The Taylor Melcher Leaky Dielectric Model

Annu. Rev. Fluid Mech. 1997. 29:27–64 Copyright c 1997 by Annual Reviews Inc. All rights reserved ELECTROHYDRODYNAMICS: The Taylor-Melcher Leaky Dielectric Model Annu. Rev. Fluid Mech. 1997. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. D. A. Saville Department of Chemical Engineering, Princeton University Princeton, New Jersey 08544 KEY WORDS: electri? ed drops and jets, suspensions, interface charge, bulk charge ABSTRACT
Electrohydrodynamics deals with ? uid motion induced by electric ? elds. In the mid 1960s GI Taylor introduced the leaky dielectric model to explain the behavior of droplets deformed by a steady ? eld, and JR Melcher used it extensively to develop electrohydrodynamics. This review deals with the foundations of the leaky dielectric model and experimental tests designed to probe its usefulness. Although the early experimental studies supported the qualitative features of the model, quantitative agreement was poor.
Recent studies are in better agreement with the theory. Even though the model was originally intended to deal with sharp interfaces, contemporary studies with suspensions also agree with the theory. Clearly the leaky dielectric model is more general than originally envisioned. INTRODUCTION The earliest record of an electrohydrodynamic experiment is in William Gilbert’s seventeenth century treatise de Magnete, which describes the formation of a conical shape upon bringing a charged rod above a sessile drop (Taylor 1969).

Nineteenth-century studies of drop dynamics revealed how radially directed forces stemming from interfacial charge offset surface tension (Rayleigh 1882), but until the 1960s most work focused on the behavior of perfect conductors, (mercury or water) or perfect dielectrics (apolar liquids such as benzene). This began to change following studies on poorly conducting liquids—leaky dielectrics—by Allan & Mason (1962). Another branch of electrohydrodynamics, electrokinetics, deals with the behavior of charged particles in aqueous electrolytes (Saville 1977, Russel et al 1989). However, there are signi? ant differences between the behavior of electrolytes and leaky dielectrics. In electrolytes, electrokinetic phenomena are dominated by effects of interface 27 0066-4189/97/0115-0027$08. 00 28 SAVILLE charge derived from covalently bound ionizable groups or ion adsorption. Near a surface charged in this fashion, a diffuse charge cloud forms as electrolyte ions of opposite charge are attracted toward the interface. A concentration gradient forms so that diffusion balances electromigration. Then, when a ? eld is imposed, processes in this diffuse layer govern the mechanics. In electrokinetics, applied ? ld strengths are small, a few volts per centimeter, whereas in electrohydrodynamics the ? elds are usually much larger. With perfect conductors, perfect dielectrics, or leaky dielectrics, diffuse layers associated with equilibrium charge are usually absent. Accordingly, development of the two subjects proceeded more or less independently. Nevertheless, the underlying processes share many characteristics. Most obvious is that electric charge and current originate with ions; therefore, charge may be induced in poorly conducting liquids even though equilibrium charge is absent.
The different treatments began to merge with the appearance of Taylor’s 1966 paper on drop deformation and Melcher & Taylor’s review of the topic (1969). Applications of electrohydrodynamics (EHD) abound: spraying, the dispersion of one liquid in another, coalescence, ink jet printing, boiling, augmentation of heat and mass transfer, ? uidized bed stabilization, pumping, and polymer dispersion are but a few. Some applications of EHD are striking. For example, EHD forces have been used to simulate the earth’s gravitational ? ld during convection experiments carried out during a space shuttle ? ight (Hart et al 1986). In this application, combining a radial electric ? eld with a temperature gradient between concentric spheres engenders polarization forces that mimic gravity. One of the more unusual appearances of EHD involves the blue haze found above heavily forested areas. BR Fish (1972) provides experimental evidence to support his proposition that the haze derives from waxy substances sprayed into the atmosphere from the tips of pine needles by high ? elds accompanying the overhead passage of electri? d clouds during thunderstorms. This review concentrates on what has come to be known as the leaky dielectric model to elucidate its structure and describe its experimental foundations. For insight into other aspects of EHD, one or more of the many reviews or monographs1 may 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 most elementary form the leaky dielectric model consists of the Stokes equations to describe ? id motion and an expression for the conservation of current employing an Ohmic conductivity. Electromechanical coupling occurs only at ? uid-? uid boundaries where charge, carried to the interface by 1 Depending on the keywords used, computer literature surveys turn up hundreds of papers on EHD since the 1960s. Annu. Rev. Fluid Mech. 1997. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. ELECTROHYDRODYNAMICS 29 Annu. Rev. Fluid Mech. 1997. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11.
