Dynamics of slender viscous dielectric liquid bridges subjected to axial AC fields

Jou nal o ELECTROSTATICS ELSEVIER Jou nal o Elec os a ics 42 (1997) 259 278 Dynamics o slende iscous dielec ic liquid b idges subjec ed o axial AC ields F.J. Ga cia a'b'*, A. Cas ellanos a, H. Gonz/dez a'c Depa amen o de Elec bnica y Elec omagne ismo, Facul ad de Fisica, Uni e sidad de Se illa, A da. Reina Me cedes s/n, 41012 Se illa, Spain bDepa amen o de Fisica Aplicada, E. U. 1. "i2 A., Uni e sidad de Se illa~ C a, U e a km 1, 41013 Se~,illa, Spain Depa amen o de Fisica Aplicada, E. S. L, Uni e sidad de Se illa, A :da. Reina Me cedes sin 41012 Se il/a, Spain Recei ed 25 No embe 1996; ecei ed in e ised o m 14 May 1997; accep ed 14 May 1997 Abs ac An analysis o slende axisymme ic liquid b idges is pe o med on he basis o one-dimen- sional models ecen ly de i ed, and gene alized he e o include he e ec o dielec ic o ces a he in e ace. The na u al equencies and s abili y c i e ia in he absence o g a i y a e ob ained. In he in iscid case, esul s a e compa ed wi h he known exac linea solu ions o he co esponding h ee-dimensional p oblem. ~ 1997 Else ie Science B.V. Keywo ds: Elec odynamics; Liquid b idge; S abili y c i e ia; Dielec ic liquid; Na u al equency 1. In oduc ion Recen ly, he e has been a enewed in e es in he loa ing zone echnique unde mic og a i y condi ions, owing o i s employmen in high-pu e monoc ys al manu ac- u ing. In his echnique, a liquid b idge is o med be ween wo solid suppo s. The s abili y o hese liquid b idges is go e ned by he su ace ension, and many pape s ha e been dedica ed o s udy his e ec (see Re . [1] and e e ences he ein). Mo e ecen ly, he e ec o an elec ic ield upon he s a ic s abili y o dielec ic liquid b idges has been conside ed bo h expe imen ally and heo e ically [2-5]. Also a linea dynamic analysis o in iscid liquid b idges subjec ed o elec ic ields has been pe o med [6]. Howe e , when we y o sol e he dynamics o iscous liquid b idges * Co esponding au ho . Tel.: -I- 34 95 4557910. 0304-3886/97/$17.00 © 1997 Else ie Science B.V. All igh s ese ed. PII S0304-3 886(97) 00 1 54-X 260 F.J. Ga cia e aL/Jou nal o Elec os a ics 42 (1997) 259-278 on he basis o he gene al h ee-dimensional (3-D) hyd odynamic equa ions, g ea di icul ies a ise. E en in he absence o elec ic ields, only pa ial esul s a e known [7-11]. In he la e case, hese di icul ies ha e been success ully ci cum en ed using one-dimensional (I-D) models [12]. In he absence o elec ic ields, he linea app oach shows ha liquid je s 1-13] as well as cylind ical liquid b idges [7] a e uns able unde axisymme ic pe u ba ions whose wa eleng h o heigh is g ea e han he pe ime e o he undis u bed column. Expe imen al obse a ions also show ha he b eaking p ocess is axisymme ic E5]. This jus i ies conside ing only axisymme ic mo ions. The smallness o he a io o he adius o he ini ial wa eleng h o heigh o he column, allows ob aining 1-D models ha g ea ly simpli y he s udy o hese columns [12, 14, 15]. Elec ical o ces ac ing upon he pola iza ion cha ges p esen a he in e ace inc ease he minimum ini ial wa eleng h o heigh below which he column is s able. Thus, mo e slende columns