Dynamics o EHD Lamina Plumes:
scaling analysis and in eg al model
P.A. V´azquez1,2,A.T.P´e ez2, A. Cas ellanos2and P. A en3
1Depa amen o de F´ısica Aplicada, E.S.I., A da. de los Descub imien os, s/n, 41092 Se illa, Spain, e-mail:p [email protected],
00-34-5-4486192, 00-34-5-4486003 (Fax)
2Depa amen o de Elec ´onica y Elec omagne ismo, Facul ad de F´
isica, A da. Reina Me cedes s/n, 41012, Se illa, Spain
3Labo a oi e d’Elec os a ique e Ma e iaux di´elec iques (C.N.R.S), G enoble, F ance
In his pape elec ohyd odynamic (EHD) plumes a e examined in he egion a om he injec -
ing elec ode and he collec o pla e, o bo h wo-dimensional and axisymme ic geome ies. The
ela i e impo ance o he conduc ion mechanisms (con ec ion, d i and diffusion o elec ic cha ge)
is analyzed. Diffusion u ns ou o be negligible compa ed o con ec ion and d i o he expe i-
men al condi ions. Bu he ans e se d i (Coulomb epulsion) is o he same o de o magni ude
han con ec ion. We find a se o h ee diffe en ial equa ions gi ing he e olu ion o he eloci y a
he cen e o he plume and he wid hs o he plume and he cha ged co e inside.
I. INTRODUCTION
Injec ion o cha ge om an elec ode in o an insula ing fluid defines a class o EHD flows. The Coulomb o ce,
=qE, ha he elec ic field, E, exe s on he space cha ge, q, induces a fluid mo ion which con ibu es o con ec
he cha ge. The e is a d as ic diffe ence be ween gases and liquids. In gases he ion d i eloci y is much highe han
he gas flow induced by he Coulomb o ce, so he ions p ac ically mo e along he elec ic field lines. The con e se
occu s in liquids whe e he ypical fluid eloci y is highe o much highe han he ion d i eloci y1. The flow
induced by he Coulomb o ce d as ically affec s he cha ge densi y dis ibu ion which in u n modifies he eloci y.
As a consequence he e is a s ong coupling be ween he cha ge densi y dis ibu ion and he eloci y field.
We conside he e he case o elec ode geome ies whe e he injec ion o cha ge a ises om sha p elec odes. In
hese cases he con as be ween gases and liquids is e en mo e ma ked. In liquids he localized injec ion induces a
flow s uc u e simila o hose o he mal plumes wi h a endency o confine he cha ge. Se e al geome ies can be
conside ed, bu in his pape we ocus ou a en ion on he wo-dimensional and axisymme ic cases. The o me
a ises when he sha p elec ode is a blade o a wi e, while he la e appea s when he injec ing elec ode is a poin
o a needle. The esul ing flow in he egion a om he elec odes (we will call his zone he asymp o ic egion) is
simila o ha o he mal plumes a ising om line o poin sou ces o hea . He e only he case o lamina flow in
liquids is examined.
The mal plumes ha e been ho oughly s udied2,3. The flow in he asymp o ic egion, o P and l numbe s, P 1,
(P =ν/κ is he P and l numbe , ν he kinema ic iscosi y o he fluid and κ he he mal diffusi i y) has a double
bounda y laye s uc u e, wi h an inne he mal laye , o wid h δT(x), and an ou e hyd odynamic laye , o wid h
δl(x), ep esen ing he olume o fluid pu in o mo ion (xis he coo dina e along he flow main di ec ion). These
he mal plumes can be desc ibed in e ms o sel -simila solu ions. The physical quan i ies o he plume scale wi h
x, and he pa ial diffe en ial equa ions can be con e ed in o a se o o dina y diffe en ial equa ions in a sel -simila
a iable, η=y/δl(x), wi h y he ans e se coo dina e.
