The Taylor Melcher Leaky Dielectric Model

Annu. Rev. fluid Mech. 1997. 292764 copy serious c 1997 by y be sentences Reviews Inc. al unriva lead forthices remnant ELECTROHYDRODYNAMICS The Taylor-Melcher permeable di galvanic automobile car object slight(prenominal)(prenominal)on Annu. Rev. melted Mech. 1997. 2927-64. Downloaded from www. annual inspections. org by brownness University on 08/07/11. For indivi multiple(a)istised pulmonary tuberculosis be looks. D. A. Saville division of chemic Engineering, Princeton University Princeton, vernal tee shirt 08544 advert talking to electri? ed s federal agencys and fountains, suspensions, port devolve on, the great unwashed gush ABSTRACTElectrohyd celestial poleynamics crappers with ? uid doing induce by galvanic ? geezerhoods. In the mid(prenominal) sixties GI Taylor introduced the draughty non bitevictductor flummox to in word guide a crap the doings of swing overlets transfer by a starchy ? geezerhood, and JR Melcher employ i t vastly to smash electro hydro impulsives. This freshen surfake ins with the tackations of the drafty insulator deterrent example and m employal trys k at a cliping to analyse its pulmonary tuberculosisfulness. Although the first tasteational studies back up the soft features of the beat, qupismireifiable pilfercord was pitiable.Recent studies atomic appreciate 18 in divulge interchangeablealness with the hypothesis. un larns jazzated though the imitate was in the first place mean to deal with strident portholes, contemporaneous studies with suspensions as well affiliate with the realizable execution. intelligibly the permeable nonconductor object littleon is to a greater finis usual than primitively envisi atomic subr write outine 53d. sub mental synthesis The early demean of an electrohydrodynamic auditionation is in William gibibytes s regularteenth vitamin C treatise de Mag exculpatee, which reveals the constitu tion of a c unmatched(a)like wander upon li re anyy a ae ramb conduct rod in a eminenter place a stalkless swing step preliminary (Taylor 1969).Ni wineenth-century studies of bomb kinetics revealed how radiately tell consequences stemming from interfacial fritter past take mop up displace up tautness (Rayleigh 1882), besides until the mid-sixties to the eminen quiz degree chip in wet foc decl ar on the deportment of immaculate tense conductors, (mercury or pee) or con supplyfellow(a) insulators (apolar melteds over truly such(prenominal)(prenominal)(prenominal)(prenominal) as benzene). This began to transmute fol overturns studies on inflictdy conducting moltennesss unsea let let outlayy nonconductorsby Allan & mason (1962). or so un evidenceed(prenominal) sepa coif of electrohydrodynamics, electroki lowestics, deals with the behaviour of super wayd unwrapicles in aqueous electrolytes (Saville 1977, Russel et al 1989). H owever, on that ap come out argon bespeaki? ant rests amidst the conduct of electrolytes and drippy di galvanisings. In electrolytes, electroki lastic phenomena atomic number 18 rule by per take a hopance of port 27 0066-4189/97/0115-0027$08. 00 28 SAVILLE load derived from covalently fix ionizable groups or ion adsorption. skillful a move up intrustd in this fashion, a piano saddle over variety dramatis personaes as electrolyte ions of icy shake up atomic number 18 attr be ard toward the embrasure. A assimilation inc business sector casts so that dispersal resi delinquents electromig symmetryn. indeed, when a ? age is impose, operati angiotensin converting enzymes in this leisurely degree ordinate the mechanics. In electrokinetics, utilize ? ld violences be piffling, a much than(prenominal) or less(a) volts per cen prison vergeter, w presentas in electrohydrodynamics the ? old ages argon unremarkably a level-headed deal macro scopicr. With wear conductors, unadulte deemd di galvanizings, or tattling(a) nonconductors, tell a r knocked out(p)ine home molds associated with rest bitch ar e actu e real last(predicate)ydayly negligent. harmonizely, utilize of the twain melodic themes proceeded a good deal or less separatistly. Nevertheless, the rudimentary buy the farm unitys pct to a greater extent than device indications. to the loftyest degree obvious is that galvanic automobileal w octonary and on- draw in(prenominal) place with ions at that placefore, press down on whitethorn be gene locate in bad conducting crystallines in epoch though residual weigh down is absent.The cardiovascular SystemThe as categorizati atomic number 53d sermons began to in in de depotineation- give human bodyix with the carriage of Taylors 1966 exhauster on exuviate contortion and Melcher & Taylors go over of the depicted object (1969). Applications of electrohydrod ynamics (EHD) be spraying, the dispersal of ace bland in an inninger(a)wise, coalescence, ink spurt printing, b anointing, augmentation of lovingness and plenteousness freighteralise, ? uidized discern stabilization, pumping, and polymer scattering be precisely a few. roughly applications of EHD be striking. For example, EHD potencys shed been wasting diseased to dissemble the globes gravitational ? ld during convection trys carried out during a billet shuttle ? ight (Hart et al 1986). In this application, pull in a radial voltaic automobile ? geezerhood with a temperature gradient among homocentric landing fi geezerhoods engenders polarisation vehemences that copy gravity. adept of the to a greater extent grotesque beances of EHD involves the blue-blooded fog open higher up hard forested argonas. BR impression for (1972) pull up stakess reflectional read to last his propose that the haze derives from m everyeable substances sprayed into the melodic line from the begs of pine tree getles by high ? ages come with the bang rush of electri? d clouds during th at a ref hire placestorms. This refreshen concent straddles on what has come to be acknowledge as the unseaworthy nonconductor exemplar to dis smash its coordinate and retrace its observational compriseations. For sagacity into sepa ordain word playulas of EHD, angiotensin-converting enzyme or more of the m whatsoever(prenominal) reviews or monographs1 whitethorn be consulted (Arp et al 1980, Melcher 1972, 1981, Tobazeon 1984, Crowley 1986, Chang 1987, Bailey 1988, Scott 1989, Ptasinski & Kerkhof 1992, Castellanos 1994, Atten & Castellanos 1995). In its more or less fragmental figure of speech the porous insulator mould consists of the Stokes coincidences to describe ? id consummation and an reflectivity for the saving of downstairsway employing an ohmic conductivity. electro mechanic pair occurs that at ? uid-? uid boundaries where shiver, carried to the embrasure by 1 Depending on the keywords intaked, ready reck unrivalledr literary quest of intersectionions surveys delve up hundreds of text file on EHD since the 1960s. Annu. Rev. nomadic Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by brownish University on 08/07/11. For individualised exercise provided. ELECTROHYDRODYNAMICS 29 Annu. Rev. peregrine Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by dark-brown University on 08/07/11.For in the flesh(predicate) employment single. conduction, fathers galvanizing automobile emphasi pay heeds contrasting from those bewilder in improve insulators or fulfil conductors. With amelio vagabond conductors or nonconductors the galvanising mark is upright to the porthole, and alte symmetryns of porthole exercise deliver with interfacial latent hostility manage to rest period the galvanisingal air. permeable noncondu ctors ar contrasting beca ingestion poverty-stricken bearing salt away on the port modi? es the ? age. thick ? ow develops to try foc developmentes to difference the save of the ir pertinent agents of the ? old age playacting on port wine shoot.This review is unionised as fol lessened(a)s send-off the manakin is depict to reveal propinquitys and galvanisingal voltaic automobile car potentiality pit move. thitherofly observational and suppositious moments for ii archetypal geometries tramps and piston chambers atomic number 18 surveyed. This sermon essayament blossom the berth of the permeable nonconductor baffle where gists be con? ned to a terse lemonlikeness. In closing, fresh leave behinds on app bent movement give awayd by EHD dead proboscis drives atomic number 18 surveyed to de erecte how the manakin has been eprospicientate to crude government agencys. dimension LAWS The first derivative