a e easible. The de i a ion o 1-D models is he e gene alized o include he e ec o dielec ic o ces a he in e ace. In his wo k, we s udy liquid b idges, ei he in iscid o iscous, on he basis o 1-D models. In Sec ion 2 he 1-D models a e gene alized o include he elec ic- ield e ec s. A linea s abili y analysis o hese models is de ailed in Sec ion 3. In Sec ion 4 some esul s a e ob ained and discussed. In o de o es he alidi y o he models in he p esence o an axial elec ic ield E, a compa ison is made wi h he exac 3-D esul s ob ained by Gonz~ lez e al. [6] o in iscid liquids. Also, some esul s a e p esen ed o iscous liquid b idges, o which no 3-D esul s a e known. Finally, he main conclusions a e d awn in Sec ion 5. 2. Equa ions Le us conside an axisymme ic liquid b idge o heigh L ancho ed o wo pa allel coaxial disks, he ancho s being o adius R. The liquid is assumed o be incomp ess- ible, wi h uni o m densi y p and iscosi y k (see Fig. 1). The liquid b idge, con ined by he su ace ension a, is supposed o be in a ze o-g a i y en i onmen . Quan i ies ha e been made dimensionless aking he scales: R o bo h he adial and axial leng hs, 'Z ?" ~e ~30 Fig. 1. Schema ic desc ip ion o an axisymme ic liquid b idge subjec ed o an AC axial elec ic ield. F.J. Ga cia e al./Jou nal o Elec os a ics 42 (1997) 259-278 261 and z; (pR3/ ) 1/2 o he ime ; [~/(pR)] 1/2 o bo h he adial and axial eloci ies, V and W; and a/R o he p essu e. The bulk equa ions and bounda y condi ions o he 3-D axisymme ic p oblem we e gi en by Mesegue [14]. The di icul y o such equa ions can be ci cum- en ed h ough he use o he so-called one-dimensional models. Ga cia and Cas ellanos ha e ecen ly de i ed a se o 1-D models in he con ex o liquid je s [ 15], which has been ex ended o liquid b idges as well [12]. These models a e a good app oxima ion o he 3-D equa ions as long as he liquid column is slende . The ela i e e o in oduced in neglec ing small e ms in he e e ed de i a ion can be exp essed as a nega i e powe o he nondimensional axial leng h scale 2, de ined as he ypical axial leng h o he sys em di ided by R. The e o e, he esul s gi en by he 1-D models a e expec ed o be good o a la ge alue o 2. No ice ha 2 does no coincide necessa ily wi h he slende ness o he b idge, A = L/(2R), a nondimen- sional numbe which measu es he leng h o he b idge. Ins ead, A is a maximum alue o 2 [12]. Two kinds o 1-D models can be conside ed, acco ding o hei dependen a iables: he mean- eloci y models and he pa abolic model. 2.1. Mean- eloci y models The mean- eloci y models ha e he shape o he in e ace = F(z, ) and he mean axial eloci y on a slice l,V(z, ) as a iables. All o hem mus sa is y he kinema ic condi ion (F2), + (FZJ,~)z = 0, (1) whe e he subsc ip s and z indica e de i a i es wi h espec o ime and he axial coo dina e, espec i ely. No app oxima ions ha e been made o ob ain his equa ion. The o he equa ion, which is de i ed om he Na ie -S okes equa ions, is cha ac e - is ic o each model. The simples and bes known I-D model o in iscid liquids is he in iscid slice model, de i ed by Lee [16] o je s, and ex ended o in iscid liquid b idges by Mesegue [14]. The ela i e e o o his model, de ined as