For personal use only. conduction, produces electric stresses different from those present in perfect dielectrics or perfect conductors. With perfect conductors or dielectrics the electric stress is perpendicular to the interface, and alterations of interface shape combined with interfacial tension serve to balance the electric stress. Leaky dielectrics are different because free charge accumulated on the interface modi? es the ? eld. Viscous ? ow develops to provide stresses to balance the action of the tangential components of the ? eld acting on interface charge.
This review is organized as follows: First the model is outlined to identify approximations and potential pitfalls. Then experimental and theoretical results for two prototypical geometries—drops and cylinders—are surveyed. This discussion will establish the status of the leaky dielectric model where forces are con? ned to a sharp boundary. In closing, recent results on motion produced by EHD body forces are surveyed to indicate how the model has been extended to new situations. BALANCE LAWS The differential equations describing EHD arise from equations describing the conservation of mass and momentum, coupled with Maxwell’s equations.
To establish a context for the approximations inherent in the leaky dielectric model, it is necessary to look on a deeper level. Then the leaky dielectric model arises naturally through a scale analysis. As noted earlier, the hydrodynamic model consists of the Stokes equations without any electrical forces; coupling to the electric ? eld occurs at boundaries, so forces from the bulk free charge must be negligible. Moreover, the electric ? eld is solenoidal. The next section examines how to establish conditions under which these approximations are appropriate. Scale Analysis and the Leaky Dielectric Model
Under static conditions, electric and magnetic phenomena are independent since their ? elds are uncoupled (Feynman et al 1964). Insofar as the characteristic time for electrostatic processes is large compared to that for magnetic phenomena, the electrostatic equations furnish an accurate approximation. When external magnetic ? elds are absent, magnetic effects can be ignored completely. From Maxwell’s equations, the characteristic time for electric phenomena, ? c ? “”o / , can be identi? ed as the ratio of the dielectric permeability2 (“”o ) and conductivity3 ( ).
For magnetic phenomena the characteristic time, ? M ? µµo `2 , is the product of the magnetic permeability, µµo , conductivity, and the square of a characteristic length. Transport process time-scales, ? P , arise rationalized Meter-Kilogram-Second-Coulomb (MKSC) system of units will be used. conductivity will be de? ned explicitly in terms of properties of the constituent ions. For the present note simply that conductivity has the dimensions of Siemans per meter, i. e. , C2 -s/kg-m3 . 3 The 2 The 30 SAVILLE Annu. Rev. Fluid Mech. 1997. 29:27-64. Downloaded from www. nnualreviews. org by Brown University on 08/07/11. For personal use only. from viscous relaxation, diffusion, oscillation of an imposed ? eld, or motion ? M . The of a boundary. Slow processes are de? ned as those where ? P ? C second inequality can be rearranged to (“/µ)1/2 “o / `(µo “o )1/2 , and since (µo “o ) 1/2 is equal to the speed of light, 3 ? 108 m/s, `(µo “o )1/2 is very small for our systems. For the electrostatic approximation to apply on a millimeterscale, the electrical relaxation time, “”o / , must be longer than 10 12 s. The inequality is satis? d easily because the conductivity is seldom larger than one micro-Siemans per meter for liquids of the sort under study here. Accordingly, the electrical phenomena are described by r · “”o E = ? e and r ? E = 0. (2) (1) E is the electric ? eld strength, and ? e is the local free charge density. Boundary conditions derived from Equations 1 and 2 using the divergence theorem and a pill-box system pning a portion of a boundary show that the tangential components of E are continuous and the normal component jumps by an amount proportional to the free charge per unit area, q, that is, k””o Ek · n = q. 3) Here k(·)k denotes the jump, “outside–inside,” of (·) across the boundary, and n is the local outer normal. Electrostatic phenomena and hydrodynamics are coupled through the Maxwell stress tensor. A simple way of seeing the relationship between Maxwell stresses and the electrical body force is to suppose that electrical forces exerted on free charge and charge dipoles are transferred directly to the ? uid. For a dipole charge 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 due 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 ], using Equations 1 and 2. The tensor 2 becomes the Maxwell stress tensor, M , ? ? ? 1 ? @” “”o 1 “”o EE E·E , 2 ” @? T upon inserting the isotropic in? uence of the ? eld on the pressure (Landau & Lifshitz 1960). ELECTROHYDRODYNAMICS 31 Using the expression for the electrical stress produces the equation of motion for an incompressible Newtonian ? uid of uniform viscosity, Du (4) = rp + r · M + µr 2 u.