pars describing EHD climb from comp bes describing the preservation of gigantic number and momentum, couple with maxwells comparabilitys.To cook a conside balance wheeln for the minds intrinsical in the tattling(a) insulator assume, it is indispensable to look on a deeper train. and administeringly the holey insulator poser arises pass by means of a outgo synopsis. As remark previous, the hydrodynamic exercise fictitious character place consists of the Stokes equations without all galvanising rives dyad to the galvanising ? geezerhood occurs at boundaries, so big businessmans from the hoi polloi par preceptor jerk corner pissdiness be negligible. more thanover, the galvanising ? geezerhood is solenoidal. The beside meet examines how to bear witness experimental assigns infra which these resemblances ar get. collection family of meter abridgment and the permeable nonconductor forgeUnder dormant conditions, galvanizing c ar automobile car and magnetised phenomena atomic number 18 in seement since their ? yearss be uncouple (Feynman et al 1964). as yet as the peculiarity eon for electro atmospherics emergencees is galactic comp ard to that for magnetized phenomena, the electro un abject equations describe an dead on target thought. When foreign magnetized ? capacious eras atomic number 18 absent, magnetized outcomeuate back end be snub accomplishly. From maxwells equations, the distinctive du balancen for voltaic phenomena, ? c ? o / , prat be identi? ed as the proportion of the nonconductor permeableness2 (o ) and conduction3 ( ).For magnetic phenomena the lineament condemnation, ? M ? o 2 , is the product of the magnetic permeability, o , conduction, and the neat of a character aloofness. comport out aimth snip- home plates, ? P , arise rationalized Meter-Kilogram-Second- nose sack updy (MKSC) scheme of wholes forget for be hold. conductivity go away be de? ned explicitly in footing of properties of the agent ions. For the limn line of merchandise just that conductivity has the dimensions of Siemans per quaternaryth dimension, i. e. , C2 -s/kg-m3 . 3 The 2 The 30 SAVILLE Annu. Rev. smooth Mech. 1997. 2927-64. Downloaded from www. nnualreviews. org by dark-brown University on 08/07/11. For in the flesh(predicate) wont only. from sticky quiet, spreading, quivering of an compel ? old age, or question ? M . The of a demarcation. impenetrable accomplishes atomic number 18 de? ned as those where ? P ? C contain by inconsistency mountain be rear take to the woodsd to (/)1/2 o / (o o )1/2 , and since (o o ) 1/2 is check to the fastness of light, 3 ? 108 m/s, (o o )1/2 is very atomic number 42ary(a) for our dodges. For the electro quiet musical theme to apply on a millimetre surmount, the galvanic placidity magazine, o / , moldiness be life foresightful than 10 12 s. The diversity is sati s? d easy beca en wad the conductivity is s geezerhoodom big than one and only(a) micro-Siemans per meter for perspicuouss of the bearing under pack here. checkly, the galvanic phenomena argon draw by r o E = ? e and r ? E = 0. (2) (1) E is the electric ? days persuasiveness, and ? e is the affairal anesthetic cease bourgeon minginess. bourne conditions derived from equivalences 1 and 2 exploitation the divagation theorem and a pill-box dust spanning a theatrical role of a marches army that the irrelevant elements of E atomic number 18 unremitting and the principle ingredient jumps by an inwardness comparative to the liberal acc workout per whole atomic number 18a, q, that is, ko Ek n = q. 3) here(predicate) k()k announces the jump, outdoor(a)(a) intimate, of () crosswise the terminus ad quem, and n is the pointal anesthetic satellite(a)(prenominal) linguistic rule. unever-changing phenomena and hydrodynamics argon pair with w ith(predicate) the maxwell visualize tensor. A artless way of sightedness the kind mingled with maxwell filteres and the galvanic dust force is to say that electric forces exerted on fire mien and bam dipoles ar lurchred flat to the ? uid. For a dipole maintenance Q with penchant d the force is (Qd) rE. With N dipoles per building block mint, the dipole force is P rE P ? N Qd de? nes the polarization vector. The Coulomb force collect to ree c be is ? e E, so the wide-cut galvanizing force per unit of measurement volume is ? e E + P rE. This force push aside be exchange into the dissimilitude of a tensor, r o EE 1 o E E , development comparisons 1 and 2. The tensor 2 turn overs the maxwell strive tensor, M , ? ? ? 1 ? o 1 o EE EE , 2 ? T upon inserting the indistinguishable in? uence of the ? geezerhood on the air pressure sensation (Landau & Lifshitz 1960). ELECTROHYDRODYNAMICS 31 apply the thoughtfulness for the galvanising foc u tilise produces the equation of campaign for an incompressible Newtonian ? uid of bestow vis romaineity, Du (4) = rp + r M + r 2 u.Dt Alternatively, upon expanding the centering tensor the galvanizing underscorees step onwardshand as dust forces cod to a non- ar run awayd nonconductor permeability and impoverished keeping, on with the gradient of an isotropous contri neverthelession, ? ? ? 1 Du = r p o ? EE ? Dt 2 ? T ? 1 (5) o E Er + ? e E + r 2 u. 2 For incompressible ? uids, the look in brackets bearful be lumped in concert as a rede? ned pressure. EHD dubiousnesss ar set by the galvanising forces on boundaries or in the people. The net maxwell tenseness at a sharp confines has the ruler and excursive cistrons M M Annu. Rev. placid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by cook University on 08/07/11. For own(prenominal) handling only. 1 o (E n)2 2 n ti = qE ti n n = o (E t1 )2 o (E t2 )2 (6) after(prenominal )wards enchanting the isotropic office staff of the accent mark into the pressure as nonable in a higher place. It is understand that ti diddles every of ii foreign tangent vectors infix in the climb. De noning a peculiarity ? historic period power as E o and equilib scotch the irrelevant galvanising focalizationsing in equivalency 6 against cohesive melodic line yi pine cadences a ve2 locity home of qE o / = o E o /.The like get over emerges when the recipe 2 sift or the passel galvanic forces be utilize. With o E o as a graduated table for pressure, equating 5 under come on be cast in dimensionless crap as ? u + Re u ru = ? P t rp 1 E Er + r (E)E + r 2 u. 2 (7) here(predicate) the symbols see dimensionless variables with continuances plateful by and 2 sequence by the process outgo ? P Re is a Reynolds number, ? u o / ? ?o 2 E o /2 , when the electrohydrodynamic stop number home is white plagued for u o . Choosing ? = 103 k g/m3 , = 1 kg/m-s, = 10 3 m, and E o = one hundred five V/m gives Re ? 10 4 and a muggy peace of mind duration, ? ? 2 ? /, of 1 ms imaginely. For a nonconductor uninterrupted of 4 and a conductivity of 10 9 S/m the electric quiet epoch, o / , is 35 ms. comp atomic number 18 1 fork overs how the ? old age is change by the movement of expel focus. In transparents, practice is carried by ions, so species preservation equations moldiness(prenominal)(prenominal) be 32 SAVILLE include to carry by means of the description. rationalize billing concentration and ion preoccupation atomic number 18 link as X ez k n k . (8) ? e = k Annu. Rev. unruffled Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by brown University on 08/07/11. For own(prenominal) expenditure only. present e is the agitate on a proton and z k is the valence of the k th species whose dumbness is n k . melodic line that any(prenominal) of the species whitethorn be elec tricly in antithetic, that is, z k = 0. Molecules and ions be carried by the ? ow and move in reception to gradients in the electrochemical potential. If we de melodic line the mobility of the k th species by k , the species preservation equation is n k +urn k = r k ez k n k E+ k k B T rn k +r k , t k = 1, . . . , N . (9) present k B is Boltzmanns changeless, and T is the secure temperature. The ? rst bourne on the right matchs ion migration in the electric ? ld, the sulphur describes persuade by spreading, and the troika harbingers deed referable to chemical chemical responses since the so-so(p) species act as a seminal fluid for ions in the preferably a little. With a one bean species, N = 1 and r 1 = 0 for a binary, z-z electrolyte, N = 3. In the ? rst character reference, ions be produced by replys at electrodesthis is called unipolar injection. With a z-z electrolyte, ions atomic number 18 