he o de o he neglec ed e ms di ided by he conse ed ones o he same na u e, is 2- 2. This model has been ecen ly gene alized o iscous liquid je s [15, 17] and b idges [12], ha ing he same ela i e e o in he iscous e ms. The iscous Lee model is gi en by F2(I/V q- WWz) = - F2(g.n)z q- 3C(FZW~)~, (2) whe e C = p/(paR) 1/2 is he Ohneso ge numbe , which shows he a io o iscous o capilla y o ces; n is he uni a y ec o no mal o he in e ace, and he mean cu a u e V. n, which gi es he capilla y p essu e jump, akes he o m .. - (I + :~) 1/: 1 +/~ " 262 F.J, Ga cia e al./Jou nal o Elec os a ics 42 (1997) 259 278 A mo e sophis ica ed app oach is he Cosse a model, de i ed by G een [18], which eads F2(I~, + WWO [1Fa(l~,z ½ -2 - - w~ + w w=)]= F2(V.n)~ + C[3(F2I~z), 1 4 - = - . -- ~ F W .... - F3F l~ -- ½(F3F~ + 3F2F2)W=]. (4) The ela i e e o o his model is 2-4 in he ine ial e ms, bu 2-2 in he iscous ones. Al hough i is mo e accu a e han he Lee model o small iscosi ies, he e o inc eases o mode a e o la ge alues o he Ohneso ge numbe . A new model, which may be called a e aged model [12], co ec s he men ioned de iciency, by es ima ing he iscous e ms ha we e inconsis en ly neglec ed in he Cosse a model. I yields Fz(I~, + WW~) [~F4(~ z ½ -2 - - w~ + w w=)L = _ F2(V.n)z + C{3(F2I~)z + 3 [(F3F~z - 3FZF~)l~z]z}. (5) This model can be ob ained by a e aging he equa ions o he pa abolic model (see below). I s ela i e e o is 2 -4 in bo h he ine ial and iscous e ms. Thus, i imp o es he Lee model o any Ohneso ge numbe . To hese equa ions, which a e common o liquid je s, some app op ia e bounda y condi ions mus be added, accoun ing o he p esence o igid walls [12, 14]. The ancho ing o he disks yields F(z = _A, ) = 1. (6) The impene abili y o he disks gi es (z = +A, ) = 0. (7) Finally, he ancho ing o he disks (6) can be pu in e ms o 1~ by means o he kinema ic condi ion (1): ~z(Z = +A, ) = 0. (8) The mean- eloci y models app oxima e he adial and axial eloci ies, wi hin a ela- i e e o o 2-2, as ollows: V( , z, ) = - ½ l~(z, ), (9) W( , z, ) = '(z, ). (10) No ice ha he condi ion o no-slip on he disks is hen ul illed by Eq. (8). The e o e, he numbe o independen bounda y condi ions a he disks is ou , he same as he di e en ial o de in z o all hese models. F.J. Ga cia e al./Jou nal o Elec os a ics 42 (1997) 259-278 263 2.2. Pa abolic model The pa abolic model [-12] p o ides an imp o ed, wo- e m adial app oxima ion o he eloci y ield, whose ela i e e o is 2-4: V( , z, ) = - ½ Wo~(Z, ) - ~ 3W2~(z, ), ll 1) W(F, 2, ) = Wo(z , ) -}- ½ ZWz(z, ). (12) I s a iables a e F, Wo, and W2, which do no depend on . In e ms o hese, # can be eadily ound by a e aging (12) on a slice. Th ee equa ions de ine his model, he i s o which is he co esponding e sion o he kinema ic condi ion: F, + ½FWoz + F~Wo + ~F3W2~ + ½F2F~W: = 0, (13) he o he wo coming om he momen um equa ions: (Wo, + WoWo~) - [¼ V2(Wo z -½ WL + WoWozz)L = - (V. n)~ + C [2 Wozz + 2 W2 1 2 -- gF W0~zzz FF~Wo.z~ 1 z - _ -~(F~ + FFz~)Woz~ + ¼F2W2=z + {FF~W2~ + (FZ~ + FFz~)W2] (14) and !W F4(W2 + WoW2z + 2 o zz + ½ WoWozzz) 1 4. = C(4FZWo~z + 24FF~Woz - 8F2W2 + ~F Wo .... - 4FEF~Wo~z + 3F4Wzz~ + 14FSF~W2~ + 8FZFZW2). (15) The ela