Dt Alternatively, upon expanding the stress tensor the electrical stresses emerge as body forces due to a non-homogeneous dielectric permeability and free charge, along with the gradient of an isotropic contribution, ? ? ? 1 @” Du = r p “o ? E·E ? Dt 2 @? T ? 1 (5) “o E · Er” + ? e E + µr 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 bulk. The net Maxwell stress at a sharp boundary has the normal and tangential components [ [ M M Annu. Rev.
Fluid Mech. 1997. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. 1 “”o (E · n)2 2 · n] · ti = qE · ti · n] · n = “”o (E · t1 )2 “”o (E · t2 )2 (6) after absorbing the isotropic part of the stress into the pressure as noted above. It is understood that ti represents either of two orthogonal tangent vectors embedded in the surface. Denoting a characteristic ? eld strength as E o and balancing the tangential electrical stress in Equation 6 against viscous stress yields a ve2 locity scale of q`E o /µ = “o `E o /µ.
The same scale appears when the normal 2 stress or the bulk electrical forces are used. With “o E o as a scale for pressure, Equation 5 can be cast in dimensionless form as ? µ @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 scaled by ` and 2 time by the process scale ? P ; Re is a Reynolds number, ? `u o /µ ? ?”o `2 E o /µ2 , when the electrohydrodynamic velocity scale is used for u o . Choosing ? = 103 kg/m3 , µ = 1 kg/m-s, ` = 10 3 m, and E o = 105 V/m gives Re ? 10 4 and a viscous relaxation time, ? ? `2 ? /µ, of 1 ms approximately. For a dielectric constant of 4 and a conductivity of 10 9 S/m the electrical relaxation time, “”o / , is 35 ms. Equation 1 shows how the ? eld is altered by the presence of free charge. In liquids, charge is carried by ions, so species conservation equations must be 32 SAVILLE included to complete the description. Free charge density and ion concentration are related as X ez k n k . (8) ? e = k Annu. Rev. Fluid Mech. 1997. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only.
Here e is the charge on a proton and z k is the valence of the k th species whose concentration is n k . Note 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 electrochemical potential. If we denote the mobility of the k th species by ! k , the species conservation equation is @n k +u·rn k = r ·[ ! k ez k n k E+! k k B T rn k ]+r k , @t k = 1, . . . , N . (9) Here k B is Boltzmann’s constant, and T is the absolute temperature. The ? rst term on the right represents ion migration in the electric ? ld, the second describes transport by diffusion, and the third denotes production due to chemical reactions since the neutral species act as a source for ions in the bulk. With a single ionic species, N = 1 and r 1 = 0; for a binary, z-z electrolyte, N = 3. In the ? rst case, ions are produced by reactions at electrodes—this is called unipolar injection. With a z-z electrolyte, ions are produced at the electrodes and by homogeneous reactions within the ? uid. Here attention is focused on liquids with charge from a single 1-1 electrolyte so that there are two homogeneous reactions.