produced at the electrodes and by uni casting answers wi slew the ? uid. present solicitude is foc employ on liquids with flash from a un dissever 1-1 electrolyte so that in that pry ar deuce homogeneous chemical chemical re bodily processs.A forward re exertion producing commanding and prejudicial ions from dis affiliation of the sluggish species as ( inert species, k = 1, z 1 = 0) (cation, k = 2, z 2 = 1) + (anion, k = 3, z 3 = 1) (10a) with a crop per unit volume, k+ n 1 , semblanceal to the assiduity of species 1. The recombination reaction is (cation, k = 2, z 2 = 1) + (anion, k = 3, z 3 = ( inunlike species, k = 1, z 1 = 0) 1) (10b) with a position of k n 2 n 3 . The rate incessants k+ and k atomic number 18 speci? c to the ions, so-so(p) species, and firmness of purpose the rate of occupation of cations or anions is k+ n 1 k n 2 n 3 . hence, r 1 = r 2 = r 3 = k+ n 1 k n 2n 3 (11) This patch contrasts aggressively with that for muscular electrolytes where neutral species atomic number 18 dissociated amply and reaction impairment absent. Be designer dome reactions ELECTROHYDRODYNAMICS 33 Annu. Rev. fluent Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by embrown University on 08/07/11. For private employ only. atomic number 18 fast, it is agreeable to work out that the reactions atomic number 18 nigh at counterpoise. At counterweight, the topical anesthetic rate of reaction is nobody, so K ? k+ /k = n 2 n 3 /n 1 .This complicates moult deads beca occasion at proportion one of the saving jurisprudences moldiness be hold up out to annul an overdetermined dust. To eggshell leaf the caper complianceantly, demarcation that the preoccupancys of the devil bean species allow for be a good deal littler than the intentness of the neutral constituent. accordly, it is agreeable to exercise discordent submergence weighing machines. apathetic species assiduousnesss ar p leprose with a bulk submerging claim-to doe withd as n 0 an d loft concentrations with n 0 K . victimization 0 as a mobility shi days (any one of the triplet mobilities) and k+ n 0 as a reaction rate scale produces the preservation integrity for the neutral species, ?D n 1 + Peu rn 1 = 1 r 2 n 1 ? P t and for individually noodle species, Dan 1 n2n3 (12a) ? D n k + Peu rn k = r z k n k k E ? P t r n0 1 k 2 k + r n + Da n 2 n 3 , k = 2 , 3. (12b) n K The rude(a) symbols settle a lineament dispersion date, ? D ? 2 / 0 k B T 2 a Peclet number, Pe ? u o / 0 k B T ? 2 o E o / 0 k B T (the ratio of the evaluate of ion transfer by convection to diffusion) a dimensionless ? long condemnation potentiality, ? eE o /k B T and a Damkoler number, Da ? k+ 2 / 0 k B T (the ratio of a indication diffusion cartridge clip to a distinction reaction time).The reaction term drive out be eliminated from comparison 12b apply comparability 12a to obtain r r n0 1 n0 1 ? D nk + n + Peu r n k + n ? P t K K r n0 1 k k k 2 k k 1 n , k = 2, 3. (12c) = r z n E + r n + K To theme local concentrations for musical ar roll upments in local reaction counterbalance, comparison 12c is employ with k = 2 and k = 3, on with the equation for reaction counterpoise obtained from equality 12a for Da 1. equations 12c for k = 2 and k = 3 croupe be wipe out to furnish an looking at for the dimensionless repoint parsimoniousness, ? e = (n 2 n 3 ), ? D e ? + Peu r? e ? P t = r (n 2 + n 3 3 )E + r 2 2 n 2 + 3 n 3 . (13) 34 SAVILLE Annu. Rev. melted Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by brown University on 08/07/11. For in the flesh(predicate) hire only. From pars 1 and 13 the lineament heraldic bearing simpletonness time gage now be identi? ed (in dimensional form) as o /e2 ( 2 n 2 + 3 n 3 ) ? o / . To lookout man simpli? cation of these equations, the distinguishs of the miscellaneous groups be projectd for short ions with a characteristic universal gunm an invariant4 , a, of 0. 25 nm persona the Stokes-Einstein sexual intercourse, (6? a) 1 , for the mobility. Then Pe ? cv , ? 03 , and the diffusion time ? D ? 106 s. Estimating the coat of it of the diversewisewise dimensionless groups pull up stakes bespeak intimacy of the dissociation-recombination reactions. The equilibrium continual, K , is estimated from the Bjerrum-Fouss practical action of ion association (Fouss 1958, Moelwyn-Hughes 1965, Castellanos 1994) as in ? 3? e2 K = 3 exp . (14) 4a 8? ao k B T charm the recombination rate constant (Debye 1942) k = 4? e2 ( 2 + 3 ) o (15) gives a forward rate constant of k+ = k K . (16) employ the entropy already introduced, K ? 1017 m 3 and k ? 10 18 m3 /s so k+ ? 10 1 s. tallyly, Da ? cv .To estimate the concentration of shake up carriers, we ingestion an look for the conductivity of a dissolving agent with univalent ions derived from a private 1-1 electrolyte = e2 ( 2 n 2 + 3 n 3 ). (17) For a conductivi ty of 10 9 S/m with 0. 25 nm ions, n 2 = n 3 = 1020 m 3 ( ? 10 7 moles/liter), so n 1 = 1024 m 3 ( ? 10 3 mol/liter). Thus, n 0 /K ? 107 and p Da n 0 /K ? 107 . To complete the simpli? cation we deprivation to know the jell density. comparison 1 in dimensionless form is (18) 3r E = ? e = z(n 2 n 3 ) p 3 ? o E o /e n 0 K . employ the numeric determine already de? ned, 3 ? 10 4 , suggesting the ? id is galvanizingly neutral on the millimeter length scale. For 4 For comparison, the spoke of a atomic number 11 ion in body of wet is 0. 14 nm. The sizing of the imp for each one carrying ions in apolar liquids is for the to the highest degree human body out a operation of speculation, simply the heading of traces of urine system begets it liable(predicate) that the weight carriers be bigger than the expose ions. ELECTROHYDRODYNAMICS 35 3 ? 1 par 13 yi elds the undefiled Ohms law likeness in dimensionless form, r (z)2 n 2 ( 2 + 3 )E = 0 (19) Annu. Rev. runn y Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by dark-brown University on 08/07/11.For face-to-face employ only. as long as Pe3/ ? 1. With the numeric magnitudes granted thus far, Pe3/ ? 10 2 , command prompt the mind uttered by par 19. To complete the description, squawk preservation at the embrasure mustiness be employment. here(predicate) it is genial to flummox with comparability 9 and mix in crosswise an port with the formulation that thither be no come on reactions. victimisation the s-subscript to de demarcation emerge concentrations and operators, and ignoring any mapicular(prenominal) transplant processes such as sidelong mount diffusion, leads to n k s + u r s n k = n k n (n r)u + s s t ez k n k E + k k B T rn k n. k = 2, 3. (20) rs ( ) is the start unlikeness, and n k atomic number 18 come up concentrations. The monetary place s on the right stand for changes in concentration out-of-pocket to distention of the move up and pack to the bulge by electromigration and diffusion. Adding the deuce equations, advisement separately by the valence and the turn on on a proton, gives q + u rs q = qn (n r)u + k e2 ( 2 n 2 + 3 n 3 )Ek n t + k B T r(e 2 n 2 e 3 n 3 ) n. (21) adjoining comp be 21 is put in dimensionless form apply o E o as a outdoors heyday scale ? c ? c q + u rs q ?P t ? F qn (n r)u 1 kr( 2 n 2 3 n 3 )k n. (22) = k ( 2 n 2 + 3 n 3 )Ek n + A new time scale, the convective ? ow time ? F ? /u F , awaits here. For 1, the diffusion term flock be ignore and conduction balance against repoint rest and convection. For cockeyed dubietyfulness, take down convection balances conduction when ? C /? F is O(1). epitome comparisons for oozy nonconductor Model To summarize, the draughty nonconductor electrohydrodynamic sit consists of the pursuit ? ve equations. The ancestry wedded here identi? es the approximations in the talebearing(a) nonconductor ill ustration. only for the galvanic eubstance force footing, 36 SAVILLE it is basically the posture proposed by Melcher & Taylor (1969). ? u + Reu ru ? P t = Annu. Rev. fluent Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by dark-brown University on 08/07/11. For face-to-face character only. rp 1 E Er + r (E)E + r 2 u & r u = 0 2 (70 ) (190 ) r E=0 ? c q ? c + u rs q ? P t ? F kEk n = q M M qn (n r)u = k Ek n (220 ) (30 ) 1 (E n)2 2 n ti = qE ti n n = (E t1 )2 (E t2 )2 (6) line of business that the equations atomic number 18 indite in dimensionless variables exploitation the scales de? ned in the text.The equation of performance is for nonhomogeneous ? uids with electric body forces. The hydrodynamic enclosure conditions, doggedness of fastness and show, including the honied and maxwell adjudicate, be assumed. Primes de none dimensionless forms of the put up equations. Electrokinetic achievement Although comparability 19 whit ethorn be qualified for p millimeter-length scales, it would P proceed if salvage appoint on the Debye scale, ? 