i e e o o he pa abolic model is 2-4 in bo h he ine ial and iscous e ms. Some app op ia e bounda y condi ions a he disks a e necessa y o sol e he p oblem. The ancho ing condi ions a he edge o he igid disks apply again: F(z = ___A, ) = 1. (16) The impene abili y o he disks implies Wo(z = +_A, ) = O, W2(z = ±A, ) = 0. (17) I he liquid is iscous, he no-slip condi ion on he disks mus be sa is ied, which gi es Woz(Z = ±A, ) = O, Wz~(Z = ±A, ) = 0. (18) Owing o he kinema ic condi ion, only eigh o he condi ions (16)-(18) a e indepen- den , he same as he di e en ial o de in z o he model when C 4: 0. 264 F.J. Ga cia e al./Jou nal o Elec os a ics 42 (1997) 259-278 Fo in iscid liquids, W2 can be decoupled om he p oblem, which is o mula ed in e ms o F and Wo, and i s di e en ial o de educes o six. In his limi he liquid is allowed o slip on he disks, and he ou condi ions gi en by Eq. (18) a e no longe alid. Ins ead, he kinema ic condi ion (13) along wi h (16) gi e us he o he wo condi ions necessa y o comple e he p oblem. The six condi ions a e hen Wo(z = +_A, ) = O, Wozz(Z = +_A, ) = O, and Wo~(Z = +_A, ) -- ~ Wozzz(Z = + A, ) = O. (19) (20) Con a y o he mean- eloci y models, he pa abolic model conse es he sensi i - i y o he 3-D equa ions and bounda y condi ions o he iscous o in iscid cha ac e o he p oblem. In pa icula , he bounda y condi ions as well as he di e en ial o de in z a e di e en depending on whe he C = 0 o no . 2.3. P esence o an axial AC elec ic ield Le us conside now ha an AC po en ial di e ence, wi h an e ec i e alue ~b o, is applied o he bounding pla es (elec odes) whe e ancho s a e welded on (see Fig. 1). Fo equencies much highe han he in e se o he cha ge elaxa ion ime, he only o ces o elec ical o igin ha a e o impo ance a e he dielec ic ones. We also assume ha he pe iod o he AC ield is much smalle han he ypical capilla y ime. The e o e, no pa ame ic esonance is expec ed [19]. The exac o mula ion o he p oblem o s a ic condi ions has been gi en in Re . [2] and o dynamic condi ions in Re . [6]. In bo h cases, he elec ic ield en e s he o mula ion o he hyd odynamic equa ions only in he no mal s ess bounda y condi ion. The elec ical p essu e has o be added o he capilla y one ollowing he eplacemen ule V n~V.n 4A2zA a 2 1 z • - [e(:q~, - ~ "2 - Fz~,~Oz)], (21) whe e ~ is he a io o he elec ic pe mi i i y o he one o he inne liquid Cln, i.e., e = 1 inside he liquid b idge and/~ = eo/~in ou side; he elec ic ield has been made dimensionless using he scale 4~o/L (so q)oR/L o he elec ic po en ial q,); A deno es he jump o a quan i y h ough he in e ace; and g = e~,~ZR/(aL z) is he elec ic Bond numbe , which shows he a io be ween he elec ical p essu e and he capilla y p essu e. The Maxwell equa ions educe o bo h he di e gence and cu l o he elec ic ield being ze o. In oducing he elec ical po en ial ~, i mus ul ill he Laplace equa ion 724 = 0, (22) as well as he app op ia e bounda y condi ions in nondimensional o m. These come om he imposi ion o he po en ial di e ence be ween he elec odes, • ( , z = A, ) = A, ~( , z = -- A, ) = - A; (23) F.J. Ga cia e al./Jou nal o Elec os a ics 42 (1997) 259-278 265 he egula i y o he po en ial along he