Coulomb-d i en low o a dielec ic liquid subjec o cha ge injec ion
by a sha p elec ode
F. J. Higue aa)
E.T.S. Ingenie os Ae ona
´u icos, Pza. Ca denal Cisne os 3, 28040 Mad id, Spain
P. A. Va
´zquez
Uni e sidad de Se illa, Dep s. Elec o
´nica y Elec omagne ismo and Fı
´sica Aplicada, 41092 Se illa, Spain
~Recei ed 21 Janua y 1999; accep ed 23 Ma ch 1999!
Injec ion o cha ge by a sha p elec ode in o a su ounding dielec ic liquid leads o Coulomb o ces
ha se he liquid in o mo ion. An analysis is p esen ed o his mo ion in a small egion a ound he
edge o he elec ode, which de e mines he injec ed cu en as a unc ion o he a elec ic po en ial
seen by his egion. By using an injec ion law app op ia e o nonpola liquids, he analysis p edic s
an elec ic cu en ha inc eases i s exponen ially and hen as he powe 7
3o he ha monic pa o
he elec ic po en ial, some imes wi h a ange o mul iplici y in be ween. © 1999 Ame ican
Ins i u e o Physics. @S1070-6631~99!01207-6#
Elec ic conduc ion h ough a pu i ied dielec ic liquid
be ween a couple o elec odes subjec ed o a high po en ial
di e ence is o en domina ed by he mo ion o cha ge ca i-
e s injec ed a one o he wo elec odes by means o elec-
ochemical eac ions. In hese cases, he p esence o a space
cha ge ~o densi y q!in he elec ic ield Eexis ing in he
liquid leads o a Coulomb o ce qE ha induces a mo ion o
he liquid. Such mo ion enhances cha ge, momen um and
hea ans e , which is o in e es in a numbe o applica ions
including elec os a ic p ecipi a o s, elec ohyd odynamic
ion-d ag pumps, and elec ohyd odynamic lamina and u -
bulen mixing; see Re s. 1 and 2 o e iews. While s ong
ep oducible injec ions can be ob ained by using elec odia-
ly ic memb anes,3,4 weak and mode a e ep oducible injec-
ions in nonpola liquids5–7 a e p obably mo e impo an in
applica ions. In he i s case he cha ge densi y a he injec -
ing elec ode is nea ly independen o he local elec ic ield
~au onomous injec ion!, whe eas i is a apidly inc easing
unc ion o he elec ic ield in he second case. The injec ion
law q/q05
$
(E/E0)1/2K1@(E/E0)1/2#
%
21, whe e E5
u
E
u
,K1
is he modi ied Bessel unc ion o second kind and o de one,
and q0and E0a e cons an s, was p oposed by Felici and
Gosse6on he assump ion ha he a e limi ing p ocess o
injec ion is he ex ac ion o ions h ough he ield-dependen
ba ie in he so called image- o ce egion on he elec ode
su ace ~Scho ky e ec !.
Coulomb-d i en con ec ion be ween plane-pa allel elec-
odes has been much s udied; see, e.g., Re s. 8 and 9 and
e e ences he ein. Sha p o peaky elec odes, on he o he
hand, lead o high elec ic ields ha a e o in e es , o con-
ce n, in applica ions. A en and Haida a10 de e mined expe i-
men ally he I-Vcha ac e is ic o a kni e-plane elec ode as-
sembly in a closed cell, using he high elec ic ield as a
means o asce ain he ole o cha ge injec ion by he sha p
elec ode in he di e en egimes ound in hei expe imen s,
and Takashima e al.11 ca ied ou simila expe imen s and
app oxima e nume ical compu a ions o kni e-plane and
needle-plane elec ode con igu a ions. In bo h cases Cou-
lomb o ce leads o an elec ohyd odynamic plume, analo-
gous o a buoyancy-induced he mal plume, a om he in-
jec ing elec ode, whose s uc u e and s abili y ha e been
s udied elsewhe e ~Re s. 12–16!. Howe e , i is he low and
cha ge dis ibu ion in a egion a ound he elec ode ip o
size o he o de o i s cu a u e adius ha de e mines he
injec ed cu en o a gi en injec ion law q5 (E) a he
elec ode su ace. This low was discussed in Re . 16 and
will be u he analyzed he e unde he assump ions ha he
mo ion o he liquid is induced locally by Coulomb o ce,
a he han by he eci cula ion possibly exis ing in he bulk
o he cell, and ha he Reynolds numbe based on he cu -
a u e adius o he elec ode a i s ip and he cha ac e is ic
eloci y o he liquid ~see below!is small. Neglec ing he
ine ia o he liquid and he di usion o he cha ge, he equa-
ions and bounda y condi ions go e ning he s eady wo-
dimensional low a ound a pa abolic elec ode ip can be
w i en in nondimensional o m as ollows
¹2
c
52
, wi h 5~u, !5
S
]
c
]
y,2
]
c
]
x
D
,~1!
