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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
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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
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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.
034106-4 Lpez-Herrera, Riesco-Chueca, and Gan-Calvo Phys. Fluids
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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.
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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
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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
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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
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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.
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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.
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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.
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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.
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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