Se e al a emp s ha e been made o apply his echnique o he EHD p oblem. The fi s s udy was ca ied ou by
Zhakin4. In his pape , he au ho neglec s he con ibu ion o Coulomb epulsion in he e olu ion o he cha ged zone,
e aining he diffusi e e m. In his way, he p oblem u ns ou o be exac ly analogous o he he mal one, wi h he
cha ge densi y qplaying he ole o empe a u e. Sel -simila solu ions o bo h wo-dimensional and axisymme ic
EHD plumes we e gi en. P´e ez e al5showed ha Coulomb epulsion is dominan compa ed o diffusion, a leas
in he usual expe imen al condi ions. V´azquez e al6analyzed bo h he mal and EHD plumes emphasazing hei
analogies and diffe ences. They showed ha he equa ions a e simila o hose o he he mal plumes in he limi
P →∞, when diffusion o cha ge, d i and Coulomb epulsion a e neglec ed compa ed o cha ge con ec ion. F om
a physical poin o iew, he wid h o he inne cha ged laye ( he hea ed laye o he he mal plumes) is ze o in his
limi . Ob iously, his is no a comple e solu ion o he p oblem, o in ha case bo h Coulomb epulsion and diffusion
o cha ge would sp ead he cha ged egion. E en mo e, o he wo-dimensional geome y he eloci y o he fluid in
he plane o symme y ends o a fini e alue o P →∞
7,8, bu o he axisymme ic case he eloci y in he axis
di e ges as √ln P 6. Wha eally happens in p ac ical si ua ions is ha he plume en e s he simila i y zone wi h
a cha ged egion o fini e wid h, δq, due o he fini e adius o he injec ing elec ode and o he effec o Coulomb
epulsion in i s icini y9. To a oid his singula i y, an effec i e P and l numbe was defined, P e =(δl/δq)2, aking
1
in o accoun ha he P and l numbe gi es he ela ion be ween he wid hs o he he mal and hyd odynamic laye ,
P =(δl/δT)2.
Takashima e al.10 ha e done a nume ical simula ion o bo h wo-dimensional and axisymme ic EHD plumes.
They used fini e diffe ences o calcula e he eloci y field and he cha ge simula ion me hod o de e mine he elec ic
field. Wi h he aim o educing he compu ing ime, hey imposed a cons an wid h o he cha ged laye , a a he
d as ic assump ion no jus ified in he pape . In his way hey ob ained nume ical alues o he eloci y and a
ela ion be ween he po en ial diffe ence and he cu en ha hey claim ag ee wi h hei expe imen s. Howe e , he
unde lying physical mechanisms a e no clea in his nume ical simula ion, and he applica ion o he calcula ions
o o he se o expe imen al da a is no s aigh o wa d. An exac nume ical solu ion o he whole p oblem is an
ex emely difficul ask, because o he diffe en leng h scales in ol ed and he coupling be ween he elec ic field, he
cha ge dis ibu ion and he eloci y field.
In eg al me hods can gi e aluable insigh s in his difficul p oblem. McCluskey and P´e ez11 used hese me hods
o he fi s ime. Imposing se e al hypo heses ob ained om he expe imen al obse a ion o EHD plumes, and
neglec ing diffusion o cha ge and Coulomb epulsion, hey ob ained exp essions o he eloci y in he symme y
plane o he plume and he wid h o he hyd odynamic laye in he asymp o ic egion. I mus be unde lined ha
hese exp essions a e simila o hose ob ained by Zhakin. An o de o magni ude analysis has been pe o med by
Mal aison e al.12 o axisymme ic and u bulen plumes ollowing he same app oach.
In he wo-dimensional case, and o weak cu en s, he effec o space cha ge is negligible in compa ison wi h
con ec ion unde ypical expe imen al condi ions. In he axisymme ic case his is no ue. A en e al.13 conside ed
he p oblem o EHD axisymme ic plumes including he effec o Coulomb epulsion wi h he use o in eg al me hods.
Making se e al hypo heses abou he eloci y p ofile, hey ob ained a se o diffe en ial equa ions o he eloci y on
he axis and he wid hs o he cha ged and hyd odynamic laye s. Howe e , hei analysis is no comple e and he
esul s depend on an ex a assump ion conce ning he a io o he cha ged and hyd odynamic laye s.