a!Elec onic mail: [email p o ec ed]
VOLUME 11, NUMBER 8 AUGUST 1999
PHYSICS OF FLUIDS
111070-6631/99/11(8)/2434/3/$15.00 2434 © 1999 Ame ican Ins i u e o Physics
¹2
1
]
q
]
x
]
]
y2
]
q
]
y
]
]
x50, ~2!
¹2
5q, wi h E5“
,~3!
“•@q~ 1
k
E!#50, ~4!
x512y2
4:
c
5
]
c
]
n5
50, q5
S
]
]
n
D
,~5!
and he a ield condi ions
c
;31/6
2J2/3 4/3~cos 2
3
u
2cos 4
3
u
!1•••
;22
35/6 J2/3 22/3 cos 2
3
u
1•••
;31/3J1/3 2/3 sin 2
3
u
1F`
h1•••
q50 a inle
6
,~6!
o (x,y)→`. He e xand ya e dis ances along he axis o
he pa abola and no mal he e o, measu ed om i s ocus and
scaled wi h wice i s cu a u e adius a he ip 0/2;
5(x21y2)1/2, and
u
5a c an(2y/x), wi h
u
5
p
/2 a x.0,
y50; and (
c
,
,
,q) a e he s eam unc ion, he o ici y,
he elec ic po en ial, and he cha ge densi y scaled wi h
(
e
Ec
2 0
2/
m
,
e
Ec
2/
m
,Ec 0,
e
Ec/ 0), espec i ely. He e
m
and
e
a e he liquid iscosi y and pe mi i i y, and Ecis a cha -
ac e is ic elec ic ield, depending on he injec ion law q
5 (E) and de ined by he condi ion (1)51 in nondimen-
sional a iables. In wha ollows he law q5exp
$
g
(E21)
%
,
wi h
g
cons an , will be used o de ini eness, hough he
esul s should be alid o any (E) inc easing apidly wi h
Ei
g
5d(ln )/dE
u
E51. Equa ions ~1!and ~2!a e he mass
and momen um conse a ion equa ions in he o ici y-
s eam unc ion o mula ion; he las wo e ms o Eq. ~2!a e
he cu l o he Coulomb o ce. Equa ion ~3!is Poisson’s law
o he elec ic po en ial and Eq. ~4!is he con inui y equa-
ion o he elec ic cha ge, which mo es wi h eloci y
1
k
E, whe e
k
is he cha ge mobili y nondimensionalized
wi h
e
Ec 0/
m
. The Reynolds numbe alluded o be o e is
Re5
e
Ec
2 0
2/
m
2, and o mula ion ~1!–~6! elies on he as-
sump ion Re!1.