A forward reaction producing positive and negative 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 rate per unit volume, k+ n 1 , proportional to the concentration of species 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 .
Thus, r 1 = r 2 = r 3 = k+ n 1 k n 2n 3 (11) This situation contrasts sharply with that for strong electrolytes where neutral species are dissociated fully and reaction terms absent. Because ionic reactions ELECTROHYDRODYNAMICS 33 Annu. Rev. Fluid Mech. 1997. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. are fast, it is convenient to imagine that the reactions are almost at equilibrium. At equilibrium, the local rate of reaction is zero, so K ? k+ /k = n 2 n 3 /n 1 .
This complicates matters because at equilibrium one of the conservation laws must be discarded to avoid an overdetermined system. To scale the problem consistently, note that the concentrations of the two ionic species will be much smaller 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, Da[n 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 /µ! 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 characteristic diffusion time to a characteristic reaction time).
The reaction term can be eliminated from Equation 12b using Equation 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, Equation 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. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. From Equations 1 and 13 the characteristic charge relaxation time can now be identi? ed (in dimensional form) as “”o /e2 (! 2 n 2 + ! 3 n 3 ) ? “”o / . To guide simpli? cation of these equations, the magnitudes of the various groups are estimated for small ions with a characteristic radius4 , a, of 0. 25 nm using the Stokes-Einstein relation, (6? µa) 1 , for the mobility. Then Pe ? 105 , ? 03 , and the diffusion time ? D ? 106 s. Estimating the size of the other dimensionless groups will require knowledge of the dissociation-recombination reactions. The equilibrium constant, K , is estimated from the Bjerrum-Fouss theory of ion association (Fouss 1958, Moelwyn-Hughes 1965, Castellanos 1994) as in ? 3? e2 K = 3 exp . (14) 4a 8? a””o k B T while the recombination rate constant (Debye 1942) k = 4? e2 (! 2 + ! 3 ) “”o (15) gives a forward 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 numerical 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 sodium ion in water 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 water makes it likely that the charge carriers are larger than the bare ions. ELECTROHYDRODYNAMICS 35 3 ? 1 Equation 13 yields the classical Ohm’s law approximation in dimensionless form, r · [(z)2 n 2 (! 2 + ! 3 )]E = 0 (19) Annu. Rev. Fluid Mech. 1997. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11.
For personal use only. as long as Pe3/ ? 1. With the numerical magnitudes given 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 surface reactions. Using the s-subscript to denote surface concentrations and operators, and ignoring any special transport 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 balanced against charge relaxation and convection. For steady 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 following ? ve equations. The derivation given 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). ?µ @u + Reu · ru ? P @t = Annu. Rev. Fluid Mech. 1997. 29:27-64. Downloaded from www. annualreviews. org by Brown 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 k”Ek · 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 written 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 velocity and stress, including the viscous and Maxwell stress, are assumed. Primes denote dimensionless forms of the parent 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 important mechanical effects. As noted earlier, charged interfaces attract counterions in the bulk ? uid and repulse co-ions on the Debye length scale.
Electric and hydrodynamic phenomena on this scale are responsible for the ubiquitous behavior of small particles in electrolytes, so it is natural to ask whether similar effects might be important here. In fact, Torza et al (1971) suggested that such effects could be responsible for the lack of agreement between the theory and their experiments on ? uid globules. To see whether the lack of agreement is due to electrokinetic effects 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 relation between charge and ? eld is represented by Equation 18 while the species conservation equations 12 ELECTROHYDRODYNAMICS 37 become ? 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. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. These equations are clearly more complex than those for Ohmic conduction, which omits entirely any accounting for individual species. Is this complexity necessary?