In his pape , we ocus ou a en ion in he beha io o lamina EHD plumes, o bo h 2D and axisymme ic
geome ies. We discuss in de ail he ela i e impo ance o he h ee mechanisms o anspo o elec ic cha ge in
plumes: diffusion, d i (including Coulomb epulsion) and con ec ion. I is shown ha in he cha ged co e and
along he ans e se coo dina e, Coulomb epulsion is o he same o de o magni ude han cha ge con ec ion and
canno be neglec ed. The inclusion o Coulomb epulsion in he equa ions p e en s hem om ha ing sel -simila
solu ions. We ex end he in eg al me hods, used by A en e al., including now he global conse a ion o ene gy. In
his way, a se o diffe en ial equa ions is ound o he eloci y a he cen e o he plumes and he wid hs o he
cha ged and hyd odynamic laye s. In hese equa ions he explici dependence o he longi udinal elec ic field wi h
xappea s. The e o e, we ha e o p o ide a unc ion E(x), along wi h he co esponding ini ial condi ions. We ha e
sol ed nume ically he ob ained equa ions o an imposed elec ic field and diffe en se s o ini ial condi ions. Simila
alues a e ob ained o he magni udes in ol ed in all cases.
II. FORMULATION OF THE PROBLEM
A. Hyd odynamic equa ions
We conside he s eady elec ohyd odynamic flow occu ing be ween a blade o a needle and a pla e a dis ance d
apa . In he fi s case, he s uc u e o he flow is 2D, and i is axisymme ic in he second case. The flow akes
he o m o a plume, o igina ed a he poin o he blade, ha impinges upon he pla e. The eloci y in he zone o
eci cula ion is much smalle han in he plume. Fo his eason one can analyze he mo ion conside ing a plume-like
flow in a non-mo ing ambien liquid. This o e all s uc u e is ske ched in figu e 1.
We examine he p oblem in he asymp o ic egion, ha is, a om bo h he injec o and he pla e. In his
asymp o ic egion he flow has a double bounda y-laye s uc u e, wi h an inne cha ged co e o ypical scale δq,and
an ou e hyd odynamic laye , o ypical scale δl. This laye ep esen s he olume o liquid pu in o mo ion by he
elec ic o ces. We ha e δq,δ
ld. The usual app oxima ions in bounda y-laye analysis a e applicable he e. Fo
he Na ie -S oke equa ion he ans e se de i a i es a e much highe han longi udinal de i a i es, and he ambien
p essu e, p0(x), is imp essed in he bounda y laye (in ou case p0is independen o x, as a consequence o neglec ing
eci cula ion, and he e o e he p essu e g adien is negligible). As a consequence he mechanical equa ions educe o
∂
∂x yku+∂
∂y yk =0,(1a)
u∂u
∂x + ∂u
∂y =ν
yk
∂
∂y yk∂u
∂y+qEx
ρ,(1b)
2
wi h k= 0 o he 2D case and k= 1 o he axisymme ic one. He e, xand ya e he longi udinal and ans e se
coo dina es (in he case o axisymme ic plumes yis he adial coo dina e), uand a e he longi udinal and ans e se
componen s o he liquid eloci y, qis he densi y o cha ge, Exis he longi udinal elec ic field, and ρand νa e he
densi y and he kinema ic iscosi y o he liquid, espec i ely. As usual in bounda y laye analysis, he conse a ion
equa ion o he ans e se componen o momen um can be igno ed14.
Following he usual analysis o he mal plumes2, we may de i e es ima es o he longi udinal eloci y, U,andδl
by s a ing ha he ine ial, iscous and o ce e ms a e o he same o de
U∼qExx
ρ1/2
,δl
x∼qExx3
ρν2−1/4
.(2)
In analogy wi h he G asho numbe defined o he mal plumes, we can define he elec ic G asho numbe , G el =
(Ux/ν)2=(qExx3/ρν2), which ep esen s he a io o elec ic and iscous o ces. These exp ession a e o o mal
a he han p ac ical in e es because qis no known a p io i. In he ollowing we u n owa d mo e sa is ac o y
exp essions.