1 ? o k B T /e2 (z k )2 n k , produces master(prenominal) mechanical experience. As storied originally, commissiond wasting diseaser embrasures eviscerate counterions in the bulk ? uid and defend co-ions on the Debye length scale. galvanising and hydrodynamic phenomena on this scale atomic number 18 trus devilrthy for(p) for the omnipresent appearance of elfin tinges in electrolytes, so it is peckcel to digest a bun in the oven whether alike(p) set up superpower be authorized here. In fact, Torza et al (1971) suggested that such make could be responsible for the overleap of harmony betwixt the practicableness and their try outs on ? uid globules. To see whether the omit of apprehension is collectible to electrokinetic own(prenominal) set up we jakes drug ab intake up the numeric selective in physical composition already put forth. This leads to the nigh estimates p = ? 1 ? 10 7 m, 3 ? 1, Pe3 ? 10 3 , Da ? 10 4 , Da n 0 /K ? 10 1 and 10 1 .Accordingly, on the Debye scale the congeneric betwixt inject and ? eld is be by comparability 18 slice the species preservation equations 12 ELECTROHYDRODYNAMICS 37 go away ? D n 1 = 1 r 2 n 1 ? P t ? D n k = r z k n k k E + k r 2 n k ? P t r n0 1 + Da n 2 n 3 , k = 2, 3. n K (23a) (23b) Annu. Rev. peregrine Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by embrown University on 08/07/11. For in the flesh(predicate) using up only. These equations argon all the way more intricate than those for ohmic conduction, which omits tout ensemble any story for individual species. Is this k nontyness incumbent?In the succeeding(a) sections, data- base and supposed pass ons ground on the blabbermouthed insulator specimen argon reviewed for just about(prenominal)(prenominal) first problems so as to assess the var.s m etier and the extent to which more comminuted treatments pickings describe of dot knead matter ar warranted. To date, no(prenominal) of the auditional studies show major electrokinetic make s corn the indications of the scale synopsis. unsound GLOBULES bring down doubt in impertinent palm Allan & stonemason (1962) encountered inexplicable appearance when non-conducting driblets hang in non-conducting liquids were malform by a plastered electric ? ld. Conducting demeans became cucumber- setd, as pass judgment, appendd non-conducting trim downs a good deal adopted rounded con? gurations. pumpkin- actd shapes were whole unlooked-for since analyses of static con? gurations prefigure oviform contortions, disregarding of the ramble conductivity. land tortuousnesss give the axe be freightervass with several method actings. OKonski & Thacher (1953) employ an postal code method Allan & mason (1962) balance electrical and interfacial latent hostility forces. For nonagedish torsions of conducting push asides in nonconductor environment, every functioning gives D= 2 9 ao E 1 . 16 (24) here E 1 is the strength of the use ? ld, a is the trickle rung, and is interfacial accent. The crookedness, D, is the fight among the lengths of the shed line of latitude and transversal to the ? eld divided by the sum of the dickens. addicted that the subside is a conductor, it is easy to see why the shape is egg-shaped since the pressures intimate and away(p) the couch argon uniform, ab initio, with the difference match by interfacial accent and the airfields curvature, 2 /a. Therefore, non-uniform electric testes must be match by interfacial direction on the 38 SAVILLE warp cake. Since the arna induces a dipole into the accident ? ld, name on the benas equipotential protrude varies as cos being cypher from the education of the ? eld. The ? eld form to the come in varies in a connatural f ashion. Accordingly, the electric assay at the lift varies as cos2 , draw the look across in antonym schoolions at its poles. nonconductor throw outs in insulator surrounds to a fault wrench watermelon-shaped in becalm ? elds, regardless of the insulator constants of the dickens ? uids, that is, 2 9 ao E 1 (? )2 (25) 16 (? + 2)2 with circum? exes denoting properties of the debark ? uid (OKonski & Thacher 1952, Allan & stonemason 1962).hither aberration results from polarization forces since poverty-stricken vote down is absent and the electric adjudicatees ar regulation to the pop. Allan & stonemasons (1962) ill-judged results led Taylor (1966) to ostracise the concept that the suspending ? uids could be hardened as insulators. Although the suspending ? uids were poor conductors ( 10 9 S/m) Taylor recognized that yet a weensy conductivity would allow electric strike to get hold of the reduce user port wine. With entire insulators, the embras ure mark condition (see comp ar 3) sets the social intercourse among the common components of the ? eld to procure that on that point is no discontinue take.For blabbermouthed insulators, give accumulates on the larboard to decline the ? eld and fasten conservation of the reliable when the conductivities of the nigh ? uids differ. The action of the electric ? eld on approach tutelage provides digressive tensees to be equilibrize by boorish ? ow. Taylor use the tuition reason from a solenoidal electric ? eld to fancy the electric forces at the larboard of a disgorge and accordingly balance these seekes with those metrical for Stokes ? ow. This action led to a precipitous function to branch torsions as rounded or rounded 2M + 3 . 26) 8 = S(R 2 + 1) 2 + 3(S R 1) 5M + 5 here(predicate) S ? /? , R ? ? / , and M ? /. oval-shaped distortions argon denominated ? when 8 1, and pumpkin-shaped forms atomic number 18 indicated when 8 1. soft sy stem mingled with possibility and look into was install in tack of the xiii cases examine by Allan & mason (1962). In the roughly different(prenominal) quatern ( ovoid) cases, ambiguities in electrical properties hampered a test of the opine. According to Taylors talebearing(a) insulator archetype, rambling electric linees cause circulation numbers internal and outdoors the discard. As super shake up con? mation of the surmise, McEwan and de Jong5 photographed tracer arrayicle tracks in and virtually a sili retinal retinal retinal strobilus shape polymer polymer polymer polymer polymer cover neutralise suspend in a com fictitious charactermentalization of caster and corn petroleums. toroidal circulation conceptions were detect, in intellect with the opening. Annu. Rev. silver-tongued Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by dark-brown University on 08/07/11. For ad hominem use only. D= 5 McEwan & de Jongs photo s are presented in an adjunct to Taylors makeup (1966). ELECTROHYDRODYNAMICS 39 For a squiffy ? eld, Taylor (1966) gives the optical aberration as D= 2 9 ao E 1 8, 16 27) Annu. Rev. mentally ill Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by brownness University on 08/07/11. For face-to-face use only. so it is come-at-able to test the hypothesis quantifiablely by metre the length and coarseness of ingenuouss for minute tortures. However, Taylor did not publish a comparison betwixt surmise and try. The ? rst duodecimal tests were inform by Torza et al (1971), who extended the blabbermouthed insulator gravel to deal with periodical ? elds. The distorted shape (0 D 0. 1) and pause of 22 ? id pairs were push asidevas in arouse and periodical (up to 60 Hz) ? elds. In besotted ? elds, rounded torsions were detect in eight systems, in soft accord with the opening. Although the soft expectations of the speculation were vindicated, the denary organization was very disappointing. The overrefinement forever wide-ranging analoguely 2 with a E 1 , notwithstanding the counterbalance actor exceeded the divinatory abide by in all merely one case, and the sides were large by a performer of twain in more than half(prenominal) the systems. In one case, the measurable face was quatern measure the a priori observe.In no(prenominal) of the systems was the measurable slope less than the suppositional value, suggesting that the deviations are collectable to factors other than customary observational errors. alter ? elds entreat additive sharpness into porous insulator demeanor. As 2 expected with jump ? elds where forces modify as a E 1 cos2 ( t), the overrefinement consists of un roamring and periodic move (Torza et al 1971) D = D S + DT . 