In the following sections, experimental and theoretical results based on the leaky dielectric model are reviewed for several prototypical problems so as to assess the model’s effectiveness and the extent 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. FLUID GLOBULES Drop Motion in External Fields Allan & Mason (1962) encountered paradoxical behavior when non-conducting drops suspended in non-conducting liquids were deformed by a steady electric ? ld. Conducting drops became prolate, as expected, but non-conducting drops often adopted oblate con? gurations. Oblate shapes were completely unexpected since analyses of static con? gurations predict prolate deformations, irrespective of the drop conductivity. Drop deformations can be analyzed with several methods. O’Konski & Thacher (1953) used an energy method; Allan & Mason (1962) balanced electrical and interfacial tension forces. For small deformations of conducting drops in dielectric surroundings, either procedure gives D= 2 9 a””o E 1 . 16 (24) Here E 1 is the strength of the applied ? ld, a is the drop radius, and is interfacial tension. The deformation, D, is the difference between the lengths of the drop parallel and transverse to the ? eld divided by the sum of the two. Given that the drop is a conductor, it is easy to see why the shape is prolate since the pressures inside and outside the drop are uniform, initially, with the difference balanced by interfacial tension and the sphere’s curvature, 2 /a. Therefore, non-uniform electric stresses must be balanced by interfacial tension on the 38 SAVILLE deformed surface. Since the sphere induces a dipole into the incident ? ld, charge on the sphere’s equipotential surface varies as cos #; # being measured from the direction of the ? eld. The ? eld normal to the surface varies in a similar 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 also become prolate in steady ? elds, irrespective of the dielectric constants of the two ? uids, that is, 2 ” 9 a””o E 1 (? “)2 (25) 16 (? + 2″)2 ” with circum? exes denoting properties of the drop ? uid (O’Konski & Thacher 1952, Allan & Mason 1962).
Here deformation results from polarization forces since free charge is absent and the electric stresses are normal to the surface. Allan & Mason’s (1962) anomalous results led Taylor (1966) to discard the notion that the suspending ? uids could be treated as insulators. Although 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 between 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 calculated from a solenoidal electric ? eld to compute the electric forces at the interface of a drop and then balanced these stresses with those calculated for Stokes ? ow. This procedure led to a discriminating function to classify deformations as prolate or oblate: 2M + 3 . 26) 8 = S(R 2 + 1) 2 + 3(S R 1) 5M + 5 Here S ? “/? , R ? ? / , and M ? µ/µ. Prolate deformations are indicated ” ? when 8 > 1, and oblate forms are indicated when 8 < 1. Qualitative agreement between theory and experiment was found in nine of the thirteen cases studied by Allan & Mason (1962). In the other four (prolate) cases, ambiguities in electrical properties hampered a test of the theory. According to Taylor’s leaky dielectric model, tangential electric stresses cause circulation patterns inside and outside the drop. As further con? mation of the theory, McEwan and de Jong5 photographed tracer particle tracks in and around a silicone oil drop suspended in a mixture of castor and corn oils. Toroidal circulation patterns were observed, in agreement with the theory. Annu. Rev. Fluid Mech. 1997. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. D= 5 McEwan & de Jong’s photos are presented in an addendum to Taylor’s paper (1966). ELECTROHYDRODYNAMICS 39 For a steady ? eld, Taylor (1966) gives the deformation as D= 2 9 a””o E 1 8, 16 27) Annu. Rev. Fluid Mech. 1997. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. so it is possible to test the theory quantitatively by measuring the length and breadth of drops for small deformations. However, Taylor did not publish a comparison between theory and experiment. The ? rst quantitative tests were reported by Torza et al (1971), who extended the leaky dielectric model to deal with oscillatory ? elds. The deformation (0 < D < 0. 1) and burst of 22 ? id pairs were studied in steady and oscillatory (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 disappointing. The deformation always varied linearly 2 with a E 1 , but the proportionality factor exceeded the theoretical value in all but one case, and the slopes were larger by a factor of two in more than half the systems. In one case, the measured slope was four times the theoretical value.
In none of the systems was the measured 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 behavior. 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 form to the other with changes in frequency. Torza et al (1971) measured the steady part of the deformation for all 22 systems in 60-Hz ? elds and obtained results similar 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 substantial margins. The transition from oblate to prolate deformation was reported for one system—a 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 lowering 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 disagreement about oscillatory ? elds could be ascribed to the omission 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 remains ? nite.
