THERMAL AND ELECTROHYDRODYNAMIC
PLUMES: A COMPARATIVE STUDY
P. A. V´azquez, A. T. P´e ez and A. Cas ellanos
Dp o. Elec ´onica y Elec omagne ismo, Facul ad de F´ısica.
A da. Reina Me cedes s/n. 41012 Se illa. Spain.
This pape deals wi h sel simila he mal and elec ohyd odynamic (EHD) plumes. The o me
a ises om ho lines o poin s, whe eas he la e a ises when sha p me allic con ou s subme ged in
non conduc ing liquids suppo high elec os a ic po en ial, esul ing in cha ge injec ion. Al hough
he mo i e o ce is buoyancy in one case and Coulomb o ce in he o he , i is shown ha he
solu ion o EHD plumes is he same as o he mal plumes in he limi o la ge P and l numbe s.
We p esen he analysis o axisymme ic plumes o la ge alues o P and l numbe , and his analysis
is subsequen ly applied o EHD plumes. The alidi y o he app oxima ions o EHD plumes is
discussed in he ligh o expe imen al da a.
I. INTRODUCTION
The mal plumes a ising om line o poin sou ces o hea ha e a ac ed he a en ion o he in es iga o s o a
numbe o yea s. These low s uc u es appea in many p ac ical p oblems and i s desc ip ion ha e been impo an
om he indus ial poin o iew. When he luid o in e es is a iscous oil, he case o high P and l numbe dese es
special conside a ion. Also plumes in high P and l numbe luids a e ele an in he con ex o con ec ion in he
Ea h’s man le.[1, 2]
The bounda y laye equa ions o na u al con ec ion, i. e., he equa ions o he mal plumes, a e known o possess
sel simila solu ions, ha educe he pa ial di e en ial equa ions o a se o o dina y di e en ial equa ions. These
sel simila solu ions a e also he s a ing poin o he s abili y analysis o he lamina plumes.[3, 4]
Al hough he p oblem o wo-dimensional plumes o e y la ge P and l numbe s ha e been s udied long ime
ago,[5] and his analysis was e ined la e ,[6] he au ho s a e no awa e o he exis ence o a simila analysis o
he axisymme ic plumes, excep o some nume ical in es iga ions.[7] The i s aim o his pape is o p esen such
analysis. We demons a e ha he eloci y o he luid a he cen al line o he plume di e ges as ln √P , p o ided
he o al hea lux is ini e.
On he o he hand elec ohyd odynamic (EHD) plumes ha e been obse ed expe imen ally.[8, 9, 11] When high
ol age is applied o elec odes wi h sha p edges o poin s imme sed in dielec ic liquids, he e is injec ion o ions
om he elec ode in o he liquid.[12] The Coulomb o ce ac ing upon he injec ed space cha ge plays he ole o
buoyancy. The e is a close analogy be ween EHD and he mal plumes. As a ma e o ac he equa ions o EHD
plumes, unde a numbe o assump ions, a e hose o he he mal ones o P → ∞.In his con ex he second aim o
his pape is o discuss he analogies and di e ences be ween EHD and he mal plumes, and o es ablish he alidi y
o he app oxima ions in ol ed in he EHD case. As sel simila solu ion o wo dimensional EHD plumes is al eady
discussed in,[11] we ocus on he axisymme ic case in his pape .
