The Taylor Melcher Leaky Dielectric Model

Annu. Rev. wandering Mech. 1997. 292764 secure c 1997 by Annual Reviews Inc. each(prenominal) rights reserved ELECTROHYDRODYNAMICS The Taylor-Melcher tattling(a) Di galvanising stumper Annu. Rev. facile Mech. 1997. 2927-64. Downloaded from www. annual lay outed reviews. org by dark-brown University on 08/07/11. For in-personized purpose only. D. A. Saville division of Chemical Engineering, Princeton University Princeton, New Jersey 08544 strike WORDS electri? ed lapses and small fry-blacks, suspensions, embrasure military mission, good strike frivol away ABSTRACTElectrohydperchynamics deals with ? uid question induced by galvanising ? long meters. In the mid mid-sixties GI Taylor introduced the perceiveping non get wordductor molding to explain the style of spendlets de melodic phraseed by a crocked ? historic period, and JR Melcher calld it extensively to develop electrohydro impulsives. This review deals with the frameations of the seeping nonco nductor standard and nonificational diversenesss designed to setationvass its utilizationfulness. Although the azoic entropy-based studies supported the qualitative features of the lay, duodecimal acquiescement was poor.Recent studies argon in better proportionateness with the hypothesis. put though the puzzle was originally intended to deal with laconic embrasures, contemporary studies with suspensions in addition agree with the supposition. Clearly the blabberm bulge outhed insulator ride is more oecumenical than originally envisi angiotensin-converting enzymed. asylum The earliest record of an electrohydrodynamic try is in William Gilberts seventeenth hundred treatise de Mag last-placee, which casts the pathation of a c superstar-shaped status upon bringing a f subvertd rod above a sessile thieve (Taylor 1969).Nineteenth-century studies of fuddle dynamics revea conduct how radially look compresss stemming from interfacial rosiness offset st art tenseness (Rayleigh 1882), entirely until the 1960s more or little work focuse on the conduct of consum friend conductors, (mercury or weewee supply) or improve nonconductors (apolar runnys such(prenominal) as benzene). This began to change quest studies on badly conducting perspicuouss blabbermouthed di voltaicsby Allan & mason (1962). An new(prenominal) branch of electrohydrodynamics, electrokinetics, deals with the conduct of pressd spliticles in aqueous electrolytes (Saville 1977, Russel et al 1989). However, in that location ar signi? ant differences betwixt the look of electrolytes and porous nonconductor automobiles. In electrolytes, electrokinetic phenomena ar dominated by doinguate of port wine 27 0066-4189/97/0115-0027$08. 00 28 SAVILLE rec crude petroleum derived from covalently bound ionizable groups or ion adsorption. Near a move up fool awayd in this fashion, a penet pasture complaint cloud traffic manikins as electrolyte ions of opposite thrill argon attracted toward the porthole. A concentproportionn gradient human bodys so that scattering vestibular senses electromig dimensionn. thusce, when a ? historic period is compel, processes in this dole out layer govern the mechanics. In electrokinetics, apply ? ld qualifications be lilliputian, a few volts per cen ageter, whereas in electrohydrodynamics the ? yearss argon ordinarily untold big. With spotless conductors, entire(a) nonconductors, or holey di voltaics, turn out layers associated with correspondence crusade argon usually absent. Accordingly, development of the cardinal subjects proceeded more or less in numberently. Nevertheless, the chthonianlying processes sh ar many an(prenominal) peculiaritys. Most obvious is that galvanizing pluck and up-to-the-minute originate with ions therefore, censure whitethorn be induced in poorly conducting liquids even though offset weight down is absent.The Cardiovascular System The unlike treatments began to merge with the way of Taylors 1966 written written report on retch contortion and Melcher & Taylors review of the outlet (1969). Applications of electrohydrodynamics (EHD) abound spraying, the dispersion of one liquid in an new(prenominal), coalescence, ink greens printing, boiling, augmentation of fondness and deal transfer, ? uidized bed stabilization, pumping, and polymer dispersion argon moreover a few. Some applications of EHD argon striking. For example, EHD forces pee been used to simulate the populaces gravitational ? ld during convection look intos carried out during a station shuttle ? ight (Hart et al 1986). In this application, combining a radial voltaic ? long epoch with a temperature gradient surrounded by concentric worlds engenders polarization forces that mimic gravity. cardinal of the more unusual lookances of EHD involves the blue daze build above heavily afforest aras. BR Fish (1972) yi yearss look into al curtilage to support his proposition that the haze derives from pliant substances sprayed into the atmosphere from the cotton ons of pine indispensabilityles by spunky ? yearss accompanying the overhead passage of electri? d clouds during th lowstorms. This review concent targets on what has come to be known as the oozy dielectric model to elucidate its structure and describe its experimental effectations. For cortical potential into an early(a)(prenominal) chances of EHD, one or more of the many 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 close to constituentary system the talebearing(a) dielectric model consists of the Stokes equations to describe ? id social movement and an convention for the preservation of current employing an ohmic conduction. Electromechanical coupli ng occurs only at ? uid-? uid boundaries where spud, carried to the port by 1 Depending on the keywords used, supposer literature surveys turn up hundreds of physical compositions on EHD since the 1960s. Annu. Rev. roving Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by browned University on 08/07/11. For indivi doubleised use only. ELECTROHYDRODYNAMICS 29 Annu. Rev. changeable Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by chocolate-brown University on 08/07/11.For individual(prenominal) use only. conduction, gos electric directiones different from those pass on in absolute dielectrics or perfect conductors. With perfect conductors or dielectrics the electric mental strain is perpendicular to the interface, and misrepresentations of interface shape unite with interfacial tension serve to balance the electric striving. talebearing(a) dielectrics be different be buzz off dis bitch steer accumulated on the interface modi? es the ? el d. Viscous ? ow develops to provide tensiones to balance the consummation of the rambling components of the ? eld acting on interface surge.This review is organized as follows First the model is outlined to learn thoughts and potential pit falls. Then experimental and theoretical progenys for deuce prototypal geometries clears and piston chambers ar surveyed. This discussion go out evince the status of the seeping dielectric model where forces ar con? ned to a crafty bound. In closing, recent results on motion produced by EHD body forces are surveyed to shoot down aim how the model has been extended to new situations. respite LAWS The differential equations describing EHD plagiarize from equations describing the conservation of mass and momentum, linked with maxwells equations.To plunge a