. Artana et al.32 included aerody-namic effects and higher stability modes using the same as-sumptions.

In this section, different growth rate curves are shown. In

growth curves, s=ResVd is plotted as a function of thewavenumber k in the instability window corresponding to theunstable branch A of Fig. 2. All dispersion curves exhibitmaximal growth rate for a particular value of the wavenum-ber, as well as upper and lower cutoff wavenumbers sk1 , k2d,which define our instability window. The fastest growingmode skm , smd is the most relevant quantity in the case ofrandom perturbation snoised. Artana et al.,32 among others,indicate that the mean diameter of the first droplets producedat the jets breakup is proportional to the inverse of km whilethe intact length of the jet sbefore detachment of the firstdropletd is proportional to the inverse of sm.

Provided that the speed difference UoU is sufficientlylarge, aerodynamic effects can no longer be neglected. Ourexploration shows that as rsWe

1/2We1/2d increases, the fol-

lowing trend is observed: the instability lobe expands, whileskm , smd increase, implying smaller droplet size at breakup.Effects are hardly noticeable when rsWe

1/2We1/2d,0.2.

The imaginary part v as a function of k is monotonouslygrowing.

FIG. 3. Jet velocity and interfacial outer field in awavelength segment for: sAd Capillary unstable mode;sBd capillary stable mode; sC and Dd recirculatingmodes sb@1, k=0.5, a=1, b=10, C=0.1, and x=0.6d.

FIG. 4. Growth rate s vs the wavenumber k for a jet with b@1, x=0.6,C=1, and b=2 and several values of the relaxation parameter a.



In the ensuing discussion, the effects of the ambient gaswill be considered negligible. In the gas-at-rest case, thisamounts to the constraint rWe,0.2, marking the onset ofthe first wind-induced breakup regime.

The influence of the conductivity and permittivity pa-rameters is explored in Figs. 4 and 5, where the growth rateis shown for a jet of moderate viscosity and electrificationsb=, C=1, and x=0.6d; Fig. 4 is plotted with b=2; in Fig.5, a=1. It is worth noting that the b influence can only beexplored provided a is finite. A similar trend is observed fora and b. Growing conductivity or permittivity results in adecrease of km and an increase of sm, implying largerbreakup wavelength and increased instability. Provided thatany of these parameters is sufficiently high, the perfect-conductor limit is reached.

These results illustrate the influence of a and b on thegrowth rate, and show a clear stabilizing trend associatedwith imperfect conductivity or permittivity. The stabilizationis more evident as the wavelength rises. Particular sensitivityto a and b is felt in the long-wavelength range. Imperfect

conductivity and permittivity ssmall enough a or bd givesrise to a slow-growth zone near k=0. It is worth noting thatpoor conductivity becomes irrelevant as the outer electrodegets closer to the jet slower values of b tend to annihilate theinfluence of b and a, bringing results close to the perfect-conductor limitd. In general, the conducting limit is rapidlyreached by increasing the permittivity b or the relaxationparameter a. In the practice, both parameters tend to be cor-related: high permittivity liquids usually exhibit a high elec-trical conductivity as well, on account of their polarity andtheir ability to dissolve ionic species. Both parameters haveno influence whatsoever on the stability range, which, as willbe shown next, is only dependent on x and b.

Figures 6 and 7 show growth rate curves for differentvalues of the electrification number x and the electrode dis-tance b. Coincident patterns of influence are observed: in-creasing the electric field x exerts a similar effect as bringingthe outer electrode closer. The electrification effect is there-fore enhanced by an encroaching electrode. Both parametershave in common a destabilizing influence, reflected in wid-ening instability lobes.

The stability range is very sensitive to electrification. Asx increases, the instability lobe expands ssee, for instance,Baudry et al.5d. Taking the V0 limit in the dispersionequation, under the assumption that k0, leads to an equa-tion for the limits sk1,k,k2d where a positive real growthrate is found:

Tsk1d = Tsk2d = 0, s68d

Tskd being the driving term defined in Eq. s64d. The upperand lower bound are plotted in Fig. 8 for different values ofx. In the unelectrified case, the Rayleigh instability limitssk1=0 and k2=1d are recovered: the capillary terms k21 are,respectively, associated with the transversal curvature sdesta-bilizingd and the longitudinal curvature sstabilizingd. Longwavelengths increase the relative importance of the first andthereby cause instability.

Electrification has an ambivalent effect on stability. Forlarge b, as discussed by Gonzlez et al.7 swho also provide a

FIG. 5. Growth rate s vs the wavenumber k for a jet with b@1, x=0.6,C=1, and a=1 and several values of the relative permittivity b.