B. Elec ic equa ions
Two equa ions desc ibe he elec ic aspec s o he p oblem: he Poisson equa ion and he conse a ion o cha ge,
in s eady condi ions,
∇·E=q
,(3a)
∇·j=0.(3b)
He e is he pe mi i i y and j he cu en densi y. Dielec ic liquids o e y poo conduc i i y a e no ohmic. Space
cha ge may appea in hese liquids unde ce ain es ic ions1. Essen ially, he space cha ge is obse able when he
ansi ime o ions be ween elec odes d2/KV (Kbeing he ionic mobili y and V he applied ol age) is smalle han
he space cha ge elaxa ion ime /σ (σbeing he conduc i i y). Unde his condi ion, he densi y cu en has h ee
componen s: he d i o ions wi h espec o he liquid, KqE, he cha ge con ec ion, quand he cha ge diffusion
−D∇q.Thus,j=q(KE+u)−D∇q, and eq. (3b) w i es, in 2D geome y,
KEx
∂q
∂x +KEy
∂q
∂y +Kq2
+u∂q
∂x + ∂q
∂y −
−D∂2q
∂x2−D∂2q
∂y2=0.(4)
The fi s e m is ela ed o he longi udinal elec ic d i , he second and hi d e ms a e due o he field c ea ed by
he space cha ge ( he Coulomb epulsion), he ou h and fi h e ms a e ela ed o he con ec ion o elec ic cha ge
due o he liquid mo ion and he las wo e ms a e due o cha ge diffusion.
C. Basic scales
F om he s uc u e o EHD plumes a mo e sa is ac o y se o es ima es can be ob ained. I he elec ic cha ge
emains confined in o he plume he cu en is (cu en pe uni leng h o 2D plumes)
J≈¯q2δq(U+K¯
Ex) wo-dimensional case,
I≈¯qπδ2
q(U+K¯
Ex) axisymme ic case, (5)
wi h ¯q,Uand ¯
Ex∼V/d he ypical scales o he densi y o cha ge, he axial eloci y and he axial elec ic field.
F om now on we assume δq o be smalle enough han δl o he p e ious exp essions o be alid. In mos EHD flows,
including plumes, he ion d i eloci y is negligible compa ed o he liquid eloci y15,K¯
ExU.Now,in eg a ing
(1b) ac oss an x-cons an plane, and s a ing ha he ine ial, iscous and elec ic e ms a e o he same o de we
ob ain
3
U∼(J2¯
E2
xx/ρ2ν)1/5
δl∼(ρν3x2/J ¯
Ex)1/5 wo-dimensional case, (6a)
U∼(I¯
Ex/ρν)1/2
δl∼(ρν3x2/I ¯
Ex)1/4axisymme ic case,(6b)
Fo 2D plumes he ypical expe imen al alues5(J=10
−8−10−7Am
−1,¯
Ex=10
6Vm
−1,ρ=10
3Kg m−3,ν=
2×10−5m2s−1,x∼10−2m) gi e u∼3−9cm/s,δ
l∼1.5−2.4 mm. Fo axisymme ic plumes12 (I=10
−8)wege
u∼1m/s,δ
l∼300 −530 μm. In he ollowing, we will use hese alues o he es ima es.
D. Cha ge anspo mechanisms
We now discuss he ela i e impo ance, in bo h geome ies, o he h ee mechanisms o anspo o cha ge in
equa ion(4), ha is, diffusion, elec ic d i and con ec ion by he liquid.
1. Diffusion s. d i
Longi udinal and ans e se diffusion mus be conside ed sepa a ely, as he leng h scales a e qui e diffe en . In he
asymp o ic egion he longi udinal scale o change o he densi y o cha ge is d, and he longi udinal elec ic field scale
is Ex∼V/d. Compa ing he e ms o (4) co esponding o diffusion and d i in he xdi ec ion we ha e
D∂2q/∂x2
KEx∂q/∂x ∼D
KV =κBT
eV ,(7)
wi h κB he Bol zmann cons an , T he absolu e empe a u e and e he elec ic cha ge o he ions. We ha e used
he Eins ein ela ion, D/K =kBT/e, o de i e (7). In usual expe imen al condi ions, (T= 300 K,V=10kV,e he
elec on cha ge) he alue o (7) is 2.5×10−61. This shows ha longi udinal diffusion is ully negligible compa ed
o d i .