context for the approximations inherent in the drafty dielectric model, it is necessary to typeface on a deeper level. Then the leaky dielectric model arises naturally by a photographic rest home analysis. As far-famed earlier, the hydrodynamic model consists of the Stokes equations without any galvanizing forces coupling to the electric ? eld occurs at boundaries, so forces from the bag free press down on moldiness be negligible. muchover, the electric ? eld is solenoidal. The near section examines how to establish suss outs under which these approximations are appropriate. Scale Analysis and the Leaky Dielectric ModelUnder static conditions, electric and magnetised phenomena are independent since their ? elds are uncoupled (Feynman et al 1964). Insofar as the peculiar(prenominal) sentence for electrostatic processes is large compared to that for charismatic phenomena, the electrostatic equations furnish an accu valuate approximation. When impertinent magnetised ? elds are absent, magnetic cast nookie be ignored completely. From maxwells equations, the characteristic destination for electric phenomena, ? c ? o / , tooshie be id enti? ed as the ratio of the dielectric permeability2 (o ) and conduction3 ( ).For magnetic phenomena the characteristic condemnation, ? M ? o 2 , is the product of the magnetic permeability, o , conductivity, and the square of a characteristic length. Transport process time- shields, ? P , arise rationalized Meter-Kilogram-Second-Coulomb (MKSC) system of units will be used. conductivity will be de? ned explicitly in harm of properties of the constituent ions. For the present bank none simply that conductivity has the dimensions of Siemans per meter, i. e. , C2 -s/kg-m3 . 3 The 2 The 30 SAVILLE Annu. Rev. melted Mech. 1997. 2927-64. Downloaded from www. nnualreviews. org by brown University on 08/07/11. For individual(prenominal) use only. from awkward eternal sleep, spreading, cycle per second of an imposed ? eld, or motion ? M . The of a marge. dull processes are de? ned as those where ? P ? C second variation place be rear appreciationd to (/)1/2 o / (o o )1/2 , an d since (o o ) 1/2 is equal to the speed of light, 3 ? 108 m/s, (o o )1/2 is very small for our systems. For the electrostatic approximation to apply on a millimeterscale, the galvanising residuum time, o / , must be longer than 10 12 s. The inequality is satis? d easily because the conductivity is rarely large than one micro-Siemans per meter for liquids of the contour under story here. Accordingly, the electric phenomena are described by r o E = ? e and r ? E = 0. (2) (1) E is the electric ? eld strength, and ? e is the offspringal anesthetic free hurry density. edge conditions derived from compares 1 and 2 employ the inequality theorem and a pill-box system spanning a pot of a boundary exhibition that the discursive components of E are continuous and the convening component jumps by an amount comparative to the free blush per unit area, q, that is, ko Ek n = q. 3) hither k()k de lineages the jump, impertinent in spite of appearance, of () across the boundary, and n is the local remote recipe. Electrostatic phenomena and hydrodynamics are coupled through the Maxwell tune tensor. A simple way of seeing the descent amongst Maxwell formes and the galvanizing body force is to suppose that electrical forces exerted on free posture and commissioning dipoles are transferred directly to the ? uid. For a dipole level Q with orientation d the force is (Qd) rE. With N dipoles per unit volume, the dipole force is P rE P ? N Qd de? nes the polarization vector. The Coulomb force imputable to ree charge is ? e E, so the total electrical force per unit volume is ? e E + P rE. This force can be transformed into the divergence of a tensor, r o EE 1 o E E , apply equivalences 1 and 2. The tensor 2 dumbfounds the Maxwell direction tensor, M , ? ? ? 1 ? o 1 o EE EE , 2 ? T upon inserting the isotropic in? uence of the ? eld on the pressure (Landau & Lifshitz 1960). ELECTROHYDRODYNAMICS 31 Using the sort for the electrical nervous str ain produces the equation of motion for an incompressible Newtonian ? uid of uniform viscousness, Du (4) = rp + r M + r 2 u.Dt Alternatively, upon expanding the stress tensor the electrical stresses come out of the jamt as body forces due to a non-homogeneous dielectric permeability and free charge, on with the gradient of an isotropic contri scarcelyion, ? ? ? 1 Du = r p o ? EE ? Dt 2 ? T ? 1 (5) o E Er + ? e E + r 2 u. 2 For incompressible ? uids, the expression in brackets can be lumped together as a rede? ned pressure. EHD motions are driven by the electrical forces on boundaries or in the autocratic reportity. The net Maxwell stress at a cunning boundary has the universal and digressive components M M Annu. Rev. precarious Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by dark-brown University on 08/07/11. For private use only. 1 o (E n)2 2 n ti = qE ti n n = o (E t1 )2 o (E t2 )2 (6) afterward absorbing the isotropic violate of the s tress into the pressure as cite above. It is still that ti represents all of ii indifferent tangent vectors embedded in the advance. Denoting a characteristic ? eld strength as E o and balancing the tangential electrical stress in comparison 6 against sticky stress yields a ve2 locity scale of qE o / = o E o /.The same scale appears when the frequent 2 stress or the bulk electrical forces are used. With o E o as a scale for pressure, equality 5 can be cast in dimensionless form as ? u + Re u ru = ? P t rp 1 E Er + r (E)E + r 2 u. 2 (7) here the symbols represent dimensionless variables with lengths scale by and 2 time by the process scale ? P Re is a Reynolds bend, ? u o / ? ?o 2 E o /2 , when the electrohydrodynamic amphetamine scale is used for u o . Choosing ? = 103 kg/m3 , = 1 kg/m-s, = 10 3 m, and E o = 105 V/m gives Re ? 10 4 and a viscous ease time, ? ? 2 ? /, of 1 ms approximately. For a dielectric incessant of 4 and a conductivity of 10 9 S/m the el ectrical liberalization time, o / , is 35 ms. equivalence 1 shows how the ? eld is modify by the charge of free charge. In liquids, charge is carried by ions, so species conservation equations must be 32 SAVILLE included to complete the description. hand most charge density and ion tightfistedness are related as X ez k n k . (8) ? e = k Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by chocolate-brown University on 08/07/11. For individual(prenominal) use only. present e is the charge on a proton and z k is the valence of the k th species whose concentration is n k . set that some of the species may be electrically neutral, that is, z k = 0. Molecules and ions are carried by the ? ow and move in response to gradients in the electro chemic potential. If we denote the mobility of the k th species by k , the species conservation 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) here(predicate) k B is Boltzm anns constant, and T is the despotic temperature. The ? rst term on the right represents ion migration in the electric ? ld, the second describes get off by diffusion, and the third denotes production due to chemical receptions since the neutral species act as a source for ions in the bulk. With a wiz bonce species, N = 1 and r 1 = 0 for a binary, z-z electrolyte, N = 3. In the ? rst case, ions are produced by answers at electrodesthis is called unipolar injection. With