FIG. 6. Growth rate s vs the wavenumber k for a jetwith b@1 and b=2. The other parameters are fixed atC=1, a=1, and b=2. Several values of the electrifica-tion number x are plotted.



physical interpretation of the dual influence of x at the crestsand valleys of the disturbanced, electric charging of the jetdestabilizes short waves while stabilizing long ones. In fact,provided that x is larger than 1 and under the condition that

b . G1US xx 1DUk=0 = expS xx 1D , s69d

the lower bound k1 rises above k=0, so that a stable region inthe range 0,k,k1 emerges ssee Fig. 6, b@1; and Fig. 8,where the lower and upper cutoff bounds are plotted as afunction of b for diverse values of xd. Nevertheless, k1 isbounded, and even in the most favorable case sb and xd,k1=G1s1d=0.595. This boundary, in the large-b limit,separates the range where electrostatic forces are stabilizing,as reported by Setiawan and Heister.4

The above observations are in agreement with recent ex-perimental and numerical studies on perfect-conductor jets17indicating that, in the range of electrification levels and We-ber numbers ensuring an axisymmetric breakup, the influ-ence of x is merely quantitative. The breakup wavelengthand frequency are sensitive to the electrification level, owingto a change in the effective surface tension operated by theelectric field. The effects of electrification sand aerodynamiccouplingd are summed in the driving term Tskd, the only partof the dispersion equation dependent on x , b , U, and Uo.However, if the liquid is an imperfect conductor, a tangentialelectric stress arises which attenuates the growth of pertur-bations, and may promote longer jets as observed in electro-spray experiments using moderate viscosity liquids in thenonwhipping jet regime.12,13,39,40

For arbitrary ground location b, a threshold value can bedefined, ks=G1s1dub, cutting off the region where electrifi-cation is stabilizing sk,ksd from the region where a desta-bilizing influence is observed sk.ksd. It is interesting to notethat b values lower than the e number sb,2.718 28d implythat, whatever the electrification, its effect is always destabi-lizing, regardless of k. Therefore, in the tight-electrode limitsb1d, increasing x gives rise to an unanimous increase ofthe growth rate.

On the other hand, the upper cutoff mode k2, also de-picted in Fig. 8, is very sensitive to electric charging, par-ticularly when the electrode gets closer to the jet. Indeed, inthe tight-electrode limit, the instability range grows asymp-totically, so that the entire wavelength range ends up beingunstable.

The long-wavelength limit can be explored in the gen-eral case by taking the limit of the dispersion equation Dwhen k0. The positive real root of D is

s < B1sb,xdk, B1sb,xd =12 fxs3 4 ln bd + 1g s70d

indicating a local linear behavior of the growth rate. Thisapproximation is only correct provided that

FIG. 7. Growth rate s vs the wavenumber k for a jetwith a=1, b=1, C=1, x=0.3, and several values of thegrounded electrode distance b.

FIG. 8. Lower and upper stability cutoff bounds k1 and k2 plotted as afunction of the ground radial distance b for different values of theelectrification.



b , expF14S3 + 1xDG = 2.117e1/4x = bosxd . s71dWhen b.bosxd, the growth rate follows a quadratic trend:

s < B2sb,xdk2,

B2sb,xd =sabC + x/2dfsx 1dln b xg

2Cs1 + 3x 4x ln bd. s72d

A low growth region is observed near k=0; the width of thisregion increases with decreasing a ,b. In order to carry outconsistent expansions of the general solution in the k!1limit, it is worth noting that kv

2=k2+V /C is of the following

order of magnitude:

kv = HOsk1/2d when b , bosxd ,Oskd when b . bosxd .J s73dThe above does not hold when C0, because the emer-

gence of an interface boundary layer modifies the small-kpatterns.

Equations s70d and s72d confirm the existence of a low-k stability region f0, k1g as illustrated by Figs. 6 and 8. In-deed, for sufficiently large values of b and x, the local valuesof s switch from positive to negative.

D. The influence of viscosity and the boundary layerat the interface in the low-viscosity limit1. High-viscosity limit

The influence of the viscosity can be illustrated by com-paring growth curves having diverse Ohnesorge numbersssee Fig. 9d. As viscosity increases, the most unstable wave-length becomes longer and the growth rate decreases, corre-sponding to an increasingly uniform and stiff flow regimeand therefore increasingly limited deformability.47 Viscositydoes not modify the stability limits, but it lowers the growthrate of any perturbation. High Ohnesorge numbers are also

linked to the addition of surfactants, all of which possess thecommon property of lowering surface tension when added toliquids in small amounts.48

In the high-viscosity limit sC@V, i.e., tv! td, where tv isthe viscous diffusion time scale and td is the disturbance timescaled, the dispersion equation yields the following simpli-fied expression for the positive real root s=ResVd:

s

Linear Stability of Axisymmetric Perturbations in Imperfectly Conducting Liquid Jets

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