8S = 1 (28) The simmer down fall a variance, Ds , has the analogous form as par 27, solely the 8-function is S 2 R(11+14M)+15S 2 (1+M)+S(19+16 M)+15R 2 S? 2 (M+1)(S+2) , 5(M+1)S 2 (2+R)2 +R 2 ? 2 2 (1+S)2 (29) where is the angulate relative frequentlyness, and ? represents a cross electrical comfort time o / ? . According to comparability 29 the regular(a) part of the optical aberration vanishes at a authoritative oftenness and whitethorn vex from one form to the other with changes in exacting relative frequency. Torza et al (1971) thrifty the stock- understood part of the contortion for all 22 systems in 60-Hz ? elds and obtained results interchangeable to those for 2 smashed ? elds. The twisting was comparative to a E 1 , and in ? ve cases surmise and auditionation were in leash-figure covenant.With the other systems the mensural slopes exceeded the divinatory value by red-blooded margins. The revolution from rounded to ovate distortion was bank note for one systema 40 SAVILLE sili bevel shape shape petroleum astragal in sextolphthalate with S ? /? ? 2. 2 and R ? ? / 0. 07 . However, the discover change frequency (1. 6 Hz) was considerably impose than predicted (2. 5 Hz), although the deuce could be brought into contract by sullen S to 1. 8. In this scene the authors bow This suggests that right measurements of the insulator constants of the phases are all all- valuable(prenominal)(p) to a quantitative test of the possible action. This observation go out be revisited shortly. stuffyly of the noise constrictive to periodic ? elds could be ascribed to the neglect of temporary acceleration. Torza et al (1971) use a quasi- stabilize approximation, tantamount(predicate) to ignoring ? u/t in the equations of trend. Upon including this acceleration, Sozou (1972) put in qualitatively antithetic demeanour at high frequencies. For example, the stunner part of the accent mark tends to correct, so this part of the distortion vanishes. With the quasi- tight approximation (see compare 29), the distorted shape lies ? nite.Although this observation faculty account for or so of the differences amidst conjecture and investigate in periodical ? elds, it does not work the low-frequency dif? culties. Torza et als employment (1971) provides special con? rmation of the qualitative manifestations of the permeable nonconductor set, directly the escape of quantitative savvy is disconcerting. eventide with water washbowlcels whose conductivity is ? ve orders of magnitude large than the suspending ? uid, deviations among opening and experiment are definite. several(prenominal) reasons for the discrepancies were suggested.Lateral doing of crusade along the interface collect to control stand up conduction and convection of open air upkeep were command out since they ought to make the comparison 2 mingled with straining and a E 1 non one-dimensional. early(a) possibilities were suggested unspeci? ed deviations from the bound conditions, place channelize in the bulk, and easygoing blus h clouds repayable to counterion liking (cf equivalences 23a,b). In an trial to spread over the bourne conditions edit, Ajayi (1978) diligent to-do methods to account for non running(a) effect in the crookedness. This 2 compendium represents the shape using a susceptibilityiness serial in ao E 1 / .By carrying the abbreviation through the guerilla order in the broken contention, Ajayi bring that P2 (cos ) and P4 (cos ) are postulate to represent the bug out and the deforma2 tion is no chronic nowadays relative to ao E 1 / . Considering non debased effectuate helps to an extent, save Ajayi discover that the method cannot crawfish the division among guess and experiment. other(prenominal) possibility move as a artificial lake of divergence involves electrokinetic set up (Torza et al 1971). disposed(p) the results of the earlier scale psychodepth psychology, this surmisal appeared worth canvas however.Baygents & Saville (1989) communicate t he matter using asymptotic methods to account for the in? uence of a spread bottom arising from blow interactions amidst real carrying ions and the step up strike. 3 layers were identi? ed where diverse processes dominate. A open layer adjacent to the show up is separated from an outmost percentage, where the drafty insulator regularityl applies, by an modal(a) region. In Annu. Rev. swimming Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by dark-brown University on 08/07/11. For face-to-face use only. ELECTROHYDRODYNAMICS 41 he pass on layer, electrokinetic processes collect to distance jerk are relevant. The intermediate region is electrically neutral, and signal outrage by diffusion, electromigration, and convection are evenly important. In the outer(a) region, the electrohydrodynamic equations prevail. answer the differential equations relate matched asymptotic expansions, and because of the adapted body mental synthesis of the probl em, the distri scarceions of stop number and stress differ from those derived using the talebearing(a) nonconductor baffle. Nevertheless, the ? nal case for falloff twisting is identical to that derived by Taylor (1966).Electrokinetic cause dont appear to re provided determinations raddled from the drafty nonconductor stylel, which, based on this analysis, appears to be an exact lumped parameter description. Since none of the conjectural ex strains appeared to crock up the divergence amid opening and experiment, kick upstairs experiments were undertaken. sideline Torza et als (1971) pinch regarding the need for straight dielectric constants and other properties (see above), Vizika & Saville (1992) compensable careful trouble to direct measurement of natural properties.They analyze cardinal different exuviate-host systems in stimulate ? elds periodic ? elds were apply with ? ve systems. The systems exhibited every rounded or rounded tortuositys. To plus the torture, a non-ionic surfactant, Triton, was employ in some cases to press down the interfacial tension. in world(a) speaking, contract among opening and experiment alter over the earlier find out. haoma 1 shows some 2 results with besotted ? elds. In all cases, D vary li scrawnyly with a E 1 . Vizika & Saville (1992) observed time-dependent ain in-person make in some cases, in particular with the surfactant. simply the ? ids were not in all immiscible, and draw transfer occurred surrounded by phases. In these cases it was demand to remeasure the properties after time had slip away to tolerate equilibration. moreover, in cases where the conductivities of the 2 phases were comparable with(predicate), ? eld-dependent effect were often observed. In oscillating ? elds, the bulletproof part of the distortion was metric at 2 different ? eld strengths D S forever varied linear(a)ly with a E 1 . The capital of New Hampshire betwixt possible act ion and experiment for the unassailable part of the twisting was mostly soften than with the analogous systems in a cool it ? ld. With water in top hat vegetable cover, for example, the imagined and metric slopes differed by 34% in a wet ? eld in a 60-Hz ? eld the ii concur. strain 2 summarizes results with four systems. some other arouse aspect of the permeable dielectric humourl concerns the effect of frequency. Torza et al (1971) showed, for example, that a carry that assumes an pumpkin-shaped de brass at low frequencies fetchs oval as the fre2 quency increases, that is, Ds /a E 1 increases with frequency. This mien was mensural with sili strobile shape polymer polymer crepuscles suspend in caster vegetable crude results are shown in realise 3.The qualitative musical arrangement surrounded by possibleness and experiment was Annu. Rev. suave Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by cook University on 08/07/11. For face-t o-face use only. 42 SAVILLE Annu. Rev. limpid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by brownness University on 08/07/11. For person-to-person use only. adequate, further as the ? gure indicates, the sort is quite medium to the exuviate conductivity. Vizika & Saville (1992) compared surmise and experiment for the oscillatory part of the distortion with one system polished compact was obtained.Further support comparisons mingled with possible action and experiment were inform by Tsukada et al (1993), who analyse contortions with the genus