II. EQUATIONS FOR THERMAL PLUMES
Na u al con ec ion esul ing om a poin sou ce o hea can be conside ed as an axisymme ic lamina , s eady
low. The go e ning con inui y, momen um and ene gy equa ions simpli y wi h he Boussinesq app oxima ion and
bounda y laye assump ions o he ollowing o m:[13]
∂ u
∂x +∂
∂ = 0 (1)
u∂u
∂x + ∂u
∂ =ν
∂
∂ ( ∂u
∂ ) + gβ(T−Ta) (2)
u∂T
∂x + ∂T
∂ =χ
∂
∂ ( ∂T
∂ ) (3)
He e xis he coo dina e along he low, he coo dina e pe pendicula o he low, uand a e, espec i ely,
he eloci ies in he xand adial di ec ions, β he coe icien o olume ic he mal expansion, ν he iscosi y, T
empe a u e, χ he he mal di usi i y and Tais he empe a u e o he ambien luid. I he ambien luid is a
2
es , he e is no any p essu e con ibu ion in equa ion (2). Viscous dissipa ion and comp essibili y e ec s ha e been
neglec ed in he ene gy equa ion.
These equa ions a e complemen ed wi h he co esponding bounda y condi ions. On one hand a he cen al line
( = 0) he adial eloci y has o be ze o, and, hence, uhas a maximum. On he o he hand eloci y and empe a u e
di e ences decay o ze o a om he plume. Tha is:
=∂u
∂ =∂T
∂ = 0 a = 0,(4)
u=T−Ta= 0 a → ∞ (5)
An addi ional condi ion is ob ained conside ing he hea lux ac oss a ho izon al plane. This lux is:
Q= 2πρcpZ∞
0
(T−Ta)u d (6)
whe e cpis he speci ic hea o he luid. In he absence o o he sou ces o hea in eg a ion o equa ion (3) on an
ho izon al plane implies he cons ancy o Q.
I is possible o de ine a s eam unc ion Ψ so ha equa ion (1) is au oma ically sa is ied:
u=1
∂Ψ
∂ and =−1
∂Ψ
∂x (7)
and we ha e now wo unknowns: Ψ and T.
The equa ions (1-3) can be ans o med in o o dina y di e en ial equa ions using he simila i y me hod. As he e is
no a ypical leng h in he axial di ec ion he eloci y and he mal p o iles scale wi h he plume wid h. Ma hema ically
his means ha eloci y and empe a u e p o iles a e he same a each xwhen exp essed as a ce ain combina ion o
xand ,η=η(x, ). In ou case i is easy o show ha
η= (G )1/4
x(8)
is an adequa e choice. In oducing simila i y unc ions and θ ha depend only on η:
Ψ = νx (η) (9)
T−Ta=ν2
gβx3G θ(η) (10)
equa ions (1-3) gi e:
′′′
η+ −1
η ′
η′
+θ= 0 (11)
(ηθ′)′+ P ( θ)′= 0 (12)
which a e now a se o o dina y di e en ial equa ions.
The associa e bounda y condi ions a e:
 ′
η′
(0) =
η(0) − ′(0)
2=θ′(0) =
η
′
(∞) = θ(∞) = 0 (13)
1 = Z+∞
0
′θ dη (14)
The quan i y G = gβx3(Ts−Ta)/ν2θ(0) is he G asho numbe . I measu es he ela i e s eng h o buoyancy
and iscous o ces (Tsis he empe a u e a he cen al line o he plume). P = ν/χ is he P and l numbe .
G is ela ed o Qby equa ion (6) which gi es:
G = gβQ
2πρν3cp
x2(15)
3
In his way he eloci y o he plume is gi en by:
u=ν
xG 1/2 ′
η(16)
and i s hickness by:
∆ = x
G 1/4(17)
Veloci y and empe a u e p o iles a e ob ained in eg a ing nume ically equa ions (11-12) using a shoo ing me hod.
The in eg a ion is ca ied ou as i i was an ini ial condi ions p oblem, wi h ′/η(0) and θ(0) known. We use a
Runge-Ku a me hod o ob ain ′/η(∞) and R ′θ dη. This p ocedu e de ines hese las alues as unc ions o ′/η(0)
and θ(0). Then, equa ions ′/η(∞) = 0 and R ′θ dη = 1 a e sol ed wi h a New on-Raphson me hod.