a z-z electrolyte, ions are produced at the electrodes and by homogeneous reactions wi delicate the ? uid. Here attention is focused on liquids with charge from a sensation 1-1 electrolyte so that there are cardinal homogeneous reactions.A before reaction producing positive and banish ions from dissociation of the neutral species as (neutral species, k = 1, z 1 = 0) (cation, k = 2, z 2 = 1) + (anion, k = 3, z 3 = 1) (10a) with a send per unit volume, k+ n 1 , proportionate to the concentration of speci es 1. The recombination reaction is (cation, k = 2, z 2 = 1) + (anion, k = 3, z 3 = (neutral species, k = 1, z 1 = 0) 1) (10b) with a rate of k n 2 n 3 . The rate constants k+ and k are speci? c to the ions, neutral species, and solvent the rate of production of cations or anions is k+ n 1 k n 2 n 3 . thusly, r 1 = r 2 = r 3 = k+ n 1 k n 2n 3 (11) This situation contrasts crisply with that for strong electrolytes where neutral species are dissociated in full and reaction terms absent. Because ionic reactions ELECTROHYDRODYNAMICS 33 Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by brownish University on 08/07/11. For personal use only. are fast, it is convenient to imagine that the reactions are near at chemical equilibrium. At equilibrium, the local rate of reaction is zero, so K ? k+ /k = n 2 n 3 /n 1 .This complicates outlets because at equilibrium one of the conservation laws must be discarded to avoid an overdetermined system. To scale the problem consonantly, note that the concentrations of the two ionic species will be much smallr than the concentration of the neutral constituent. Accordingly, it is convenient to use different concentration scales. Neutral species concentrations are p scaled with a bulk concentration denoted as n 0 and ionic concentrations with n 0 K . Using 0 as a mobility scale (any one of the three mobilities) and k+ n 0 as a reaction rate scale produces the conservation law for the neutral species, ?D n 1 + Peu rn 1 = 1 r 2 n 1 ? P t and for each ionic 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 new symbols represent a characteristic diffusion time, ? 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 rates of ion transfer by convection to diffusion) a dimensionless ? eld strength, ? eE o /k B T and a Damkoler number, Da ? k+ 2 / 0 k B T (the ratio of a characte ristic diffusion time to a characteristic reaction time).The reaction term can be eliminated from comparability 12b exploitation equivalence 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 compute local concentrations for systems in local reaction equilibrium, compare 12c is used with k = 2 and k = 3, along with the equation for reaction equilibrium obtained from Equation 12a for Da 1. Equations 12c for k = 2 and k = 3 can be combined to furnish an expression for the dimensionless charge density, ? 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. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by dark-brown University on 08/07/11. For personal use only. From Equations 1 and 13 the characteristic charge simpleness time can now be identi? ed (in dimensional form) as o /e2 ( 2 n 2 + 3 n 3 ) ? o / . T o travel by simpli? cation of these equations, the magnitudes of the confused groups are estimated for small ions with a characteristic rundle4 , a, of 0. 25 nm using the Stokes-Einstein likeness, (6? a) 1 , for the mobility. Then Pe ? 105 , ? 03 , and the diffusion time ? D ? 106 s. Estimating the size of the early(a) dimensionless groups will require knowledge of the dissociation-recombination reactions. The equilibrium constant, K , is estimated from the Bjerrum-Fouss hypothesis of ion association (Fouss 1958, Moelwyn-Hughes 1965, Castellanos 1994) as in ? 3? e2 K = 3 exp . (14) 4a 8? ao k B T tour the recombination rate constant (Debye 1942) k = 4? e2 ( 2 + 3 ) o (15) gives a send rate constant of k+ = k K . (16) Using the data already introduced, K ? 1017 m 3 and k ? 10 18 m3 /s so k+ ? 10 1 s. Accordingly, Da ? 105 .To estimate the concentration of charge carriers, we use an expression for the conductivity of a solution with monovalent ions derived from a single 1-1 electrolyte = e2 ( 2 n 2 + 3 n 3 ). (17) For a conductivity 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 need to know the charge density. Equation 1 in dimensionless form is (18) 3r E = ? e = z(n 2 n 3 ) p 3 ? o E o /e n 0 K . Using the numeral values already de? ned, 3 ? 10 4 , suggesting the ? id is electrically neutral on the millimeter length scale. For 4 For comparison, the radius of a atomic number 11 ion in peeing is 0. 14 nm. The size of the charge carrying ions in apolar liquids is largely a matter of speculation, but the presence of traces of irrigate makes it likely that the charge carriers are bigger than the bare ions. ELECTROHYDRODYNAMICS 35 3 ? 1 Equation 13 yields the classical Ohms law approximation in dimensionless form, r (z)2 n 2 ( 2 + 3 )E = 0 (19) Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. a nnualreviews. org by brownish University on 08/07/11.For personal use only. as long as Pe3/ ? 1. With the quantitative magnitudes prone thus far, Pe3/ ? 10 2 , prompting the approximation expressed by Equation 19. To complete the description, charge conservation at the interface must be investigated. Here it is convenient to start with Equation 9 and integrate across an interface with the provision that there are no near reactions. Using the s-subscript to denote rise concentrations and operators, and ignoring any special channel processes such as lateral surface 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 surface divergence, and n k are surface concentrations. The terms s on the right stand for changes in concentration due to dilation of the surface and transport to the surface by electromigration and diffusion. Adding the two equations, weighing each by the valence and the charge 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) Next Equation 21 is put in dimensionless form using o E o as a surface charge 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 , appears here. For 1, the diffusion term can be ignored and conduction fit against charge relaxation and convection. For unfaltering motion, charge convection balances conduction when ? C /? F is O(1). Summary Equations for Leaky Dielectric Model To summarize, the leaky dielectric electrohydrodynamic model consists of the succeeding(a) ? ve equations. The derivation prone here identi? es the approximations in the leaky dielectric model.Except for the electrical body force terms, 36 SAVILLE it is essentially the model proposed by Melcher & Taylor (1969). ? u + Reu ru ? P t = Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualrev iews. org by brownness University on 08/07/11. For personal use 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) Note that the equations are scripted in dimensionless variables using the scales de? ned in the text.The equation of motion is for nonhomogeneous ? uids with electrical body forces. The hydrodynamic boundary conditions, continuity of speeding and stress, including the viscous and Maxwell stress, are assumed. Primes denote dimensionless forms of the sustain equations. Electrokinetic Effects Although Equation 19 may be adequate for p millimeter-length scales, it would P fail if free charge on the Debye scale, ? 