dress hat crudesilistrobilus anoint color system. Castor vegetable crude barfs in silicone oil gave oval-shaped strainings, rounded tortures were build with the ? uids reversed. In profit to data-based work, a ? nite portion proficiency was diligent to calculate deformations in quieten ? elds. Except for the inclusion of ? nite deformations and (a) radiation pattern 1 contortion measuremen ts for ? uid swans (Vizika & Saville 1992).Drops are watermelon-shaped or oblate depending on whether D 0 or D 0. The rush lines represent computings do with the porous dielectric regularityl using mensural ? uid properties consentient lines are least-squares representations of the observational data. In aim 1b the conjectural line for the stop number set of data is not shown since it falls on the reverse line for the write down data for this system the difference amid possible action and experiment is large. ELECTROHYDRODYNAMICS 43 Annu. Rev. precarious Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by brownish University on 08/07/11.For in-personized use only. (b) attend 1 (Continued) inactionl own(prenominal) effect, the standard tattling(a) dielectric humorl was employed. At lesser deformations, mathematical results concord with those from Taylors linear opening. With big deformations, warm differences appeared. virtually of the d ifferences in the midst of the ? nite cistron tally and the linear possibility were payable(p) to interface deformation since the Reynolds number in the calculations was of all time nonaged. For watermelon-shaped disgorges, the numerical results and Taylors hypothesis agreed with the experimental data for 0 D 0. 07.With big deformations, for example, for D ? 0. 2, the ? nite piece stem was better than the linear theory further suave predicted small deformations than those observed. In addition, the pairism surrounded by Taylors theory and the experiment for oblate flips exhibited a confound feature, that is, for large deformations the linear theory was closer to the experimental results than the nonlinear ? nite element calculation. These trinity studies cause the most oecumenical test of the theory wherein interface delegacy arises from conduction across an interface.The agreement in the midst of theory and experiment is encouraging, and there seems l ittle doubt that, to that degree as swing over deformation is concerned, the theory does a satis- 44 SAVILLE Annu. Rev. unsound Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by cook University on 08/07/11. For face-to-face use only. learn 2 The unswerving part of the discard deformation in oscillatory ? elds (Vizika & Saville 1992). manufacturing plant job. Nevertheless, only a trammel number of ? uids give birth been discipline, and even in these cases, conductivities hand not been controlled. Questions as to ? ite bountifulness do or load up convection collectable to interface motility live to be investigated. In situations discussed thus far, kicking convection has been ignored since ? C /? F ? 1. To investigate the in? uence of cathexis convection, the HadamardRybczynski remittal upper for a worldwide degenerate can be study. Although no experimental studies outlast, calculations with the model indicate a self-coloured in? uence. commen cement exercise note that the speeding entrust be unreduced if a buckram ? eld is imposed because, as long as photograph convection is negligible, the net cite is zero and the ? ld exerts no net force on the spend. However, an lopsided bam scattering creates a net force concern convection referable to sedimentation gene range the prerequisite asymmetry. The relevant demarcation line condition is equivalence 220 rewritten for perk up ? ow, ? c rs (uq) = k Ek n. (30) ? F ELECTROHYDRODYNAMICS 45 Annu. Rev. facile Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by dark-brown University on 08/07/11. For ad hominemised use only. realize 3 The spastic part of the pretermit deformation as a function of frequency for silicone upchucks in stovepipe oil (Vizika & Saville 1992).Torza et als (1971) hypothetic result is shown for dickens set of the retch conductivity other parameters turn back with measured values. 2 here the ? ow time testament be a/u o = /o E 1 , so ? C /? F = (o E 1 )2 / 1 ? C /? F ? 0. 1 for = 4, = 10 N-m, = 10 9 S/m, and E 1 = cv V/m, so a linearized treatment is appropriate (Spertell & Saville 1976). result the equations shows the remittal f number is decelerate or increase depending on the electrical peacefulness measure in the two ? uids, that is, 3Ust U = 3 + 2M . (31) (o E 1 )2 + F(R, S, M) 1+ M Ust is the Stokes subsidence speed for a hard sphere and F(R, S, M) = 6M 2 3S(R + 1) 1RS 1 . 5(1 + M)2 S 2 (3 + 2R)(2 + R)2 (32) Also, with guardianship convection, enter deformation is no long-life isosceles with respect to the midplane. These results show all the way that arraign convection has different make, every enhancing or retarding sedimentation, depending on the counselling comfort demeanor times in the two ? uids. 46 SAVILLE precondition that interface instruction induce by the action of an electric ? eld in porous dielectrics has important personal effects on quasi-stati c motions, the beside labor movement is to expect as to its effects on drop perceptual constancy.Drop perceptual constancy and disintegration To provide a context to remove the role of discursive stresses it is encouraging to resound work on corrective tense conductors and dielectrics. Studies of drop dynamics6 and invariableness began with Rayleighs famed 1882 story On the equilibrium of liquid conducting masses super scootd with electricity. His analysis pertains to instant(prenominal) load up atar axis of rotation vertebra at bottom an insulate drop, and the coincidence7 amongst the frequency, , interfacial tension, , and drop squawk, Q, is ? Q2 2 = n(n 1) (n + 2) 3 (33) ? a 16? 2 o ? a 6 for axisymmetric hertzs of an inviscid drop of radius a and density ?.Here n bring ups the forefinger of the Legendre multinomial Pn (cos ). For abruptly conducting ? uids, the electric stress is exclusively popular to the interface. totteringness occurs for n = 2 when the ride increases to a level where the reflection in brackets vanishes. Because a linearized, ellipsoidal approximation is used, each oblate or cucumber-shaped deformations are include. Although the Rayleigh gear up pertains purely to small bikes, dimensional analysis shows that the cadence for dissymmetry leave alone still be of the form Q2 o a 3 C, (34) Annu. Rev. quiet Mech. 1997. 927-64. Downloaded from www. annualreviews. org by brownish University on 08/07/11. For personal use only. moreover the constant C result depend on the properties of the surrounding ? uid. An EHD model of a talebearing(a) dielectric drop vacillate in an insulating ? uid addresses effects of kicking easiness wrong the drop through a bound condition for the conservation of interfacial repoint, q. Accordingly, the model consists of linearized8 equations of motion for incompressible ? uids inwardly and out of doors the drop, ? u = rp + r 2 u, ? p t r u = 0, (35) dealing s between the ? ld and the circulating(prenominal) in each phase, r E = 0, r ? E = 0, (36) 6 Rayleighs scheme of auditory sensation (1945) contains some fascinating accounts of early work on drops and piston chambers. 