The elec ic ield away om he injec ing elec ode is
domina ed by he space cha ge and i s image by he elec-
odes. This cha ge is con ined o a hin s eak a ound he
symme y plane by he ac ion o he con e ging low, which
in u n is domina ed by he Coulomb o ce ac ing on he
cha ged s eak. The a ield condi ions ~6! esul om he
balance o he Coulomb o ce and he iscous shea o ce on
he sides o he s eak. The i s o ce is ExJ/u, whe e, ad-
ancing ha
u
u
@
k
u
“
u
when x@1, J5limx→`
*
2`
`uqdy
is he elec ic cu en pe uni dep h o he elec ode ~ o be
ound as pa o he solu ion!, so ha J/uis he cha ge den-
si y in eg a ed ac oss he s eak, and Exis he elec ic ield a
he symme y plane, due mainly o he cha ged s eak and i s
image by he injec ing elec ode. This ield can be seen o be
o o de J/u. The second o ce is o o de u/x, and he
balance o he wo equi es u5O(J2/3x1/3) and Ex
5O(J1/3/x1/3). Since ¹2
5¹2
5¹4
c
50 ou side he
s eak, hese asymp o ic beha io s lead o he ha monic and
biha monic unc ions in ~6!.17 In he second e m o he
asymp o ic expansion o
in ~6!
h5221/2@(x21y2)1/2
1x#1/221 is an exac solu ion o he p oblem wi h q50 and
he pa ame e F`measu es he ha monic ield seen by he
egion o conce n. Finally
]
/
]
n5(
]
/
]
x11
2y
]
/
]
y)/(1
1y2/4)1/2 deno es he de i a i e no mal o he elec ode in
~5!.Equa ions ~1!–~4!we e ew i en in pa abolic coo di-
na es, disc e ized using second o de ini e di e ences, and
sol ed wi h a pseudo ansien me hod. Values o
g
anging
om 10 o 30 we e used in mos o he compu a ions. These
alues a e ep esen a i e o he limi
g
→`, in which he
ex eme sensi i i y o he cha ge densi y a he elec ode wi h
he elec ic ield leads o wo egions on he elec ode su -
ace: an injec ion egion whe e q.0 and E51, and he es
o he elec ode, whe e q50 and E,1. Figu e 1 shows Jas
a unc ion o F` o se e al alues o he nondimensional
mobili y
k
and o
g
.
As can be seen in Fig. 1, he elec ic cu en begins
inc easing e y apidly wi h F`, p esen s mul iplici y in a
ange o F`i
g
is su icien ly la ge, and se les o a powe
law when F`→`. On dec easing Jin he middle b anch, he
educ ion o he ield due o he space cha ge and i s image is
nea ly balanced by an inc ease o he ha monic ield, so ha
he ield on he elec ode dec eases e y li le, as equi ed by
he injec ion law when
g
is la ge. ~This desc ip ion and he
shape o he cu e depend on he manne he wo con ibu-
ions o he ield a e de ined, which is no unique.!In he
s eep lowe b anch he ha monic ield domina es. The esul s
o Fig. 1 a e alid e en i he cha ged s eak is nonhomoge-
neous o uns able beyond he egion analyzed he e, inso a
as he cha ac e is ic ime o i s e olu ion is la ge compa ed
wi h he local esidence ime (
m
/
e
Ec
2in dimensional a i-
ables!. Then F`is a slowly a ying unc ion o ime and a
S-shaped esponse may lead o jumps and hys e esis o he
cu en .
In he absence o space cha ge, he elec ic po en ial is
F`
hand he maximum elec ic ield is Emax5F`/2, a he
elec ode ip, so ha he condi ion Emax51 de ining he onse
o cha ge injec ion o
g
→`amoun s o F`52. This should
be he limi ing posi ion o he lowe u ning poin s in Fig. 1.
In addi ion, since
h;x1/2 away om he injec ing elec ode,
FIG. 1. Jas a unc ion o F` o
g
520 and
k
50.2, 0.5, 1, and 2, inc eas-
ing as indica ed by he a ow ~solid!, and o
k
51 and
g
510 ~do ed le !
and
g
530 ~do ed igh !. The dashed s aigh lines on he igh side a e he
asymp o es J5c
k
2/3F`
7/3 o
k
50.5, 1, and 2, wi h c51.3.