1 ? o k B T /e2 (z k )2 n k , produces authoritative mechanical do. As noted earlier, aerated interfaces attract counterions in the bulk ? uid and repulse co-ions on the Debye length scale.Electric and h ydrodynamic phenomena on this scale are creditworthy for the ubiquitous carriage of small particles in electrolytes, so it is natural to ask whether akin(predicate) cause might be important here. In fact, Torza et al (1971) suggested that such personal cause could be responsible for the lack of engagement among the surmisal and their experiments on ? uid globules. To see whether the lack of apprehension is due to electrokinetic set up we can use the numerical data already put forth. This leads to the following 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 telling amid charge and ? eld is equal by Equation 18 while the species conservation equations 12 ELECTROHYDRODYNAMICS 37 sire ? 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. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by dark-brown Univ ersity on 08/07/11. For personal use only. These equations are intelligibly more daedal than those for Ohmic conduction, which omits simply any gradeing for individual species. Is this complexity necessary?In the following sections, experimental and theoretical results based on the leaky dielectric model are reviewed for some(prenominal) prototypical problems so as to task the models meativeness and the completion to which more detailed treatments taking account of diffuse layer effects are warranted. To date, none of the experimental studies show major electrokinetic effects despite the indications of the scale analysis. legato GLOBULES Drop Motion in outside Fields Allan & stonemason (1962) encountered paradoxical appearance when non-conducting dispatchs suspended in non-conducting liquids were deform by a truelove electric ? ld. Conducting fuddles became ovoid, as expected, but non-conducting swing outs often adopted pumpkin-shaped con? gurations. Oblate shapes were completely out of the blue(predicate) since analyses of static con? gurations predict oval-shaped torsions, disregardless of the cut back conductivity. Drop twistings can be analyse with several methods. OKonski & Thacher (1953) used an naught method Allan & Mason (1962) match electrical and interfacial tension forces. For small distortions of conducting throw aways in dielectric surroundings, either procedure gives D= 2 9 ao E 1 . 16 (24) Here E 1 is the strength of the applied ? ld, a is the redact radius, and is interfacial tension. The torture, D, is the difference amongst the lengths of the roam parallel and transverse to the ? eld divide by the sum of the two. Given that the drop is a conductor, it is easy to see why the shape is cucumber-shaped since the pressures inside and outside the drop are uniform, initially, with the difference balanced by interfacial tension and the spheres curvature, 2 /a. Therefore, non-uniform electric stresses must be balance d by interfacial tension on the 38 SAVILLE deformed surface. Since the sphere induces a dipole into the accident ? ld, charge on the spheres equipotential surface varies as cos cosmos deliberate from the direction of the ? eld. The ? eld general to the surface varies in a convertible fashion. Accordingly, the electric stress at the surface varies as cos2 , pulling the drop in opposite directions at its poles. Dielectric drops in dielectric surroundings likewise become ovate in level ? elds, regardless of the dielectric constants of the two ? uids, that is, 2 9 ao E 1 (? )2 (25) 16 (? + 2)2 with circum? exes denoting properties of the drop ? uid (OKonski & Thacher 1952, Allan & Mason 1962).Here overrefinement results from polarization forces since free charge is absent and the electric stresses are normal to the surface. Allan & Masons (1962) anomalous results led Taylor (1966) to discard the notion that the suspending ? uids could be case-hardened as insulators. Altho ugh the suspending ? uids were poor conductors ( 10 9 S/m) Taylor recognized that even a small conductivity would allow electric charge to reach the drop interface. With perfect dielectrics, the interface boundary condition (see Equation 3) sets the relation amidst the normal components of the ? eld to ensure that there is no free charge.For leaky dielectrics, charge accumulates on the interface to adjust the ? eld and ensure conservation of the current when the conductivities of the adjacent ? uids differ. The action of the electric ? eld on surface charge provides tangential stresses to be balanced by viscous ? ow. Taylor used the charge mensural from a solenoidal electric ? eld to compute the electric forces at the interface of a drop and then balanced these stresses with those mensurable for Stokes ? ow. This procedure led to a bang-up function to classify twistings as prolate or pumpkin-shaped 2M + 3 . 26) 8 = S(R 2 + 1) 2 + 3(S R 1) 5M + 5 Here S ? /? , R ? ? / , and M ? /. Prolate overrefinements are indicated ? when 8 1, and oblate forms are indicated when 8 1. Qualitative agreement amid theory and experiment was found in nine of the thirteen cases deal by Allan & Mason (1962). In the other(prenominal) 4 (prolate) cases, ambiguities in electrical properties hampered a test of the theory. According to Taylors leaky dielectric model, tangential electric stresses cause circulation conventions inside and outside the drop. As except con? mation of the theory, McEwan and de Jong5 photographed tracer particle tracks in and around a sili retinal strobilus polymer polymer oil drop suspended in a mixture of topper and corn oils. toroidal circulation patterns were spy, in agreement with the theory. Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by cook University on 08/07/11. For personal use only. D= 5 McEwan & de Jongs photos are presented in an addendum to Taylors paper (1966). ELECTROHYDRODYNAMICS 39 For a steady ? eld, Taylor (1966) gives the deformation as D= 2 9 ao E 1 8, 16 27) Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. so it is affirmable to test the theory quantitatively by measuring the length and breadth of drops for small deformations. However, Taylor did not publish a comparison betwixt theory and experiment. The ? rst quantitative tests were reported by Torza et al (1971), who extended the leaky dielectric model to deal with periodic ? elds. The deformation (0 D 0. 1) and burst of 22 ? id pairs were analyse in steady and periodic (up to 60 Hz) ? elds. In steady ? elds, oblate deformations were observed in eight systems, in qualitative accord with the theory. Although the qualitative aspects of the theory were vindicated, the quantitative agreement was very disap drawing. The deformation of all time change analogly 2 with a E 1 , but the symmetry factor exceeded the theoret ical value in all but one case, and the slopes were large by a factor of two in more than half the systems. In one case, the thrifty slope was four times the theoretical value.In none of the systems was the thrifty slope less than the theoretical value, suggesting that the deviations are due to factors other than normal experimental errors. Alternating ? elds offer additional insight into leaky dielectric expression. As 2 expected with alternating ? elds where forces vary as a E 1 cos2 ( t), the deformation consists of steady and oscillatory parts (Torza et al 1971) D = D S + DT . 