Although this observation might account for some of the differences between theory and experiment in oscillatory ? elds, it does not resolve the low-frequency dif? culties. Torza et al’s study (1971) provides additional con? rmation of the qualitative aspects of the leaky dielectric model, but the lack of quantitative agreement is disconcerting. Even with water drops whose conductivity is ? ve orders of magnitude larger than the suspending ? uid, deviations between theory and experiment are substantial. 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 effort to address the boundary conditions issue, Ajayi (1978) employed perturbation methods to account for nonlinear effects in the deformation. This 2 analysis represents the shape using a power series in a””o E 1 / .
By carrying the analysis through the second order 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 a””o E 1 / . Considering nonlinear effects helps to an extent, but Ajayi observed that “[the method] cannot remove the discrepancy between theory and experiment. ” Another 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 further.
Baygents & Saville (1989) addressed the matter using asymptotic methods to account for the in? uence of a diffuse layer arising from coulomb interactions between current carrying ions and the surface charge. Three layers were identi? ed where different processes dominate. A diffuse layer adjacent to the surface is separated from an outer region, where the leaky dielectric model applies, by an intermediate region. In Annu. Rev. Fluid Mech. 1997. 29:27-64. 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 diffusion, electromigration, and convection are equally important. In the outer region, the electrohydrodynamic equations prevail. Solving 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 don’t appear to contradict conclusions drawn from the leaky dielectric model, which, based on this analysis, appears to be an exact lumped parameter description. Since none of the theoretical extensions appeared to resolve the divergence between theory and experiment, further experiments were undertaken. Following Torza et al’s (1971) suggestion regarding the need for accurate dielectric constants and other properties (see above), Vizika & Saville (1992) paid careful attention to direct measurement of physical properties.
They studied 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. Figure 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 generally better than with the same systems in a steady ? ld. With water in castor oil, for example, the calculated 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. Another 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. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. 42 SAVILLE Annu. Rev. Fluid Mech. 1997. 29:27-64. 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 studied deformations with the castor oil–silicone 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 element technique was employed to calculate deformations in steady ? elds. Except for the inclusion 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 dashed 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 upper 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. 29:27-64. 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 Taylor’s 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 results and Taylor’s 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 smaller deformations than those observed. In addition, the agreement between Taylor’s 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 comprehensive 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. 29:27-64. 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). factory job. Nevertheless, only a limited number of ? uids have been studied, and even in these cases, conductivities have not been controlled. Questions as to ? ite amplitude effects or charge convection due to interface motion remain 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 spherical drop can be studied. Although no experimental studies exist, calculations with the model indicate a substantial in? uence. First note that the velocity will be unaltered 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 asymmetric charge distribution creates a net force; charge convection due to sedimentation 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. 29:27-64. Downloaded from www. annualreviews. org by Brown University on 08/07/11. For personal use only. Figure 3 The unsteady part of the drop deformation as a function of frequency for silicone drops in castor oil (Vizika & Saville 1992).