axis o symme y and i s asymp o ic alue a om i , ~b(0, z, ) ini e, lim ~b( , z, ) = z; (24) he con inui y o he angen ial componen o he elec ic ield h ough he in e ace, Aq~ = 0; (25) and he con inui y o he no mal componen o he displacemen ield in he absence o ee su ace cha ge, A[e(- 4)~ + F~q~=)] = 0. (26) 3. Linea dynamics Gonzb, lez e al. [6] ha e sol ed he linea ized 3-D p oblem associa ed wi h he cylind ical in iscid liquid b idge subjec ed o an AC elec ic ield. A simila ea men can be applied o he 1-D models shown abo e. This p ocedu e allows us o check he e o o he solu ions p o ided by such models. Fu he mo e, he linea 1-D p oblem can be sol ed o iscous liquids in he same manne as o in iscid ones. The same is no ue o he 3-D bounda y p oblem, which is much mo e di icul o sol e in he iscous case, because o he addi ional bounda y condi ions a he disks [9]. The s a ic cylind ical solu ion is cha ac e ized by ha ing ze o eloci y, cons an alues o he p essu e and shape o he in e ace, and he elec ic po en ial o a plane capaci o . Le q be he g ea es ini ial de ia ion o he shape o he in e ace om he cylind ical one. The ollowing pe u ba i e solu ion is p oposed: F = 1 + q , I~ = ~W, Wo = qWo, W2 = ~]W2, ~ = Z + qq~, (27) whe e he pe u ba i e pa ame e /is e y small ( /~ 1). I his solu ion is in oduced in he abo e equa ions and e ms o o de q2 o highe a e neglec ed, he esul ing p oblem is linea . The e o e, i is expec ed a ime dependency o he o m [~(z, ), Wo(Z, ), w2(z, ), (z, ), c~( , z, )] = Re {em [w(z), ~o(Z), ~2(z), (z), ~o( , z)] }, (28) whe e w, ~o, ~2, , and ~ a e complex; and (2 = ~ + ko is a complex eigen alue, whose eal and imagina y pa s ep esen he g ow h ac o ~ and he oscilla ion equency co, espec i ely. Eqs. (22)-(26) lead o a linea p oblem o he pe u ba ion o he elec ic po en ial ~, gi en by q$,, + 1 q~ + ~zz = 0, (29) g q~( , _ A) = 0, (30) 266 F.J. Ga cia e al./Jou nal o Elec os a ics 42 (1997) 259-278 ~(0, z) ini e, lim ~( , z) = 0, ~oo Aq] = 0, a(~&) = (zx~) ', whe e he p imes s and o o al de i a i es wi h espec o z. The mos gene al solu ion o he sys em o Eqs. (29)-(33) is ~)( , z) = ~ A,~,( )sin[x,(z + A)], n-1 whe e ~n( ) = ~Ko(xn )/Ko(xn) (Io(xn )/Io(Xn) (31) (32) (33) (34) and (35) No ice ha Eq. (34) couples he elec ic p oblem o he mechanical one. 3.1. Mean- eloci y models P oceeding simila ly, he applica ion o Eqs. (27) and (28) o he kinema ic condi- ion (1) allows us o ob ain =-~', (38) which is alid o he Lee, Cosse a , and a e aged models. The linea coun e pa s o Eqs. (2), (4), and (5) can be ob ained by applying he same p ocedu e o hese equa ions, and o he de ini ion o he capilla y p essu e (3) as well. Eq. (38) allows elimina ing he a iable , which leads o he ollowing linea 1-D momen um equa ion: e2~ I + c1~" + Co~ = - 2Zs'2A(~z), (39) whe e he coe icien s Co, ca, and c2 depend on 2 and C. Thei alues o each model a e as ollows. Lee: Co = 2 22, cl = 1 - 6C 2, c2 = 1; (40) n~ x, - 2A (37) i ~>l, i .