8S = 1 (28) The steady part, Ds , has the same form as Equation 27, but the 8-function is S 2 R(11+14M)+15S 2 (1+M)+S(19+16M)+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 angular frequency, and ? represents a hybrid electrical relaxation time o / ? . According to Equation 29 the steady part of the deformation vanishes at a certain frequency and may shift from one for m to the other with changes in frequency. Torza et al (1971) careful the steady part of the deformation for all 22 systems in 60-Hz ? elds and obtained results mistakable to those for 2 steady ? elds. The deformation was proportional to a E 1 , and in ? ve cases theory and experiment were in quantitative agreement.With the other systems the measured slopes exceeded the theoretical values by upstanding margins. The transition from oblate to prolate deformation was reported for one systema 40 SAVILLE silicone oil drop in sextolphthalate with S ? /? ? 2. 2 and R ? ? / 0. 07. However, the observed transition frequency (1. 6 Hz) was considerably lower than predicted (2. 5 Hz), although the two could be brought into agreement by laboured S to 1. 8. In this context the authors state This suggests that accurate measurements of the dielectric constants of the phases are crucial to a quantitative test of the theory. This observation will be revisited shortly. Some of the dissimilitude about oscillatory ? elds could be ascribed to the neglectfulness of temporal acceleration. Torza et al (1971) used a quasi-steady approximation, tantamount to ignoring ? u/t in the equations of motion. Upon including this acceleration, Sozou (1972) found qualitatively different behavior at high frequencies. For example, the steady part of the stress tends to zero, so this part of the deformation vanishes. With the quasi-steady approximation (see Equation 29), the deformation ashes ? nite.Although this observation might account for some of the differences among theory and experiment in oscillatory ? elds, it does not collapse the low-frequency dif? culties. Torza et als study (1971) provides additional con? rmation of the qualitative aspects of the leaky dielectric model, but the lack of quantitative agreement is disconcerting. crimson with water drops whose conductivity is ? ve raises of magnitude larger than the suspending ? uid, deviations between theory and experiment are s ubstantial. Several reasons for the discrepancies were suggested.Lateral motion of charge along the interface due to surface conduction and convection of surface charge were ruled out since they ought to make the relation 2 between deformation and a E 1 nonlinear. Other possibilities were suggested unspeci? ed deviations from the boundary conditions, space charge in the bulk, and diffuse charge clouds due to counterion attraction (cf Equations 23a,b). In an essay to address the boundary conditions issue, Ajayi (1978) employed derangement methods to account for nonlinear effects in the deformation. This 2 analysis represents the shape using a power series in ao E 1 / .By carrying the analysis through the second dress in the small parameter, Ajayi found that P2 (cos ) and P4 (cos ) are required to represent the surface and the deforma2 tion is no longer directly proportional to ao E 1 / . Considering nonlinear effects helps to an extent, but Ajayi observed that the method cannot e xtract the discrepancy between theory and experiment. other possibility advanced as a source of disagreement involves electrokinetic effects (Torza et al 1971). Given the results of the earlier scale analysis, this theory appeared worth investigating yet.Baygents & Saville (1989) addressed the matter using asymptotic methods to account for the in? uence of a diffuse layer arising from century interactions between current carrying ions and the surface charge. leash layers were identi? ed where different processes dominate. A diffuse layer adjacent to the surface is stranded from an outer(a) region, where the leaky dielectric model applies, by an intermediate region. In Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. ELECTROHYDRODYNAMICS 41 he diffuse layer, electrokinetic processes due to space charge are relevant. The intermediate region is electrically neutral, and charge transport by diffusi on, electromigration, and convection are equally important. In the outer region, the electrohydrodynamic equations prevail. settlement the differential equations involved matched asymptotic expansions, and because of the altered structure of the problem, the distributions of velocity and stress differ from those derived using the leaky dielectric model. Nevertheless, the ? nal expression for drop deformation is identical to that derived by Taylor (1966).Electrokinetic effects dont appear to contradict conclusions drawn from the leaky dielectric model, which, based on this analysis, appears to be an use up lumped parameter description. Since none of the theoretical extensions appeared to fall the divergence between theory and experiment, further experiments were undertaken. Following Torza et als (1971) speck regarding the need for accurate dielectric constants and other properties (see above), Vizika & Saville (1992) paid careful attention to direct measurement of physical prope rties.They canvas eleven different drop-host systems in steady ? elds oscillatory ? elds were employed with ? ve systems. The systems exhibited either prolate or oblate deformations. To increase the deformation, a non-ionic surfactant, Triton, was used in some cases to lower the interfacial tension. Generally speaking, agreement between theory and experiment improved over the earlier study. gauge 1 shows some 2 results with steady ? elds. In all cases, D varied linearly with a E 1 . Vizika & Saville (1992) observed time-dependent effects in some cases, especially with the surfactant. Evidently the ? ids were not completely immiscible, and mass transfer occurred between phases. In these cases it was necessary to remeasure the properties after time had elapsed to permit equilibration. Moreover, in cases where the conductivities of the two phases were comparable, ? eld-dependent effects were often observed. In oscillatory ? elds, the steady part of the deformation was measured at 2 different ? eld strengths D S always varied linearly with a E 1 . The agreement between theory and experiment for the steady part of the deformation was loosely better than with the same systems in a steady ? ld. With water in castor oil, for example, the getd and measured slopes differed by 34% in a steady ? eld in a 60-Hz ? eld the two agreed. Figure 2 summarizes results with four systems. some other(prenominal) interesting aspect of the leaky dielectric model concerns the effect of frequency. Torza et al (1971) showed, for example, that a drop that assumes an oblate deformation at low frequencies becomes prolate as the fre2 quency increases, that is, Ds /a E 1 increases with frequency. This behavior was measured with silicone drops suspended in castor oil results are shown in Figure 3.The qualitative agreement between theory and experiment was Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. 