The p o iles so ob ained a e shown in Figu es 1 and 2 o di e en P and l numbe s. The eloci y a he cen al
line = 0 is p opo ional o ′(0)/η. This alue, along wi h θ(0) is lis ed in Table I o di e en P and l numbe s.
This able is consis en wi h o he au ho s’ esul s.[7]
III. THE CASE OF LARGE PRANDTL NUMBER
We a e specially in e es ed in axisymme ic plumes when P → ∞,because i is in his limi ha he mal and
EHD plumes beha e in he same way. In his limi , he hickness o he he mal bounda y laye is ze o, and his
allows us o sol e he p oblem wi hou sol ing he ene gy equa ion.
Equa ion (12) can be in eg a ed o gi e:
θ(η) = θ(0) exp −P Zη
0
( )
d (18)
Condi ion /η − ′/2 = 0 a η= 0 is ul illed only i (0) = ′(0) = 0.The e o e, when η << 1 :
(η) = ′′(0)η2/2 + O(η3) (19)
And we ge o θ:
θ(η) = θ(0) exp −P ′′ (0)
4η2(20)
I is clea ha θ(η) goes o 0 a a dis ance o o de η=δ ∼1/√P and he alue o θ(0) can be ob ained om
equa ion (14): θ(0) = P /2.Fo P → ∞,buoyancy ac s only along he axis, and is negligible o any η6= 0.
We seek now o an addi ional bounda y condi ion in he limi P → ∞ ha allows us o sol e he p oblem using
only equa ion (11) o η > 0. In eg a ing (11), using (19), neglec ing e ms o o de ǫand aking in o accoun ha in
he limi P → ∞ is Rǫ
0ηθ dη = 1/ ′′ (0) we ob ain:
′′(ǫ)− ′
η|η=ǫ=−1
′/η |η=0 (21)
No egula unc ion o η > 0 can ul ill his exp ession unless ′/η goes o in ini y a η= 0.
Le us analyze close he way in which ′/η di e ges. Fo high enough P he eloci y is cons an in he he mal
bounda y laye , as he he mal laye δ is much smalle han he momen um bounda y laye (see igu e 3). The e o e
i is possible o hink o a dis ance ǫmuch smalle han 1 bu ye g ea e han δ so as o assume he ollowing eloci y
p o ile:
′(η)
η=αi η < δ
h(η) i δ < η < ǫ (22)
By con inui y is α=h(δ ).Ou side o he he mal laye is
′′′ + ( −1)  ′
η′
= 0 (23)
4
Bu ≃ ′′(0)η2/2,so << 1 o η < ǫ, and equa ion (23) can be app oxima ed by:
′′′ − ′
η′
= 0 (24)
In eg a ing once we ha e: ′′ −( ′/η) = λ, and his gi es an equa ion o h(η) :
ηh′=λ(25)
whose solu ion is h=λln η. The e o e:
′(η)
η=λln δ i η < δ
λln ηi δ < η < ǫ (26)
Equa ion (21) gi es now λ2=−1/ln δ so
′
η=p−ln δ (27)
inside he he mal laye .
Since δ ≃1/√P is:
′
η≃√ln P (28)
and he eloci y a he axis o he plume di e ges as √ln P . Table I gi es he esul s o he nume ical solu ion o
equa ions (11-12) o di e en P and l numbe s. The bes i o hose alues o ′/η is:
′
η|0= 0.38 + 0.67√ln P (29)
in comple e ag eemen wi h he analysis. The conclusion is ha he anspo o a ini e hea lux equi es an in ini e
eloci y a he axis o P → ∞.
E en hough he eloci y di e gence is e y weak, i is in con as wi h he wo–dimensional case. In ha case,[5]
he eloci y is ini e in he limi P → ∞ and i is:
u(x, y) = 1.42ν
xG 1/2(30)
The G asho numbe is ela ed o he hea lux pe uni leng h in ans e se di ec ion by:
G = Qgβx3
2√2ρcpν3!4/5
(31)
and he eloci y in he cen al plane is ini e p o ided Q emains ini e.