%< 1; (36) F.J. Ga cia e al./Jou nal o Elec os a ics 42 ¢1997) 259-278 267 Cosse a : co = 2g? 2, cl = 1 -- 6C 2 - ¼ 22, C 2 = 1 + ¼C 2; (41) a e aged: 1 2 C0 = 2(2 2, c1 1 - 6C 2 - J2 , c,_ = 1. ~42) Finally, he ou bounda y condi ions a he disks (7) and (8) become ~'(_+ A) = 0, w'(+A) = 0. (43) Al hough he e olu ion o a liquid b idge is an ini ial- alue p oblem wi h espec o ime, we do no conside ini ial condi ions he e. Ins ead, we add ess ou in e es o a modal analysis, om which a coun able in ini e se o eigen alues 2m as well as hei co esponding eigen unc ions a e ob ained. The inal solu ion o a pa icula p oblem, cha ac e ized by i s ini ial condi ions, would be an app op ia e supe posi ion o hese eigenmodes. The linea p oblem, gi en by Eqs. (29)-(34) and (38)-(43), has well-de ined pa i y wi h espec o z, o he equa ions and bounda y condi ions a e in a ian unde he change o sign o z. The e o e, he eigenmodes can be classi ied in ei he an isymme ic o symme ic, acco ding o he pa i y o he shape o he in e ace wi h espec o z. No e ha bo h he mean axial eloci y and he elec ic po en ial ha e opposi e pa i y o ha o he shape o he in e ace, since applying a de i a i e wi h espec o z o any quan i y changes i s pa i y. Owing o he analogy in he ea men o bo h kinds o modes, he analysis p esen ed below only deals wi h he an isymme ic ones. A e wa ds, a ecipe is p o ided o ob ain he analogous esul s o he symme ic eigenmodes. Using Eq. (35) o e alua e he igh -hand side o Eq. (39) yields - 2g 2A(e,~z:) = 2AzI2(Ae) ~ A,x2sin[x,(z + A)], n= 1 (44) which is he inhomogeneous e m o he di e en ial equa ion (39). The e o e, a gene al solu ion o his inhomogeneous p oblem is ~ = w h + ~ p, (45) whe e w h is he gene al solu ion o he homogeneous p oblem, i.e., Eq. (39) wi h Z = 0; and } p is a pa icula solu ion o he inhomogeneous one. The gene al solu ion o he homogeneous p oblem is 2 ~h = ~ [s~Cj cosh(/cjz) + N'j sinh( cjz)], (46) j=l whe e _+ ~cj (wi h j = 1, 2) a e he ou complex oo s o he biquad a ic equa ion C2 K:4 q'- Cl K2 q- C O = O. (47) 274 F.~ Ga cia e al./Jou nal o Elec os a ics 42 (1997) 259-278 0.8 ~,l i i i ~ d ~ i i i i i i , J 0.6- ,~, F- ~ _ 0.2 0.4 -- "~"~..C = 0 Ac ~ - c=o.2"~-.~ , /~ o.1 0.2 -- "~-,.., .-.,'~ C= 0.2 _ co o.o ~ ~ o.o c~ -0.2 ~- C=O -0.4 -0.1 --~-C = 0.2 iO~ 6 7" iO~ 2 -0.8 I I I I I I J i i i [ i i i i i 2 3 4 5 A Fig. 6. Oscilla ion equency (dashed) and g ow h ac o (solid lines) o he i s mode e sus he slende - ness, o ,8 = 0.58 and X = 2.4, as gi en by he a e aged model. The labels on he cu es show he Ohneso ge numbe (C = 0 and C = 0.2). In o de o show he e ec o iscosi y in he p esence o an elec ic ield, a plo o he i s eigen alue (m = 1) e sus A is p esen ed in Fig. 6, o in iscid (C = 0) and iscous (C = 0.2) liquids. These cu es ha e been compu ed wi h he a e aged model, o /~ = 0.58 and ;( = 2.4. Obse e ha he iscosi y has no in luence on he s abili y o he b idge, since he c i ical slende ness Ac is no a ec ed by he alue o C. Howe e , he dynamics changes d ama ically. When he liquid is iscous, he s able zone o he mode (A < Ac) sepa a es in o wo zones o di e en beha io . Fo sho enough b idges, he e is a zone o damped oscilla ions (c~ < 0, co ~ 0). G ea e alues o A lead o lowe - equency oscilla ions, un il a second zone o pu e damping (c~ < 0, co = 0) is eached. The mo e iscous he liquid is, he sho e he liquid b idge mus be o