The ans e se leng h scale o he cha ge densi y change is δq, he ypical wid h o he cha ged co e. The ans e se
elec ic field mainly a ises om he spa ial cha ge densi y. In o de o es ima e i , we model his cha ge densi y as an
infini e laye o wid h 2δqand he ypical uni o m densi y o cha ge ¯q, in he 2D case, and as an infini e cylinde o
adius δqin he axisymme ic case. Using he Gauss law we ob ain Ey∼¯qδq/. F om (5) he ans e se elec ic field
is
Ey∼J/2U wo-dimensional,
I/πUδqaxisymme ic.(8)
Compa ing he ans e se d i and diffusion in (4) we ha e
D∂2¯q/∂y2
KEy∂¯q/∂y ∼2UkBT/eJδq wo-dimensional,
πUkBT/eI axisymme ic.(9)
F om he abo e es ima es
D∂¯q/∂y
¯qKEy∼10−6/δq wo-dimensional,
3×10−5axisymme ic.(10)
The conclusion is ha diffusion effec s a e negligible compa ed o d i i δqis g ea e han a ew μmin he2Dcase,
and o any alue o δqin he axisymme ic case. Hence, diffusion will be neglec ed in he ollowing de i a ion.
Rema k ha , i he e would be o he physical si ua ions o which, in he 2D case, cha ge diffusion would be
dominan o e d i he p oblem would be comple ely analogous o he he mal case. This si ua ion was conside ed
by Zhakin4. The elec ic Schmid numbe , Scel =ν/D, hen plays he ole o he P numbe . Since Scel =ν/D ∼
105−106, we could apply he sel -simila solu ions ound elsewhe e5,6.
4
2. D i s. con ec ion
Conside now he ela i e impo ance o d i and con ec ion e ms in (4) a he bounda y o he cha ged egion.
Taking y∼δqas ans e se leng h scale, om he con inui y eq. (1a) he scale o ans e se eloci y is ∼(δq/x)U.
Fo he elec ic field we deduced in II D 1, Ex∼V/d,Ey∼¯qδq/. Thus, he o de o magni ude o he diffe en e ms
is K¯qV/xd o he longi udinal d i , K¯q2/ o he Coulomb epulsion and U¯q/x o he con ec ion.
Compa ing he longi udinal e ms due o d i and con ec ion we ge
KEx(∂q/∂x)
u(∂q/∂x)∼KV/d
U1,(11)
in he usual expe imen al condi ions. So longi udinal d i can be ully neglec ed.
Fo he ans e se componen s we ha e
KEy(∂q/∂y)
u(∂q/∂x)∼Kx¯q
U .(12)
An es ima ion o ¯qis needed. The densi y o cha ge can be exp essed, in he absence o diffusion, in e ms o he
ime du ing which Coulomb epulsion ac s, i.e. he ime equi ed o he ions o go om he injec o o he poin o
coo dina es (x, y). Neglec ing cha ge diffusion, he cha ge conse a ion equa ion in s eady condi ions can be w i en1:
(KE+u)·∇q=−Kq2
⇒dq
d =−Kq2
,(13)
He e d/d is a o al de i a i e associa ed o he esul ing eloci y (KE+u) o cha ge ca ie s. Eq. (13) is eadily
in eg a ed o ob ain
q( )= qi
1+ /τ
.(14)
He e, qiis he densi y o cha ge a he injec o , τ =/Kqi he ypical e olu ion ime. I xδq,weha ein heco e
(x, y)∼ (x, 0) ∼x/U, because KExU.In hisway,¯qcan be aken as he densi y o cha ge a he cen e o he
plume, ¯q(x)=q(x, y =0)=q0(x). We define Ω(x) as he quo ien o he wo cha ac e is ic imes
Ω= (x)/τ ∼Kqix/U. (15)
I Ω 1, (14) gi es q0(x)≈qiand (12) is e y small. In his case, Coulomb epulsion is negligible in he sp eading
o he cha ged egion and (4) w i es
u∂q
∂x + ∂q
∂y =0.(16)
The p oblem is o mally analogous o ha o he mal plumes in he limi P →∞and sel -simila solu ion a e
concei able.