IV. EHD PLUMES
Insula ing liquids s essed by in ense elec ic ields a e subjec ed o s ong injec ion o cha ge om me allic sha p
poin s o edges. The injec ed ions a e ac ed upon by he elec ic ield, and h ough collisions wi h he neu al molecules
gi e ise o s ong con ec i e mo ions o liquid. The low so de eloped is simila o a he mal plume, he Coulomb
o ce playing he ole o buoyancy. In he ollowing we will analyze he simila i ies and di e ences be ween he mal
and EHD plumes.
5
A. EHD equa ions
The gene al s eady s a e equa ions o a pe ec insula o liquid wi h pe mi i i y ǫ, kinema ic iscosi y νand
densi y ρa e:[14]
∇·E=q
ǫ(32)
∇·j= 0 (33)
∇·u= 0 (34)
(u·∇)u=−1
ρ∇p+ν∇2u+q
ρE(35)
he e Eis he elec ic ield, q he cha ge densi y, j he cu en densi y, u he liquid eloci y and p he p essu e.
The cu en densi y is gi en by:
j=KqE+qu−D∇q(36)
whe e Kis he ion mobili y and D he di usion coe icien . Usually in mos EHD p oblems di usion is neglec ed.[10]
B. Axisymme ic EHD plumes
The wo-dimensional EHD plumes ha e been conside ed in o he pape s,[8, 11] so we will es ic he e o he
axisymme ic case. This is he EHD low occu ing be ween a poin and a pla e elec ode a dis ance dapa . Ou
conce n is wi h he cen al egion o his EHD low: a om he injec ion sou ce and om he s agna ion poin on
he opposi e elec ode. In his egion he low is expec ed o ha e de eloped sel simila p o iles o eloci y and cha ge
because he d i e m KEq in he cu en is much smalle han he con ec ion one qu. We also conside ha he
elec ic ield is cons an and p ima ily xdi ec ed, and ha he e ec s o eci cula ing luid a e negligible. Simila i y
solu ions a e also possible o elec ic ields wi h a po en ial dependency in he low di ec ion.[15]
Wi h all hese assump ions equa ions (32-35) in cylind ical coo dina es (x, ) a e:
∂ u
∂x +∂
∂ = 0 (37)
u∂u
∂x + ∂u
∂ =ν
∂
∂ ( ∂u
∂ ) + 1
ρqE (38)
u∂q
∂x + ∂q
∂ = 0 (39)
wi h bounda y condi ions simila o hose o he he mal plume:
=∂u
∂ =∂q
∂ = 0 a = 0,(40)
u=q= 0 a → ∞ (41)
whe e is he dis ance om he axis o he plume. The elec ical cu en Imus be cons an a any sec ion o he
plume, hen I= 2πR∞
0qu d is cons an .
Equa ion (39) dese es some commen s. F om (36) and (33) i is,
∇·j=K(E·∇)q+ (u·∇)q+Kq2
ǫ= 0.(42)
Consequen ly, equa ion (39) is alid i he i s e m, d i , and he hi d e m, Coulomb epulsion, a e negligible
compa ed o he second one, con ec ion. The d i e m has wo con ibu ions
K(E·∇)q=KEx
∂q
∂x +KE
∂q
∂ ∼KV
d
q
x+KE
q
a.(43)
He e ais he adius o he cha ged zone, Vis he applied ol age and d he dis ance be ween he poin and he
opposi e elec ode. The longi udinal con ibu ion KEx∂q/∂x is negligible compa ed o u∂q/∂x, since in mos EHD
lows he obse ed eloci ies a e usually an o de o magni ude g ea e han he ion d i eloci ies.[8, 10] On he o he

6
hand he adial pa KE q/a is o he same o de o magni ude han Coulomb epulsion, as i is eadily deduced om
es ima ing he adial ield E as he one p oduced by a cylinde o adius a:E ∼qa/2ǫ.