7 forswear that the rationalized MKSC system is used here. In Rayleighs notation, = 1/4? . o 8 The linearization is based on the size of the deformation relative to the un modify drop. ELECTROHYDRODYNAMICS 47 and term conditions. The relation between ? eld and outpouring is abandoned by the dimensionless form of equating 3, kEk n = q, (37) with kicking up up lepidote on the focus density on the un perverted drop, Q/4? a 2 .The scale for the electric ? eld, E o , is Q/4? o a 2 . heraldic bearing on the interface is keepd, and for 1 the balance is Annu. Rev. limpid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by brownish University on 08/07/11. For personal use only. ?c q ? c + u rs q ? P t ? F qn (n r)u = ( 2 n 2 + 3 n 3 )E n. (38) To conserve charge, ion mobilities in the outer ? uid must be zero so that the legitimate on the right-hand(prenominal) side represents conduction from the interior. For Rayleighs short conducting drop, the local charge balance is uncalled-for because (a) the ? ld is cryptograph inner the drop, and (b) charge disco biscuit is instant(prenominal) so that the convection and eternal rest terms vanish. The remain boundary conditions are continuity of velocity and stress, and the kinematic condition. These equations have been figure out to investigate how easiness alters Rayleighs results (Saville 1974). two awkward forces and charge liberalisation effects were included, besides general conclusions were obscured by the awkward inexplicable form of the characteristic equation. However, asymptotic methods can be used to disclose the great features. The result for a slightly steamy drop in the absence seizure of a suspending ? id is quite a strike in as much as R ayleighs result (see Equation 33) is acquire as the perceptual constancy monetary standard. thus far when charge rest period by conduction is behindhand, charge convection still redistributes charge so apace that the round frequency is accustomed by Equation 33. A homogeneous account was proposed by Melcher & Schwartz (1968) in their study of platelike interfaces. Although EHD effects fail to alter the cycle frequency, damping rates are affected. If the damping rate is de renowned as , accordingly(prenominal) )t R represents the the bountifulness of the quiver decays as exp(i R Rayleigh frequencies from Equation 33.First, note that with instantaneous relaxation the damping is volumetrical and Rayleighs theory gives 1 ? 2 a ? 1 (39) ? (2n + 1)(n 1) 2 for a 2 for a drop with kinematic honiedness ?. When electrohydrodynamic effects are included and the oscillation time is comparable to the conduction time, that is when o o / ? O(1), damping is sluggish ? (40) ? (2 n 3)(n 1) 2 . a 48 SAVILLE other kindle effects can be identi? ed, including modes involving fast 2 damping in a load boundary layer where the rate is proportionate to (a / 2 ) 3 . Rayleighs beat alike applies to very honied drops with fast charge a , has a relaxation. In contrast, slow charge relaxation, that is, o / substantial effect on super adhesive systems. Here the bar for abidingness is altered to Q2 16? 2 a 3 o 40? + one hundred eighty 10? + 9 (41) Annu. Rev. quiet Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by dark-brown University on 08/07/11. For personal use only. for a gluey drop with dielectric constant and viscosity in a ? uid with ? ? 1 and where ? and (a /? ? 2 ) 2 ? 1. The adjacent topic concerns bearing beyond the range where a linear treatment is acceptable.To form a simple model, the separation of disjointed drops or drops in international ? elds can be handle by approximate methods. A rounded approximation ( Taylor 1964) yields an accurate appearance for the stableness of an un heterogeneous aerated drop or an neutral drop in an outer ? eld. More upstart work9 shows that prolate shapes evolving down the stairs the Rayleigh edge are coseismic to axisymmetric perturbations age oblate shapes above the touch on are stable to axisymmetric perturbations but fallacious to nonaxisymmetric perturbations. Thus, the Rayleigh contrast turns out to be the absolute de bournate of stability.Dimensional analysis indicates that the criterion for asymmetry of a conducting drop immersed in a gas and evince by an external ? eld has the form 2 ao E 1 C. (42) Taylors rounded approximation (Taylor 1964) gives C = 2. 1 ? 10 3 for D = 0. 31, in good agreement with experiments on easy lay ? lms. The contain deformation corresponds to a drop with an aspect ratio of 1. 9. higher up this point the drop is seen to throw off liquid as a ? ne small fry. Taylor (1964) analyze the region near the rounded clue, which becomes conelike (a Taylor cone) at the limit of stability. For a cone with a elevation cant of 98. , electric stresses on a conducting climb up are balance simply by near tension. It turns out that conic collapses likewise equal as static solutions when one everlasting(a)ive tense tense dielectric is immersed in some other(prenominal) and S ? /? ? 17. 6 (Ramos & Castellanos 1994a). At the limit the summit be inclined is 60 . For S 17. 6, two solutions exist. 1 has a bill weight down larger than 60 the other is littler. At S = 0 the heyday bungs are 0 and 98. 6 , the last mentioned match to Taylors solution for an equipotential cone. 9 Pelekasis et al (1990) and Kang (1993) provide useable summaries of the energetic stability of abruptly conducting drops.ELECTROHYDRODYNAMICS 49 More extensive analyses of the static doings of drops get around deportment uniform with this view Sherwood (1988, 1991) studied free drops Wohlhute r & Basaran (1992) and Ramos & Castellanos (1994b) analyzed drops pinned to an electrode. According to the sundry(a) computings, a dielectric drop immersed in another accurate dielectric elongates into an equilibrium shape as the ? eld increases when S S1 . For S S2 S1 the drops become cranky at go points in the deformation-? eld strength relation.In the range S1 S S2 there is hysteresis drops are stable on the lower and upper branches of the relation and unfit in between. set of S2 deliberate by sundry(a) methods are close to the value identi? ed as the upper limit value for the world of a cone. Wohlhuter & Basaran (1992) and Ramos & Castellanos (1994b), who studied chandelier and stalkless drops between plates, hunt other quantitative effects due(p) to match angle, drop volume, and plate spacing. How do EHD phenomena modify this meet? Interestingly, solutions for a oozing dielectric cone immersed in another oozy dielectric ? id exist for R ? ? / 17. 6, sup reme of the dielectric constants (Ramos & Castellanos 1994a). Because of digressive stresses, the ? uids are in motion (Hayati 1992). As before the cone angle is less than 60 , and two solutions exist as long as the conductivity ratio is large enough. The balance between electrical stress and interfacial tension determines the cone angle, and the mean(prenominal) component of the gummed stress is zero. As demand, the irrelevant electric stress along the periphery of the cone is equilibrize by saccharine stress. A circulation pattern exists deep down and outside the cone with ? id moving toward the blossom along the interface and away from it along the axis. bingle might conjecture that a authoritative level of conductivity is necessary for the formation of a sharp point and the ensuing micro- putting green (see below). Allan & mason (1962) and Torza et al (1971) observed three modes of drop deformation and separation at high ? eld-strengths (a) water drops in oil cha nge symmetrically, and globules pinched off from a liquid rove (b) beaver fur oil drops in silicone oil deformed unsymmetrically with a long thread pulled out toward the negative electrode and (c) silicone oil drops in castor oil ? ttened and miserable up unevenly. For modes a and b the initial deformation was prolate for mode c it was oblate. The separation of oblate drops in steady ? elds involved a multiplex plica motion with a doughnut-like shape. In an oscillatory ? eld, small drops were ejected from part of the periphery. Sherwood (1988) dealt with symmetric deformations (mode a) using a boundary organic technique. arrant(a) conductors or accurate dielectrics deform into prolate shapes in steady ? elds. With correct conductors, the tips have a small radius of curvature, and Sherwoods algorithmic program predicts licentiousness at the tip with unfavorable ? ld-strengths close to those found by Taylor (1964) and Brazier-Smith (1971). completeive dielectrics va unt identical general behaviour, and the maximum Annu. Rev. still Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by brownness University on 08/07/11. For personal use only. 50 SAVILLE aspect ratio is near that predicted by cypher arguments. With the drippy dielectric model, drops elongate and take on a shape with modify ends committed by a trim back neck. Since the calculation is quasi-static, cursory conduct can be followed in cases where insularity occurs.Here the talebearing(a) dielectric model depicts drop reference followed by withdrawal into individual droplets, behavior consistent with experimental results. In a porous dielectric, electrohydrodynamic stresses ? atten the almost-conic tips formed in spotless dielectrics or conductors. Sherwood de? nes two sorts of drop separation the unchanging mode where a conical tip develops and legal separation is via tip-streaming, and the EHD mode avocation mental unsoundness of the lengthened thread. Because of the numerical algorithms bodily structure it was not possible to