I Ω 1, q0(x) akes he asymp o ic alue q0(x)∼/K ∼U/Kx. The e o e, (12) is o o de 1, and Coulomb
epulsion and con ec ion a e o he same o de . In his case (4) w i es
Kq2
+u∂q
∂x +(KEy+ )∂q
∂y =0.(17)
Eqs. (16) o (17), wi h (1a)-(1b), define he dynamics o EHD plumes. We mus now es ima e he alue Ω in
expe imen al condi ions.
3. Es ima ion o Ω
The c ucial pa ame e is he ini ial densi y o cha ge, qi. I can be ela ed o he cu en because, a he elec odes,
he liquid eloci y anishes, and conduc ion is due only o d i . We conside fi s he 2D case.
Modeling he elec ode as an hype bole, and in he absence o space cha ge, he elec ic field can be calcula ed
by means o con o mal mapping16.I gi esEel =√2V/πd1/2 1/2
0, 0being he adius o cu a u e o he elec ode.
5

The injec ion o cha ge occu s when Eel eaches a h eshold alue, Es. The space cha ge modifies he elec ic field
on he injec o and, simila ly o wha occu s o co ona effec in gases17, o V>V
S he s eady elec ic field in he
icini y o he needle ip keeps nea ly cons an alues (co esponding o he ha monic field o V=Vs) independen ly
o he cu en . F om expe imen s in mine al and silicone oils5we ge Es∼0.1MVcm
−1. In o de o ge he o de
o magni ude o he injec ed cha ge we supposed he blade o injec in o an angle 0 <θ<θ
0(see figu e 2). In he
icini y o he su ace o he poin , he liquid mus be a es , and only he d i can anspo he elec ic cha ge.
Fo θ0=π/2 he cu en pe uni leng h is J=qiπ 0KEs.Taking 0=5μm, and K=10
−9m2V−1s−1one ob ains
qi∼J
π 0KEs∼6×10−2Cm
−3J=10
−8Am
−1,
6×10−1Cm
−3J=10
−7Am
−1.(18)
Simila conside a ions can be made o axisymme ic plumes. The ha monic field is calcula ed app oxima ing he
needle by an hype boloid( Laplace equa ion be ween an hype boloid and a pla e16,18). The field a he ip o he
needle is Eel =2V/( 0ln(4d/ 0)). F om expe imen s in mine al oils13,Es∼4MVcm
−1. This alue o Esis simila
o ha ob ained om expe imen s in cyclohexane and liquid a gon19,20. In o de o ge he o de o magni ude o qi,
we simpli y u he he geome y modeling he poin by a hemisphe e o adius 0.Takingθ0=π/2, he cu en is
I∼2π 2
0KEsqi.Wege
13 (K=10
−9m2V−1s−1)
qi∼I
2π 2
0KEs∼160 C m−3I=10
−8A,
1600 C m−3I=10
−7A.(19)
F om (19) and (18) he alue o he pa ame e Ω can be es ima ed o bo h geome ies, plana (Ωp) and axisymme ic
(Ω ). Taking he ypical alues es ima ed in II C we ge , o x=1cmand =2,
Ωp∼0.3J=10
−8Am
−1,
3J=10
−7Am
−1.Ω ∼90 I=10
−8A,
900 I=10
−7A.(20)
We see ha only in he 2D case, and o low alues o he in ensi y o he cu en , he effec o Coulomb epulsion
can be neglec ed.