see oscilla ions. Finally, in he uns able zone (A > Ac), he changes in oduced by he iscosi y a e only quan i a i e: he g ow h ac o ~ dec eases as C inc eases. In he abo e-ci ed expe imen al con igu a ion [5] he iscosi y o he liquid leads o a alue o he Ohneso ge numbe C = 4.6. H~ ea e , we will es ic ou discussion o ha alue. When such a b idge is s able, i 'is obse ed in he expe imen s ha all dis u bances a e ape iodically damped. Besides, he b eaking is app eciably slowed down by he e ec o iscosi y. No oscilla ions ha e been obse ed in any case. In Fig. 7 he g ow h ac o o he i s an isymme ic mode is plo ed e sus A o C = 4.6,/~ = 0.58, g = 0 and 2.4, as gi en by each 1-D model. No e ha , as expec ed, he s abili y limi s a e he same as in he in iscid case. Fo he s udied ange o slende ness, he eigen alue is eal, which means ha his mode does no oscilla e. When he b idge is s able, all dis u bances a e ape iodically damped. When i is uns able, he p edic ed g ow h ac o is eal, bu signi ican ly smalle han o in iscid liquids. This explains he slowing down ha we ha e obse ed in he b eaking o liquid b idges in ou labo a o y, using he expe imen al se up desc ibed in Re . [5]. No ice ha he esul s gi en by he di e en models a e close, he di e ences gi ing an es ima e o he e o s. Fo ou iscous liquid b idge, he esul s o he Lee model a e F.J. Ga cia e al./'Jou nal o/Elec os a ics 42 (1997) 259 278 275 0.04 0.02 0.00 (3I, -0.02 -0.04 -0.06 ........ ~-~ I -0.08 , i _ _ 2 3 10 W¢" [ .......... ,00 0- I 0.015 - - -- i 0.010 4.3 4.5 4.7 F ..... T-- 4 5 6 7 8 9 A Fig. 7. G ow h ac o o he i s an isymme ic mode as a unc ion o he slende ness, o C = 4.6, # = 0.58, Z = 0 and Z = 2.4. Gi en by he Lee and a e aged (sho dashed), Cosse a (poin -dashed), and pa abolic (solid lines) models. A de ail is magni ied. 0.8 0.4 0.0 i i -I .0 -0.5 0.0 z/A Fig. 8. Shape o a hal o he in e ace o he i s an isymme ic mode (m = 11 e sus z'A. o .4 = 10, # = 0.58, and Z = 0, , ..., 6 as gi en by he pa abolic model o C = 4.6. he same as he ones o he a e aged model, wi hin he plo accu acy. Fo la ge C wi hou elec ic ield [12], he 3-D solu ion is be ween he a e aged and pa abolic ones when A > Ac. The Cosse a model clea ly unde es ima es he co ec alues o ~, owing o he inconsis ency in he de i a ion o i s iscous e ms [15]. Al hough he 3-D solu ion is no a ailable o Z -¢ 0, a de ail o he 1-D p edic ions is included in he igu e, which shows simila ela i e posi ions o he cu es. The di e ences gi e us an es ima e o he e o o he 1-D app oxima ions. The e ec o he elec ic ield on he shape o he in e ace o iscous liquids is shown in Fig. 8. In he absence o elec ic ield, he e ec o iscosi y is o inc ease he 276 F.J. Ga cia e al./Jou nal o Elec os a ics 42 (1997) 259 278 ypical axial leng h 21-12]. Fo C = 4.6, i is e iden ha 2 -- A. In his case, inc easing Z canno a ec 2. Ins ead, he main e ec o he elec ic ield on he b idge shape is ha he posi ions o he g ea es and smalles c oss sec ions a e close o he disks. This is also obse able o any alue o C as long as 2 -~ A. 5. Conclusions A se o new 1-D models o iscous je s, also ex ended o liquid b idges, has been gene alized o include he e ec o