In conclusion, equa ion (39) holds i Coulomb epulsion is negligible compa ed o con ec ion. This condi ion can
be es ima ed as:
Kq2/ǫ
(u·∇)q∼Kq2/ǫ
uq/x ∼KIx
πǫa2u2≪1,(44)
whe e he elec ic cu en is I∼qπa2u. We discuss below he alidi y o his app oxima ion in some epo ed
expe imen s.
Wi h he ollowing simila i y a iables and unc ions:
η=IE
2πρν31/4
x1/2(45)
Ψ = νx (η) (46)
q=I
2πνxg(η) (47)
he esul ing equa ions a e:
′′′
η+ −1
η ′
η′
+g= 0 (48)
( g)′= 0 (49)
wi h bounda y condi ions:
 ′
η′
(0) =
η(0) − ′(0)
2=g′(0) =
η
′
(∞) = g(∞) = 0 (50)
1 = Z+∞
−∞
′g dη (51)
These equa ions o he EHD plume a e he same as o he he mal plume in he limi P → ∞.Bu we know om
sec ion III ha his limi does no exis , being he eloci y di e gen . So in o de o ha e a meaning ul physical esul
we need o econside ou model.
As he he mal solu ion shows, he eloci y is ini e, al hough la ge, o ini e la ge P and l numbe . The e o e he
i s idea is o ake in o accoun he equi alen o he P and l numbe in he elec ical p oblem. This numbe will be
ν/D, being D he di usion coe icien o he elec ic cha ge. The inclusion o di usion would esol e ma hema ically
he singula i y.[15] Bu his is no physically sound. Typically D≃10−11m2/s, and he expansion o he cha ged
laye due o di usion would be D/u ≃10−11m. This alue is o be compa ed o he adius o he injec ing poin ,
abou 10 mic ons, ha gi es an o de o magni ude o he adius o he cha ged zone. Mo eo e in he simila i y
egion he d i e m is negligible in compa ison o he con ec i e anspo o cha ge ( ypically K∼10−9m2/V s and
E∼2×106V/m, so KE ∼2×10−3m/s while u∼1m/s).
Wha eally happens in p ac ical si ua ions is ha he plume en e s in o he simila i y zone wi h a cha ged egion
o ini e adius, in opposi ion o he assump ion o a poin sou ce o cha ge. This is due, i s , o he ini e adius
o he ip and, second, o he s ong Coulomb epulsion nea he injec ing ip, ha expands he cha ged egion o a
ce ain hickness ha will be la ge han he adius o cu a u e o he ip. The e o e in he EHD case he singula i y
is a oided in oducing a ini e adius o he cha ge densi y. The app oxima ion will be be e he hinne his adius,
o he wise equa ion (33) has o be sol ed o de e mine he cha ge dis ibu ion.
We ha e sol ed nume ically equa ions (48-49) assuming ha
g=g0 o η < a
0 o η > a (52)
The eloci y and he cha ge a he cen e line a e, using he bes i o he nume ical da a:
( ′
η)η=0 = 0.51 + 0.65p−ln a2, g0= 0.60a−2(53)
7
TABLE I: Veloci y and empe a u e a he cen al line o di e en P and l numbe s in axisymme ic plumes.
As i is clea om his equa ion and (29) we may go om he he mal o he EHD plume jus subs i u ing P wi h
a−2.The physical eason is he ex en o he olume o ce. Ou side he limi adius 1/√P he e is no any buoyancy
o ce in he he mal plume, nei he is he e any elec ic o ce ou side he adius ain he EHD case.
Veloci y and hickness o he plume a e:
u= ′
η(0) IE
2πρν 1/2
(54)
∆ = 2πρν3
IE 1/4
x1/2(55)
I we de ine an elec ical G asho numbe G = qEx3/ρν2g(0), he eloci y and he hickness ha e he same
exp ession as in he he mal case, equa ions (16) and (17). This elec ical G asho numbe is ela ed o he elec ic
cu en by:
G = IEx2
2πρν3(56)
an exp ession o be compa ed wi h i s he mal coun e pa , equa ion (15).We ha e summed up he esul s in able II.