E. Bounda y condi ions
The bounda y condi ions a e gi en by he geome y o he p oblem. A he cen e line o he plumes, he dis i-
bu ions o longi udinal eloci y and cha ge ha e a maximum and he ans e se eloci y mus anish. On he o he
hand, he longi udinal eloci y a om he plume anishes. We ha e, o bo h geome ies,
∂u
∂yy=0
=∂q
∂yy=0
= |y=0 = 0; (21a)
u|y→∞ = 0 (21b)
A s eady s a e, he in ensi y o cu en pe uni leng h J, in wo-dimensional plumes, and he in ensi y cu en Iin
axisymme ic plumes, mus be a cons an a any plane no mal o he axis o he plume. Using again ha KExU,
J=+∞
−∞
qudy, (22a)
I=2π+∞
0
quy dy (22b)
6
III. INTEGRAL ASYMPTOTIC SOLUTIONS
When Coulomb epulsion is negligible, sel -simila solu ions o (1a)-(1b) in bo h geome ies ha e been ound in he
case o cons an elec ic field, he cha ge emaining confined in o an infini ely hin laye o cylinde , so ha δqis
aken o be null6. In he 2D case we ob ain
u(x, y)=4J2E2
ρ2ν1/5
x1/5 (η),(23a)
δl(x)=16ρν3
JE 1/5
x2/5.(23b)
He e η=y/δl(x) is he sel -simila a iable and (η) is he sel -simila s eam unc ion. In he axisymme ic case
u(x, y)=IE
2πρν 1/2 
η,(24a)
δl(x)=2πρν3
IE 1/4
x1/2.(24b)
These solu ions canno desc ibe he in e nal s uc u e o he cha ged co e. E en mo e, in he axisymme ic case,
in he limi δq→0, he eloci y ends o infini y6. The effec o Coulomb epulsion mus be conside ed o ully
unde s and he EHD plumes s uc u e.
Sel -simila solu ions a e no longe possible when he Coulomb epulsion e m in (17) is no neglec ed. Now we
p esen an in eg al me hod analysis, gene alizing he p e ious wo ks by McCluskey and P´e ez11 andA ene al.
13,
ha can desc ibe he beha io o he p incipal magni udes o EHD plumes.
A. 2D plumes
I δq<δ
l, we can define an a e age eloci y um(x)so ha
J=+∞
−∞
qudy =Qp(x)um(x),(25)
wi h Qp(x)=∞
−∞ qdy he cha ge pe uni leng h. So umis he a e age alue o he axial eloci y o he liquid in o
he cha ged co e. When δqδl,umcoincides wi h he eloci y a he cen e o he plume. Ou aim is o find he
equa ions ela ing δq,δland um.
In eg a ing (1b), wi h k= 0, wi h espec o y om y=−∞ o y=∞, using (1a) and (21a), a e some
manipula ions, leads o
∞
−∞
∂u2
∂x dy =∞
−∞
qEx
ρdy. (26)
I δqis small enough, he a ia ions o he longi udinal elec ic field as a unc ion o he coo dina e ycan be neglec ed
so ha , o a fi s app oxima ion, Ex(y,x)=E0(x) o |y|≤δq. Then, he igh -hand side e m in (26) can be
exp essed in e ms o Qp(x). Mul iplying (26) by umwe ge
um
d
dx ∞
−∞
u2dy =JE0
ρ.(27)
and exp esses ha he a ia ion o o al momen um in a plane ans e se o he flow equa es he o al o ce pe uni
leng h exe ed by he imposed elec ic field on he liquid. No e ha he iscous e m does no con ibu e o he
balance because i is an in e nal o ce.
Conse a ion o kine ic ene gy allows us o ake he iscous effec s in o accoun . Mul iplying (1b) by uand
manipula ing i in he same way we ob ain
7
d
dx ∞
−∞
1
2u3dy =−ν∞
−∞ ∂u
∂y2
dy +JE0
ρ.(28)
This equa ion exp esses he balance be ween he kine ic ene gy o he liquid, he iscous losses and he powe injec ed
by he elec ic o ce. Finally, le us econside equa ion (25),
J=Qum≈2¯q(x)δq(x)um(x)⇒δqum≈J
2¯q(x).(29)
Fo y= 0 eq. (17) gi es
um
∂¯q
∂x =−K¯q2
⇒um
∂
∂x 1
¯q=K
,(30)
because KExum.Fo ¯q(x), he pa ial de i a i e can be subs i u ed by he o al de i a i e. Hence, applying
umd/dx o (29) we ge
um
d
dx (δqum)=KJ
2.(31)
Eqs. (31), (27) and (28) define he p oblem. In o de o ob ain a se o diffe en ial equa ions we ha e o exp ess
he in eg als in (27) and (28) in e ms o umand δl. This can be pe o med by imposing a eloci y p ofile. I δqis
clea ly lowe han δl he elec ic o ce emains confined in o a e y na ow egion inside he plume, so ha we can
app oxima e he dis ibu ion u(x, y) by a pseudo sel -simila eloci y p ofile u(x, y)=um(x) (η), wi h η=y/δl(x),
(0) = 1 and (η)≤0∀η. The e o e, we ob ain h ee diffe en ial equa ions o um(x), δl(x)andδq(x),
um
d
dx um2δl=1
B1
JE0(x)
ρ,(32a)
B2
d
dx um3δl=−νB3
um2
δl
+JE0(x)
ρ,(32b)
um
d
dx (umδq)=JK
2.(32c)
The cons an s a e
B1=∞
−∞
2(η)dη, (33a)
B2=∞
−∞
1
2 3(η)dη, (33b)
B3=∞
−∞
( )2(η)dη. (33c)
In o de o es ima e he alues o B1,B2and B3, a eloci y p ofile mus be gi en. Choosing (η)=exp(−η)asa
plausible p ofile leads o B1=1,B2=1/3, B3= 1. Any o he choice sa is ying he bounda y condi ions gi es simila
alues o he coefficien s.