2435Phys. Fluids, Vol. 11, No. 8, Augus 1999 Coulomb-D i en Flow o a Dielec ic Liquid...
he onse ol age be ween he elec odes is V15O(R1/2),
whe e R@1 is he dis ance be ween he elec odes scaled
wi h 0.
In he asymp o ic limi F`→` he cha ge is con ined o
a hin laye on he elec ode su ace and o a s eak on he
symme y plane. Le qand
d
be he cha ac e is ic cha ge
densi y in he laye on he elec ode and he hickness o his
laye , qsand
d
s he co esponding magni udes in he s eak
a ~nondimensional!dis ances o o de uni y om he elec-
ode, and E5O(F`)@1 he cha ac e is ic elec ic ield.
The balance o Coulomb o ce and iscous o ce in oduced
be o e equi es now Eqs
d
s; , whe e is he cha ac e is ic
eloci y o he liquid. This eloci y ends o ze o a he elec-
ode su ace, being o o de cl5
d
and essen ially angen
o he su ace in he cha ged laye on he elec ode, whe e he
eloci y no mal o he su ace is o o de n5 cl
d
5
d
2
~ om he con inui y equa ion!. This la e eloci y pushes he
cha ge owa d he su ace agains he d i eloci y
k
E,
which is di ec ed away om he su ace. The balance o he
wo e ec s de e mines he hickness o he cha ged laye ,
p o iding he ela ion n;
k
E,o q
s
d
s
d
2
;
k
. On he o he
hand, conse a ion o he injec ed cha ge equi es J; qs
d
s
; clq
d
, whe e q;E/
d
om Poisson’s law in he cha ged
laye @whe e he elec ic ield mus dec ease om O(E)in
he ou e egion o essen ially O(1) a he su ace, specially
i
g
is la ge#. Thus qs
d
s;(q
d
) cl / ;E
d
, and elimina ing
qs
d
sbe ween his ela ion and he one coming om he bal-
ance o con ec ion and d i eloci y yields
d
;(
k
/E)1/3.
Then q;
k
1/3E4/3, ;
k
1/3E5/3, and J;
k
2/3E7/3. The las e-
sul is consis en wi h he powe law ela ions be ween he
cu en and he ol age ound in some expe imen s. A com-
pa ison wi h he nume ical solu ion o Eqs. ~1!–~6!is p e-
sen ed in Fig. 1, whe e he dashed lines a e J5c
k
2/3F`
7/3 ,
wi h c51.3 chosen o i he nume ical solu ions.
The main ea u es o he J–Vcha ac e is ic o he cell
p edic ed by he p esen model unde s a iona y condi ions
and
g
@1 can now be ou lined. Fi s , a ol age V.V1
5O(R1/2) is equi ed o injec ion. Second, since he i s
e m o he expansion o
in ~6!g ows wi h as e han he
second, he excess ol age necessa y o keep a s a iona y
cu en Jis V2V15O(J1/3R2/3), so ha J!1 while (V
2V1)!R2/3 and J@1 when V@R2/3. In he i s case he
ha monic ield domina es in he egion !1/J2@ om ~6!
wi h F`5O(1)], which sh inks when Vbecomes o
O(R2/3). In he second case F`5O(J3/7)@ aking
k
5O(1)], which is also la ge, and he ha monic ield domi-
na es in he expanding egion !J4/7. Thi d, his egion ends
up co e ing he whole cell when J5O(R7/4), co esponding
o V5O(R5/4), and he ela ion J5O(V3/R2) changes o J
5O(V7/3/R7/6) when V@R5/4. No ice ha F`
5O@max(1,J3/7)#is small compa ed wi h he alue i would
ha e in he absence o space cha ge, o O(V/R1/2), o any
ol age sa is ying V1!V!R5/4: The space cha ge is sc een-
ing he injec ing elec ode almos comple ely in his ange o
ol ages. The s abili y o he s a iona y low and he e ec s
o he ine ia o he liquid will be analyzed elsewhe e.