In o de o p o ide he eade wi h a comple e o e iew o he mal and EHD plumes, we ha e included in his able
he esul s o he wo–dimensional case.[11, 13]
Some expe imen s ha e been ca ied ou in poin -plane as well as in blade o pla e geome y in ou g oup and
elsewhe e.[8, 9, 11, 16]
In blade o plane geome y he epo ed measu emen s ag ee wi h he equa ions lis ed in able II. Resul s o silicone
oil o iscosi y 50 cen is okes and densi y ρ= 930Kg/m3a e:[11] E= 106V/m, J = 2 ×10−7A/m and x∼1cm.
which gi es u∼9cm/s and ∆ ∼4mm. The co esponding Reynolds numbe u∆/ν is 7.3. The plume is expec ed o
be lamina , as obse ed.
In axisymme ic cases s able je s ha e no ye been obse ed.[9, 17] We ha e[17] I= 4 ×10−8A, E = 3 ×105V/m
and x∼30mm. Equa ions (54) and (55) gi e u∼1m/s and ∆ ∼1mm so he Reynolds numbe is he e (νwas
27 ×10−6m2/s) close o 40. Simila alues ha e been ob ained in ou labo a o y. F om his da a i is clea ha
condi ion (44) is no ul illed in hese expe imen s. The ole o Coulomb epulsion seems o be impo an in he
de e mina ion o he cha ged egion hickness, bo h nea he ip whe e i expands e y quickly and a om he
elec odes. Whe he he e could be expe imen al condi ions in which a s eady lamina axisymme ic EHD plume is
obse able o no is a ma e o in es iga ion.
V. CONCLUSION
We ha e shown ha a pa icula solu ion o bounda y laye equa ions desc ibes bo h EHD plumes and he mal
plumes o e y la ge P and l numbe . Fo axisymme ic plumes he eloci y a he axis becomes in ini y o P → ∞,
bu in any case his di e gence is e y weak, i goes as ln √P . This esul is no di ec ly use ul o he EHD case.
The analogue o P and l numbe in EHD is he a io be ween he di usion coe icien and iscosi y. Bu di usion is
no an impo an e ec in EHD. An app oxima e solu ion o EHD axisymme ic plumes can be ound supposing ha
he cha ged egion has a ini e, al hough small, adius. The in e se o he cha ge laye c oss sec ion plays he ole o
an e ec i e P and l numbe . The eloci y o he EHD plume inc eases wi hou limi as his hickness dec eases.
VI. ACKNOWLEDGMENTS
This wo k was ca ied ou wi h inancial suppo om DGICYT (Spanish Go e nmen Agency) PB93-1182 and
Fundacin Cma a.
8
P ( ′/η)(0) θ(0)
0.7 0.938 0.482
10 1.40 5.63
1021.79 53.1
1032.13 520.
1042.41 5152.
9
TABLE II: Summa y o o mulae o EHD and he mal plumes. Qand Jdeno e, in he plane case, hea lux pe uni leng h
and elec ic cu en pe uni leng h, espec i ely.
Plane plume Axisymme ic plume
The mal Elec ical The mal Elec ical
G asho G = Qgβx3
2√2ρcpν3!4/5
G = JEx3
2√2ρν34/5
G = Qgβx2
2πρν3cp
G = IEx2
2πρν3
Thickness ∆ = 4
G 1/4
x∆ = 1
G 1/4
x
Veloci y u=2ν
xG 1/2 ′(η)u=ν
xG 1/2 ′(η)
η
Tempe a u e
Cha ge
T−Ta=ν2G θ(η)
gβx3q=ρν2G g(η)
Ex3T−Ta=ν2G θ(η)
gβx3q=ρν2G g(η)
Ex3
[1] W. J. Mo gan, “Con ec ion Plumes in he Lowe Man le,” Na u e 230, 43 (1971).