dielec ic o ces. A linea s abili y analysis based on hese models has been pe o med. In he in iscid case, a quan i a i e compa ison o he 1-D esul s wi h he 3-D ones show he ela ionship be ween he e o o such models and he ypical axial leng h 2. Fo slende enough b idges, 2 inc eases wi h he elec ic ield, and he e o dec eases acco dingly. In he iscous case in he p esence o elec ic ields, no 3-D solu ions a e a ailable. Simila ly o he case )~ = 0, he linea esul s ob ained wi h he 1-D models p edic ha he iscosi y e ec does no change he s abili y c i e ia. On he con a y, he dynamics is g ea ly a ec ed, as iscosi y inhibi s he oscilla ions o s able liquid b idges and slows down he up u e o uns able ones, in quali a i e ag eemen wi h he expe imen s done in ou labo a o y. I can be concluded om he compa ison wi h he known exac 3-D linea solu ions, ha he 1-D models a e qui e adequa e o deal wi h he i s s ages o de elopmen o pe u ba ions. In gene al, he main aspec s conce ning he ela i e e o o hese models wi hou elec ic ield emain alid in i s p esence. The accu acy o hese models imp o es in he p esence o an AC axial elec ic ield, since he la e makes he ypical axial leng h inc ease. Besides, hey allow us o s udy he dynamics o iscous liquid b idges subjec ed o axial elec ic ields, which o he momen has de ied a 3-D app oach. Expe imen s a e now unde way o compa e he g ow h a e o he ins abili y in he linea s age wi h he p edic ions made by hese 1-D models. Acknowledgemen s The au ho s a e g a e ul o Angel Sanz and F ancisco Medina o help ul dis- cussions. This wo k has been suppo ed by he Spanish Di eccidn Gene al In e minis- e ial de Ciencia y Tecnologia unde con ac PB93-1182. Nomencla u e ~',, d, sgj, ~, Co, Cl, c2, in, h,, ~,, x,, ~c, Kj , wo, auxilia y cons an s, coe icien s, and unc ions linea coun e pa s o F, I/P, Wo, W2, spa ial dependence o , ~, Wo, We, q5 F.J. Ga cia e al./Jou nal o Elec os a ics 42 (1997) 259-278 277 C F L m n V-n R V W Wo, W2 2 6Q A ~;in ~;0 17 2 ~Lmax A Ac P O" ~b qo 0 Z u9 O, Qm Ohneso ge numbe (p/(paR) ~/z) nondimensional shape o he in e ace leng h o he liquid b idge index o modes ec o no mal o he in e ace nondimensional capilla y p essu e jump nondimensional adial coo dina e adius o he ancho s o he liquid b idge nondimensional ime nondimensional adial eloci y nondimensional axial eloci y nondimensional mean axial eloci y nondimensional eloci y a iables in he pa abolic model nondimensional axial coo dina e nondimensional g ow h ac o nondimensional pe mi i i y o he ou e zone (eo/~i,) ela i e e o o he eigen alue 2 applied o a quan i y, jump ac oss he in e ace nondimensional pe mi i i y pe mi i i y o he liquid pe mi i i y o he ou e zone small ampli ude o pe u ba ions ypical nondimensional axial leng h nondimensional wa eleng h o maximum g ow h in a je slende ness (L/2R) c i ical alue o he slende ness dynamic iscosi y o he liquid densi y o he liquid su ace ension o he in e ace nondimensional elec ic po en ial po en ial di e ence be ween elec odes elec ic bond numbe (Sin~2R/aL 2) nondimensional oscilla ion equency complex eigen alue (~ + i a) Re e ences [1] J.M. 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