In summa y, an analysis has been ca ied ou o he
c eeping low induced by Coulomb o ces in a dielec ic liq-
uid a ound a sha p wo-dimensional elec ode injec ing
cha ge wi h a law q5 (E) ha e lec s he high sensi i i y
o cha ge injec ion wi h he elec ic ield. The analysis leads
o a cu en -ha monic ield ela ionship (J2F`) ha may be
mono onically inc easing o S-shaped, ea u es an injec ion
onse when
g
→`, and becomes a powe law J;F`
7/3 in he
space cha ge-limi ed injec ion limi F`→`. An analogous
analysis o pa aboloidal elec odes leads o he same 7
3law
up o loga i hms.
ACKNOWLEDGMENTS
We a e g a e ul o P o s. A. Cas ellanos and A. T. Pe
´ ez
o use ul discussions. This wo k was suppo ed by DGICYT
g an PB95-0008.
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IEEE T ans. Elec . Insul. 26, 1201 ~1991!.
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jec ed space cha ge in insula ing liquids,’’ IEEE T ans. Dielec . Elec .
Insul. 3,1~1996!.
3G. B ie
` e and J. P. Gosse, ‘‘Injec ion d’ions dans les liquides polai es,’’ J.
Chim. Phys. 65, 725 ~1968!.
4N. J. Felici and R. E. Tobaze
´on, ‘‘Cha ge ca ie elimina ion and p oduc-
ion by elec o-dialy ic polyme s in con ac wi h dielec ic liquids,’’ J.
Elec os . 11, 135 ~1981!.
5A. Dena , B. Gosse, and J. P. Gosse, ‘‘Ion injec ions in hyd oca bons,’’ J.
Elec os . 7, 205 ~1979!.
6N. J. Felici and J. P. Gosse, ‘‘Injec ion d’ions pa des e
´lec odes me
´ al-
iques dans les hyd oca bu es liquides de e
´sis i i e
´e
´le ee
´,’’ Re . Phys.
Appl. 14, 629 ~1979!.
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~1985!.
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Phys. Fluids 13, 1948 ~1970!.
9R. Chico
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~1997!.
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dielec ic liquids in a kni e-pla e elec ode assembly,’’ IEEE T ans. Elec .
Insul. 20, 187 ~1985!.
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con igu a ions in ans o me oil and liquid ni ogen,’’ IEEE T ans. Elec .
Insul. 23, 645 ~1988!.
12F. M. J. McCluskey and A. T. Pe
´ ez, ‘‘The elec ohyd odynamic plume
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27, 334 ~1992!.
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poin -plane geome y,’’ IEEE 11 h ICDL, Baden, Swi ze land ~1993!.
14A. T. Pe
´ ez, P. A. Va
´zquez, and A. Cas ellanos, ‘‘Dynamics and linea
s abili y o cha ged je s in dielec ic liquids,’’ IEEE T ans. Ind. Appl. 31,
761 ~1995!.
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16P. A. Va
´zquez, ‘‘Dina
´mica y es abilidad de penachos elec ohid odina
´mi-
cos,’’ Doc o Thesis, Uni e sidad de Se illa, Spain ~1998!.
17The scaling and he bounda y condi ions a he elec ode (
u
'0) imply
;A 2/3 sin 2
3
u
and
c
;B 4/3(cos 2
3
u
2cos 4
3
u
), whence Ex5
]
/
]
u
u
5
p
5321/2A/ 1/3,u5 21
]
c
/
]
u
u
u
5
p
531/2B 1/3, and he iscous s ess on he
sides o he s eak is 2 22
]
2
c
/
]
u
2
u
u
5
p
54
3B/ 2/3. On he o he hand, us-
ing he exp ession J/u5J/(31/2B 1/3) o he cha ge in he s eak, elec o-
s a ics gi es Ex5J/(2B 1/3) and he Coulomb o ce ExJ/u5J2/(2
331/2B2 2/3). The cons an s Band Aused in ~6! hen ollow by equa ing
his o ce o he iscous o ce and by equa ing he wo exp essions o Ex
abo e.
2436 Phys. Fluids, Vol. 11, No. 8, Augus 1999 F. J. Higue a and P. A. Vazquez