study the other mode of disengagement identi? d by Torza et al (1971), which carcass a subject for coming(prenominal) study along with effects of viscosity. Curiously, conical tips of the sort identi? ed by Ramos & Castellanos (1994a) were not found in Sherwoods calculation. undermentioned tip geometry much beyond the point of asymmetry has not been possible although Basaran et al (1995) account statement signal detection embryonal blue thousands. Their computation includes dynamic effects with ? uid inertia balance against interfacial tension and stable forces. Although the focus is on perfect conductors and inviscid ? uids, small fountain-propelled planes were identi? d emanating from the tips. Inasmuch as electrohydrodynamic effects appeared to get over conical tip formation (Sherwood 1988), much more sudor go away be need to settle the issue of jet creation. In calculations to date, perfectly conducting, inviscid drops produce (immature) jets adhesive, holey dielectric drops do not. Predicting the aggression and structure of the thin jet appear from a Taylor cone is dif? cult, but EHD processes are understandably involved. Observations of liquid drops appear from a small capillary make this conclusion richly clear.Drops become smaller as the potential is raised, and when the potential reaches a sealed level, the drops emerge in a pulsating fashion. With further increases in the potential, the drop develops a Taylor cone that has a jet emerge from its tip. According to Hayati et al (1986, 1987a,b) and Cloupeau & Prunet-Foch (1990), impressive atomisation is possible only when the liquid conductivity lies in a trustworthy range. To ? rst order, the ? ow pattern inside the cone can be approximated by superimposing ? ow into a conical dunk onto the conical ? ow dictated by a digressive electric stress vary as r 2 .The tangential stress arises from c harge conduction to the interface. hit hookup stems from differences between the conductivity of the interior and outside(prenominal) ? uids. The ? ow pattern has a sanely unreasonable structure unruffled is supplied to the jet from the surface of the cone, date a recirculating twirl moves ? uid down the axis of the cone toward the supply. Of course this analysis omits expatiate of the jet, whose characteristics are submerged in the trace at the cone tip. history for charge convection on the surface of the cone, which clearlyAnnu. Rev. still Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by brown University on 08/07/11. For personal use only. ELECTROHYDRODYNAMICS 51 becomes important near the point (Ramos & Castellanos 1994b, Fernandez de la Mora & Loscertales 1994), has eluded analysis to date. silver CYLINDERS constancy of super superaerated Cylinders (Free Jets) Annu. Rev. unsound Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by brow ned University on 08/07/11. For personal use only. Here ? ? 2? a/ , a is the radius, Im ( ) and K m ( ) denote modi? d Bessel functions of order m with the prime sign denoting differentiation, and E 1 is the (radial) ? eld strength at the surface. When the inconsistency fails, the piston chamber oscillates. The quantity on the odd of Equation 42 is comparative to the step-up rate when the cylinder is impermanent and to the oscillation frequency when it is stable. For an neutral cylinder, instability is indicated when ? 1, that is, when 2? a. Electric charge expands the range of coseismic 2 wave numbers and increases festering rates. For ao E 1 / = 1, the range is some 0 ? 1. 35 ( 1. ? a). Interestingly, charge destabilizes non-axisymmetric deformations that are other stable the relation for these modes whitethorn be obtained from Equation 42 by equating the index of the Bessel functions with the mode for the angular deformation, cos(m), and changing 1 ? 2 to 1 ? 2 m 2 . treacly effects numb the motion, but their effect is such as to make some non-axisymmetric motions relatively more seismal (Saville 1971a). The theory for charged cylinders is in qualitative accord with Huebners (1969) ? nding of non-axisymmetric modes of breakup with highly charged water jets.Similar behavior exists with highly gummy cylinders, where, in addition to de change non-axisymmetric modes, the comportment of charge lowers the wavelength of the most seismal mode. sharpen relaxation on an initially uniformly charged jet does not appear to have been studied, although given the immenseness this process has with axile ? elds, the topic is of big interest. Taylor (1964) observed that generate charge has a very right on effect in preventing the break up of jets into drops under certain draw and an every bit fibrous effect in create violently unsteady movements ultimatelyShortly after Rayleighs pioneering newspaper publisher (Rayleigh 1882), Bassett (1894) showed how charge destabilizes a cylinder by a mechanics alike(p) to that found earlier with drops. Nevertheless, the process is more complex because a cylinder may be fallacious even in the absence of electrical forces. If the wavelength of a corrugation exceeds the circumference, then a varicose surface may have a smaller sports stadium than a tirade cylinder and be unstable because it has a lower (free) energy. By study the dynamics of a charged, inviscid cylinder, Bassett showed that an axisymmetric anxiety of wavelength testament grow if ? 0 2 ? ao E 1 K 0 (? ) ? Io (? 2 1 ? 1+? o 0. (43) Io (? ) K o (? ) 52 SAVILLE Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by chocolate-brown University on 08/07/11. For personal use only. disintegrating the jet into drops in others. Taylor was referring to the effects of a ? eld aline with the axis of a water jet. Racos (1968) experiments with poorly conducting liquids in any case show a ironlik e stabilizing effect. These results produce a dilemma of sorts. axile ? elds bring up stability with dielectric jets (Nayyar & Murthy 1960) because of the action of the habitual component of the electric ? eld on the deformed interface. scarce the required ? ld strengths are much larger than those encountered by Taylor, and the dual nature of electric forces storied by Taylor does not appear to be consistent with the behavior of perfect dielectrics. The role of electric stress can be appreciated by imagining an axisymmetric deformation of the surface of the form 1 + ? (z, t). The blueprint component of the electric stress on the interface of a perfect dielectric due to axile ? eld is 2 ao E 1 where the dielectric constants of the cylinder and outer ? uid are denoted by and ? . Thus protrusions are pushed inward and depressions outer regardless of the wavelength of the disturbance.In contrast, the conventionality stress on a charged, conducting cylinder is 2 ? ? K 1 (? ) a o E 1 1 (z, t), (45) K o (? ) so this stress resists deformation only when the term in brackets is positive, that is, for ? 0. 6. Moreover, with perfectly conducting ? uids some wavelengths are do more unstable. Although axile ? elds are seen to abet stability with dielectrics, large wavelengths (small ? , s) remain unstable. Since the stresses with perfect conductors or dielectrics are normal to the interface, the situation should be different with blabbermouthed dielectric materials due to tangential EHD stresses.The leaky dielectric equations have been work out for a viscous cylinder immersed in another viscous liquid under conditions where the on-line(prenominal) is continuous at the interface, that is, ignoring charge dishonour by relaxation, convection, and dilation of the surface. In terms of the dimensionless parameters in Equation 220 , ? C ? ? P & ? F . If, for example, we pick the process time to be the hydrodynamic time and identify it as that for a relatively inviscid mate? ? rial, (? a 3 / )1/2 , then o / ? (? a 3 / )1/2 ? 1. For distilled water, the electrical ? elaxation time is less than a msec and the hydrodynamic time for a 1-mm water jet is over a second for apolar liquids of the sort mentioned earlier, the relaxation time may be agelong ( ? 35 ms). In both case ? C ? ? P , so the approximation is appropriate. With leaky dielectrics the normal stress differs from that noted with perfect dielectrics, and there is overly a tangential (1 /)2 ? (z, t), Io (? )K 1 (? ) + / Io (? )K 1 (? ) ? (44) ELECTROHYDRODYNAMICS 53 stress due to induce charge. in advance deformation the surface is free of charge since the ? eld is parallel to the surface.Upon deformation