[2] J. A. Whi ehead, “Fluid Models o Geological Ho spo s,” Annual Re iew o Fluid Mechanics 20, 61 (1988).
[3] L. Pe a and B. Gebha , “On he S abili y o Lamina Plumes: Some Nume ical Solu ions and Expe imen s,” In e na ional
Jou nal o Hea and Mass T ans e 14, 975 (1971).
[4] B. Gebha , Y. Jalu ia, R. L. Mahajan, and B. Sammakia, Buoyancy-Induced Flows and T anspo (Hemisphe e Publishing
Co po a ion, New Yo k, 1988).
[5] D. B. Spalding and R. G. C uddace, “Theo y o he S eady Lamina Buoyan Flow Abo e a Line Hea Sou ce in a Fluid
o La ge P and l Numbe and Tempe a u e–Dependen Viscosi y,” In e na ional Jou nal o Hea and Mass T ans e 3, 55
(1961).
[6] H. K. Kuiken and Z. Ro em, “Asymp o ic solu ion o plume a e y la ge and small P and l numbe s,” Jou nal o Fluid
Mechanics 45, 585 (1971).
[7] K. V. Rao, B. F. A maly, and T. S. Chen, “Mixed Con ec ion Plumes A ising F om a The mal Poin Sou ce,” T ansac ions
o he ASME 107, 720 (1985).
[8] F. M. J. McCluskey and A. T. P´e ez, “The Elec ohyd odynamic Plume Be ween a Line Sou ce o Ions and a Fla Pla e,”
IEEE T ansac ions on Elec ical Insula ion 27, 334 (1992).
[9] B. Mal aison, P. A en and A. T. P´e ez, “Panaches cha g´es ´esul an de l’injec ion d’ions dans un liquide isolan pa une
lame ou une poin e plac´ee en ace d’un plan,” Jou nal de Physique III F ance 4, 75 (1994).
[10] A. Cas ellanos, “Coulomb-d i en Con ec ion in Elec ohyd odynamics,” IEEE T ansac ions on Elec ical Insula ions 26,
1201 (1991).
[11] A. T. P´e ez, P. A. V´azquez, and A. Cas ellanos, “Dynamics and Linea S abili y o Cha ged Je s in Dielec ic Liquids,”
IEEE T ansac ions on Indus y Applica ions 31, 761 (1995).
[12] M. Haida a and P. A en, “Role o EHD Mo ion in he Elec ical Conduc ion o Liquids in a Blade-Plane Geome y,”
IEEE T ansac ions on Indus y Applica ions 21, 709 (1985).
[13] Y. Jalu ia, Na u al Con ec ion Hea and Mass T ans e (Pe gamon P ess, Ox o d, 1980).
[14] P. A en and R. Mo eau, “S abili ´e ´elec ohyd odynamique des liquids isolan s soumis ´a une injec ion unipolai e,” Jou nal
de M´ecanique 11, 471 (1972).
[15] A. Zhakin, “Elec ocon ec i e Je s in Liquid Dielec ics,” Mekhanica Zhigkos y i Gaza. Iz es ia Academy Science USSR
6, 13 (1984).
[16] T. Takashima, R. Hanaoka, I. R., and A. Oh subo, “I-V Cha ac e is ics and Liquid Mo ion in Needle- o-Plane and Razo
Blade- o-Plane Con igu a ions in T ans o me Oil and Liquid Ni ogen,” IEEE T ansac ions on Elec onic Ins umen a ion
23, 645 (1988).
[17] B. A en, P.; Mal aison and M. Zahn, “Elec ohyd odynamic Plumes in Poin -Plane Geome y,” P oceedings 1993 IEEE
ICDL 534 (1993).