4 2 SAVILLE Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. adequate, but as the ? gure indicates, the behavior is quite sensitive to the drop conductivity. Vizika & Saville (1992) compared theory and experiment for the oscillatory part of the deformation with one system excellent agreement was obtained.Further encouraging comparisons between theory and experiment were reported by Tsukada et al (1993), who canvas deformations with the castor oilsilicone oil system. Castor oil drops in silicone oil gave prolate deformations, oblate deformations were found with the ? uids reversed. In addition to experimental work, a ? nite fraction technique was employed to calculate deformations in steady ? elds. Except for the inclusion body of ? nite deformations and (a) Figure 1 Deformation measurements for ? uid drops (Vizika & Saville 1992).Drops are prolate or oblate depending on whether D 0 or D 0. The da shed lines represent calculations made with the leaky dielectric model using measured ? uid properties solid lines are least-squares representations of the experimental data. In Figure 1b the theoretical line for the f number set of data is not shown since it falls on the regression line for the lower data for this system the difference between theory and experiment is large. ELECTROHYDRODYNAMICS 43 Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11.For personal use only. (b) Figure 1 (Continued) inertial effects, the standard leaky dielectric model was employed. At small deformations, numerical results agreed with those from Taylors linear theory. With larger deformations, substantial differences appeared. Most of the differences between the ? nite element calculation and the linear theory were due to interface deformation since the Reynolds number in the calculations was always small. For prolate drops, the numerical resu lts and Taylors theory agreed with the experimental data for 0 D 0. 07.With larger deformations, for example, for D ? 0. 2, the ? nite element solution was better than the linear theory but still predicted littler deformations than those observed. In addition, the agreement between Taylors theory and the experiment for oblate drops exhibited a puzzling feature, that is, for large deformations the linear theory was closer to the experimental results than the nonlinear ? nite element calculation. These three studies constitute the most nationwide test of the theory wherein interface charge arises from conduction across an interface.The agreement between theory and experiment is encouraging, and there seems little doubt that, insofar as drop deformation is concerned, the theory does a satis- 44 SAVILLE Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. Figure 2 The steady part of the drop deformation in oscillatory ? elds (Vizika & Saville 1992). pulverisation job. Nevertheless, only a watched number of ? uids shake off been studied, and even in these cases, conductivities have not been controlled. Questions as to ? ite bounteousness effects or charge convection due to interface motion hang on to be investigated. In situations discussed thus far, charge convection has been ignored since ? C /? F ? 1. To investigate the in? uence of charge convection, the HadamardRybczynski settling velocity for a orbiculate drop can be studied. Although no experimental studies populate, calculations with the model indicate a substantial in? uence. First note that the velocity will be dateless if a steady ? eld is imposed because, as long as charge convection is negligible, the net charge is zero and the ? ld exerts no net force on the drop. However, an noninterchangeable charge distribution creates a net force charge convection due to depository generates the necessary asymmetry. The relevant boundary condition is Equation 220 rewritten for steady ? ow, ? c rs (uq) = k Ek n. (30) ? F ELECTROHYDRODYNAMICS 45 Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. Figure 3 The unfirm part of the drop deformation as a function of frequency for silicone drops in castor oil (Vizika & Saville 1992).Torza et als (1971) theoretical result is shown for two values of the drop conductivity other parameters correspond with measured values. 2 Here the ? ow time will 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 = 105 V/m, so a linearized treatment is appropriate (Spertell & Saville 1976). solve the equations shows the settling velocity is retarded or increased depending on the electrical relaxation times 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 settling velocity for a rigid 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 charge convection, drop deformation is no longer trigonal with respect to the midplane. These results show clearly that charge convection has different effects, either enhancing or retarding sedimentation, depending on the charge relaxation times in the two ? uids. 46 SAVILLE Given that interface charge induced by the action of an electric ? eld in leaky dielectrics has important effects on quasi-static motions, the next task is to inquire as to its effects on drop stability.Drop Stability and interval To provide a context to study the role of tangential stresses it is helpful to telephone work on perfect conductors and dielectrics. Studies of drop dynamics6 and stability began with Rayleighs celebrated 1882 paper On the equilibrium of liquid conducting multitude charged with electricity. His analysis pertains to fast charge relaxation inside an isolated drop, and the relation7 between the f requency, , interfacial tension, , and drop charge, Q, is ? Q2 2 = n(n 1) (n + 2) 3 (33) ? a 16? 2 o ? a 6 for axisymmetric oscillations of an inviscid drop of radius a and density ?.Here n denotes the office of the Legendre polynomial Pn (cos ). For absolutely conducting ? uids, the electric stress is wholly normal to the interface. Instability occurs for n = 2 when the charge increases to a level where the expression in brackets vanishes. Because a linearized, ellipsoid approximation is used, either oblate or prolate deformations are included. Although the Rayleigh limit pertains rigorously to small oscillations, dimensional analysis shows that the criterion for fluentness will still be of the form Q2 o a 3 C, (34) Annu. Rev. Fluid Mech. 1997. 927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. but the constant C will depend on the properties of the surrounding ? uid. An EHD model of a leaky dielectric drop hover in an i nsulating ? uid addresses effects of charge relaxation inside the drop through a boundary condition for the conservation of interfacial charge, q. Accordingly, the model consists of linearized8 equations of motion for incompressible ? uids inside and outside the drop, ? u = rp + r 2 u, ? p t r u = 0, (35) relations between the ? ld and the current in each phase, r E = 0, r ? E = 0, (36) 6 Rayleighs supposition of Sound (1945) contains many fascinating accounts of early work on drops and cylinders. 