1. Solu ions o some pa icula dependences o he elec ic field
I we assume ha he elec ic field depends on xas E0(x)= ¯
E(x/d)m,weob ain
um=CuJ2¯
E2
ρ2νd2m1/5
x(1+2m)/5,(34a)
8
δl=Clρν3dm
J¯
E1/5
x(2−m)/5,(34b)
δq=Cq
K
Jρ4ν2d4m
¯
E41/5
x(3−4m)/5,(34c)
δq
δl
=Cql
K
J2ρ3d3m
ν¯
E31/5
x(1−3m)/5.(34d)
The cons an s a e
Cu=B2
3
n1B1
(n1B1−n2B2)1/5
,(35a)
Cl=n1B1B3
3
(n1B1−n2B2)31/5
,(35b)
Cq=B2
3
32n1n5
3B1
(n1B1−n2B2)1/5
,(35c)
Cql =(n1B1−n2B2)4
32n2
1n5
3B2
1B31/5
,(35d)
wi h
n1=(4−3m)/5,n
2=1+m, n3=(4−2m)/5.(36)
In he case m= 0 (uni o m elec ic field), he exp essions (34a) and (34b) a e qui e analogous o hose ob ained
neglec ing Coulomb epulsion. The e o e, o 2D EHD plumes, including Coulomb epulsion e ms does no affec he
hyd odynamic bounda y laye . Bu now he cha ge bounda y laye is cha ac e ized. No e ha δq a ies wi h xin a
way clea ly diffe en om δland om he e olu ion p edic ed by Zhakin when only diffusion is e ained.
Taking he same alues used in II C, wi h m=0,weob aina x=1cm: um∼8cms
−1,δ
l∼2.2mm,δ
q∼
4μm,δ
q/δl∼2×10−3. Recalling (10), we see ha he o de o magni ude o δqis big enough o he effec o cha ge
diffusion being negligible.
B. In eg al solu ions o axisymme ic EHD plumes
The p e ious analysis o 2D EHD plumes can be easily ex ended o axisymme ic plumes. He e, he flow is
conside ed again o ha e a double bounda y laye s uc u e, wi h an inne cha ged co e o adius δq(x) and an ou e
hyd odynamic laye o adius δl(x), wi h δq<δ
l. The o al cha ge pe uni leng h in a plane a cons an xis
Q(x)=2π+∞
0yq dy, and he o al cu en is I=2π+∞
0yqudy ≈Q(x)um(x). F om (1b) and (14), ollowing he
same s eps han in §III A, we ob ain he equa ions ha desc ibe he dynamics o axisymme ic EHD plumes
um
d
dx ∞
0
yu2dy =IE0(x)
2πρ ,(37a)
d
dx ∞
0
1
2yu3dy =−ν∞
0
y∂u
∂y2
dy +IE0(x)
2πρ ,(37b)
um
d
dx umδ2
q=IK
π .(37c)
9
FIG. 5. E olu ion o he hickness o he hyd odynamic laye , δland he cha ged laye , δq, o diffe en ini ial alues o λ,
wi h an imposed elec ic field. Linea and loga i hmic scales in x/d a e used.
16