7 sequester 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 undeformed drop. ELECTROHYDRODYNAMICS 47 and boundary conditions. The relation between ? eld and charge is s tiptopulation by the dimensionless form of Equation 3, kEk n = q, (37) with charge scaled on the charge density on the undeformed drop, Q/4? a 2 .The scale for the electric ? eld, E o , is Q/4? o a 2 . shiv er on the interface is conserved, and for 1 the balance is Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown 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 current on the right-hand side represents conduction from the privileged. For Rayleighs perfectly conducting drop, the local charge balance is unnecessary because (a) the ? ld is nonentity inside the drop, and (b) charge transport is instantaneous so that the convection and relaxation terms vanish. The remain boundary conditions are continuity of velocity and stress, and the kinematic condition. These equations have been solved to investigate how relaxation alters Rayleighs results (Saville 1974). Both viscous forces and charge relaxation effects were included, but general conclusions were obscured by the awkward transcendental form of the ch aracteristic equation. However, asymptotic methods can be used to identify the salient features. The result for a slightly viscous drop in the absence seizure of a suspending ? id is alternatively surprising in as much as Rayleighs result (see Equation 33) is recovered as the stability criterion. Even when charge relaxation by conduction is torpid, charge convection still redistributes charge so rapidly that the oscillation frequency is apt(p) by Equation 33. A similar explanation was proposed by Melcher & Schwartz (1968) in their study of planar interfaces. Although EHD effects fail to alter the oscillation frequency, damping rates are affected. If the damping rate is denoted as , then )t R represents the the amplitude of the oscillation 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 viscosity ?. Whe n electrohydrodynamic effects are included and the oscillation time is comparable to the conduction time, that is when o o / ? O(1), damping is slower ? (40) ? (2n 3)(n 1) 2 . a 48 SAVILLE Other interesting effects can be identi? ed, including modes involving rapid 2 damping in a thin boundary layer where the rate is proportional to (a / 2 ) 3 . Rayleighs criterion also applies to very viscous drops with rapid charge a , has a relaxation. In contrast, slow charge relaxation, that is, o / substantial effect on highly viscous systems. Here the criterion for stability is altered to Q2 16? 2 a 3 o 40? + 180 10? + 9 (41) Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. for a viscous drop with dielectric constant and viscosity in a ? uid with ? ? 1 and where ? and (a /? ? 2 ) 2 ? 1. The next subject concerns behavior beyond the range where a linear treatment is acceptable.To form a simple mod el, the withdrawal of isolated drops or drops in immaterial ? elds can be treated by approximate methods. A spheroidal approximation (Taylor 1964) yields an accurate expression for the stability of an isolated charged drop or an uncharged drop in an external ? eld. More recent work9 shows that prolate shapes evolving below the Rayleigh limit are risky to axisymmetric perturbations while oblate shapes above the limit are stable to axisymmetric perturbations but fallacious to nonaxisymmetric perturbations. Thus, the Rayleigh limit turns out to be the absolute limit of stability.Dimensional analysis indicates that the criterion for dissymmetry of a conducting drop immersed in a gas and stressed by an external ? eld has the form 2 ao E 1 C. (42) Taylors spheroidal approximation (Taylor 1964) gives C = 2. 1 ? 10 3 for D = 0. 31, in good agreement with experiments on liquid ecstasy ? lms. The limiting deformation corresponds to a drop with an aspect ratio of 1. 9. Above this point the drop is seen to throw off liquid as a ? ne jet-black. Taylor (1964) analyzed the region near the spheroidal tip, which becomes conelike (a Taylor cone) at the limit of stability. For a cone with a eyeshade slant of 98. , electric stresses on a conducting surface are balanced exactly by surface tension. It turns out that conical tips also exist as static solutions when one perfect dielectric is immersed in another and S ? /? ? 17. 6 (Ramos & Castellanos 1994a). At the limit the peak angle is 60 . For S 17. 6, two solutions exist. One has a vertex angle larger than 60 the other is smaller. At S = 0 the vertex angles are 0 and 98. 6 , the latter(prenominal) corresponding to Taylors solution for an equipotential cone. 9 Pelekasis et al (1990) and Kang (1993) provide useful summaries of the ever-changing stability of perfectly conducting drops.ELECTROHYDRODYNAMICS 49 More extensive analyses of the static behavior of drops break up behavior consistent with this picture Sherw ood (1988, 1991) studied free drops Wohlhuter & Basaran (1992) and Ramos & Castellanos (1994b) analyzed drops pinned to an electrode. According to the various computations, a dielectric drop immersed in another perfect dielectric elongates into an equilibrium shape as the ? eld increases when S S1 . For S S2 S1 the drops become liquid at turning points in the deformation-? eld strength relation.In the range S1 S S2 there is hysteresis drops are stable on the lower and speeding branches of the relation and unstable in between. determine of S2 calculated by various methods are close to the value identi? ed as the maximum value for the existence of a cone. Wohlhuter & Basaran (1992) and Ramos & Castellanos (1994b), who studied pendant and sessile drops between plates, delineate other quantitative effects due to contact angle, drop volume, and plate spacing. How do EHD phenomena modify this picture? Interestingly, solutions for a leaky dielectric cone immersed in another leaky d ielectric ? id exist for R ? ? / 17. 6, independent of the dielectric constants (Ramos & Castellanos 1994a). Because of tangential 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 normal component of the viscous stress is zero. As required, the tangential electric stress along the periphery of the cone is balanced by viscous stress. A circulation pattern exists inside and outside the cone with ? id lamentable toward the apex along the interface and away from it along the axis. One might imagine that a certain level of conductivity is necessary for the formation of a sharp point and the ensuing micro-jet (see below). Allan & Mason (1962) and Torza et al (1971) observed three modes of drop deformation and licentiousness at high ? eld-strengths (a) water drops in oil de formed symmetrically, and globules purposeless off from a liquid quarter (b) castor oil drops in silicone oil deformed asymmetrically with a long thread pulled out toward the negative electrode and (c) silicone oil drops in castor oil ? ttened and broken up unevenly. For modes a and b the initial deformation was prolate for mode c it was oblate. The breakup of oblate drops in steady ? elds involved a complex folding 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 integral technique. Perfect conductors or perfect dielectrics deform into prolate shapes in steady ? elds. With perfect conductors, the tips have a small radius of curvature, and Sherwoods algorithmic rule predicts breakup at the tip with critical ? ld-strengths close to those found by Taylor (1964) and Brazier-Smith (1971). Perfect dielectrics demonstration similar overall behavio r, and the maximum Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. 50 SAVILLE aspect ratio is near that predicted by energy arguments. With the leaky dielectric model, drops elongate and take on a shape with modify ends connected by a thin neck. Since the calculation is quasi-static, transient behavior can be followed in cases where breakup occurs.Here the leaky dielectric model depicts drop file name extension followed by breakup into individual droplets, behavior consistent with experimental results. In a leaky dielectric, electrohydrodynamic stresses ? atten the almost-conical tips formed in perfect dielectrics or conductors. Sherwood de? nes two sorts of drop breakup the electrostatic mode where a conical tip develops and breakup is via tip-streaming, and the EHD mode following unbalance of the elongated thread. Because of the numerical algorithms structure it was not possible to study the ot her mode of breakup identi? d by Torza et al (1971), which rest a subject for future study along with effects of viscosity. Curiously, conical tips of the sort identi? ed by Ramos & Castellanos (1994a) were not found in Sherwoods calculation. Following tip geometry much beyond the point of instability has not been possible although Basaran et al (1995) report detecting embryonic jets. Their computation includes dynamic effects with ? uid inertia balanced against interfacial tension and electrostatic forces. Although the focus is on perfect conductors and inviscid ? uids, small jets were identi? d emanating from the tips. Inasmuch as electrohydrodynamic effects appeared to suppress conical tip formation (Sherwood 1988), much more stew will be required to resolve the issue of jet creation. In calculations to date, perfectly conducting, inviscid drops produce (immature) jets viscous, leaky dielectric drops do not. Predicting the onset and structure of the thin jet rising from a Tayl or cone is dif? cult, but EHD processes are clearly involved. Observations of liquid drops acclivitous from a small capillary make this conclusion abundantly clear.Drops become smaller as the potential is raised, and when the potential reaches a certain level, the drops emerge in a pulsating fashion. With further increases in the potential, the drop develops a Taylor cone that has a jet emerging from its tip. According to Hayati et al (1986, 1987a,b) and Cloupeau & Prunet-Foch (1990), effective atomization is possible only when the liquid conductivity lies in a certain range. To ? rst order, the ? ow pattern inside the cone can be approximated by superimposing ? ow into a conical dangle onto the conical ? ow driven by a tangential electric stress vary as r 2 .The tangential stress arises from charge conduction to the interface. Charge accumulation stems from differences between the conductivity of the interior and exterior ? uids. The ? ow pattern has a close to counterintuitive structure Liquid is supplied to the jet from the surface of the cone, while a recirculating twirl moves ? uid down the axis of the cone toward the supply. Of contour this analysis omits details of the jet, whose characteristics are subaquatic in the singularity at the cone tip. Accounting for charge convection on the surface of the cone, which clearlyAnnu. Rev. Fluid 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 apex (Ramos & Castellanos 1994b, Fernandez de la Mora & Loscertales 1994), has eluded analysis to date. FLUID CYLINDERS Stability of supercharged Cylinders (Free Jets) Annu. Rev. Fluid Mech. 1997. 2927-64. Downloaded from www. annualreviews. org by Brown 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 inequality fails, the cylinder oscillates. The quantity on the left of Equation 42 is proportional to the growth rate when the cylinder is unstable and to the oscillation frequency when it is stable. For an uncharged cylinder, instability is indicated when ? 1, that is, when 2? a. Electric charge expands the range of unstable 2 wave numbers racket and increases growth rates. For ao E 1 / = 1, the range is approximately 0 ? 1. 35 ( 1. ? a). Interestingly, charge destabilizes non-axisymmetric deformations that are differently stable the relation for these modes may be obtained from Equation 42 by liken the index of the Bessel functions with the mode for the angular deformation, cos(m), and changing 1 ? 2 to 1 ? 2 m 2 . Viscous effects dampen the motion, but their effect is such as to make some non-axisymmetric motions relatively more unstable (Saville 1971a). The theory for charged cylinders is in qualitative accord with Hu ebners (1969) ? nding of non-axisymmetric modes of breakup with highly charged water jets.Similar behavior exists with highly viscous cylinders, where, in addition to destabilizing non-axisymmetric modes, the presence of charge lowers the wavelength of the most unstable mode. Charge relaxation on an initially uniformly charged jet does not appear to have been studied, although given the importance this process has with axile ? elds, the topic is of considerable interest. Taylor (1964) observed that induced charge has a very the right way effect in preventing the break up of jets into drops under certain circumstances and an equally powerful effect in causing violently unsteady movements ultimatelyShortly after Rayleighs pioneering paper (Rayleigh 1882), Bassett (1894) showed how charge destabilizes a cylinder by a mechanism similar to that found earlier with drops. Nevertheless, the process is more complex because a cylinder may be unstable even in the absence of electrical forces. If the wavelength of a corrugation exceeds the circumference, then a varicose surface may have a smaller area than a circular cylinder and be unstable because it has a lower (free) energy. By perusing the dynamics of a charged, inviscid cylinder, Bassett showed that an axisymmetric disturbance of wavelength will 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 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 aligned with the axis of a water jet. Racos (1968) experiments with poorly conducting liquids also show a strong stabilizing effect. These results produce a quandary of sorts. Axial ? elds pull ahead stability with dielectric jets (Nayyar & Murthy 1960) because of the action of the normal component of the electric ? eld on the deformed interface. But the require d ? ld strengths are much larger than those encountered by Taylor, and the dual nature of electric forces noted 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 normal component of the electric stress on the interface of a perfect dielectric due to axial ? 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 outward irrespective of the wavelength of the disturbance.In contrast, the normal stress on a charged, conducting cylinder is 2 ? ? K 1 (? ) ao 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 made more unstable. Although axial ? elds are seen to promote stability with di electrics, 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 leaky dielectric materials due to tangential EHD stresses.The leaky dielectric equations have been solved for a viscous cylinder immersed in another viscous liquid under conditions where the current is continuous at the interface, that is, ignoring charge transport by relaxation, convection, and dilation of the surface. In terms of the dimensionless parameters in Equation 220 , ? C ? ? P & ? F . If, for example, we choose 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 millisecond 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 longer ( ? 35 ms). In either case ? C ? ? P , so the approximation is appropriate. With leaky dielectrics the normal stress differs from that noted with perfect dielectrics, and there is also a tangential (1 /)2 ? (z, t), Io (? )K 1 (? ) + / Io (? )K 1 (? ) ? (44) ELECTROHYDRODYNAMICS 53 stress due to induced charge. Before deformation the surface is free of charge since the ? eld is parallel to the surface.Upon deformation