Torza et al’s (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 = µ/””o E 1 , so ? C /? F = (“”o E 1 )2 /µ · 1 ? C /? F ? 0. 1 for ” = 4, µ = 10 N-m, = 10 9 S/m, and E 1 = 105 V/m, so a linearized treatment is appropriate (Spertell & Saville 1976). Solving 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 µ
Ust is the Stokes settling velocity for a rigid sphere and F(R, S, M) = 6M 2 [3S(R + 1) 1][RS 1] . 5(1 + M)2 S 2 (3 + 2R)(2 + R)2 (32) Also, with charge convection, drop deformation is no longer symmetric 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 Breakup To provide a context to study the role of tangential stresses it is helpful to recall work on perfect conductors and dielectrics. Studies of drop dynamics6 and stability began with Rayleigh’s celebrated 1882 paper “On the equilibrium of liquid conducting masses charged with electricity. ” His analysis pertains to instantaneous charge relaxation inside an isolated drop, and the relation7 between the frequency, ! , 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 index of the Legendre polynomial Pn (cos #). For perfectly 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, spheroidal approximation is used, either oblate or prolate deformations are included. Although the Rayleigh limit pertains strictly to small oscillations, dimensional analysis shows that the criterion for instability will still be of the form Q2 “o a 3 > C, (34) Annu. Rev. Fluid Mech. 1997. 9:27-64. 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 oscillating in an insulating ? 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, ? µ @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 Rayleigh’s Theory of Sound (1945) contains many fascinating accounts of early work on drops and cylinders. 7 Recall that the rationalized MKSC system is used here. In Rayleigh’s 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 given by the dimensionless form of Equation 3, k”Ek · 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 . Charge on the interface is conserved, and for 1 the balance is Annu. Rev. Fluid Mech. 1997. 29:27-64. 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 interior. For Rayleigh’s perfectly conducting drop, the local charge balance is unnecessary because: (a) the ? ld is nil inside the drop, and (b) charge transport is instantaneous so that the convection and relaxation terms vanish. The remaining boundary conditions are continuity of velocity and stress, and the kinematic condition. These equations have been solved to investigate how relaxation alters Rayleigh’s results (Saville 1974). Both viscous forces and charge relaxation effects were included, but general conclusions were obscured by the awkward transcendental form of the characteristic equation. However, asymptotic methods can be used to identify the salient features. The result for a slightly viscous drop in the absence of a suspending ? id is rather surprising in as much as Rayleigh’s result (see Equation 33) is recovered as the stability criterion. Even when charge relaxation by conduction is slow, charge convection still redistributes charge so rapidly that the oscillation frequency is given 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 volumetric and Rayleigh’s theory gives 1 ? 2 a ? 1 (39) ? (2n + 1)(n 1) 2 for a ?? 2 for a drop with kinematic viscosity ?. When 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 . Rayleigh’s 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. 29:27-64. 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 µ in a ? uid with ” ? ? 1 and µ where µ ? µ and (a /? ? 2 ) 2 ? 1. The next topic concerns behavior ?? beyond the range where a linear treatment is acceptable.
To form a simple model, the breakup of isolated drops or drops in external ? 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 unstable to axisymmetric perturbations while oblate shapes above the limit are stable to axisymmetric perturbations but unstable to nonaxisymmetric perturbations. Thus, the Rayleigh limit turns out to be the absolute limit of stability.
Dimensional analysis indicates that the criterion for instability of a conducting drop immersed in a gas and stressed by an external ? eld has the form 2 a”o E 1 > C. (42) Taylor’s spheroidal approximation (Taylor 1964) gives C = 2. 1 ? 10 3 for D = 0. 31, in good agreement with experiments on soap ? 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. Taylor (1964) analyzed the region near the spheroidal tip, which becomes conical (a “Taylor cone”) at the limit of stability. For a cone with a vertex angle 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 vertex 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 corresponding to Taylor’s solution for an equipotential cone. 9 Pelekasis et al (1990) and Kang (1993) provide useful summaries of the dynamical stability of perfectly conducting drops.
ELECTROHYDRODYNAMICS 49 More extensive analyses of the static behavior of drops disclose behavior consistent with this picture: Sherwood (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 unstable 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 upper branches of the relation and unstable in between. Values 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 dielectric ? 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 moving toward the apex along the interface and away from it along the axis. One might conjecture 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 breakup at high ? eld-strengths: (a) water drops in oil deformed symmetrically, and globules pinched off from a liquid thread; (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 Sherwood’s algorithm predicts breakup at the tip with critical ? ld-strengths close to those found by Taylor (1964) and Brazier-Smith (1971). Perfect dielectrics display similar overall behavior, and the maximum Annu. Rev. Fluid Mech. 1997. 29:27-64. 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 fattened 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 elongation 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 instability of the elongated thread. Because of the numerical algorithm’s structure it was not possible to study the other mode of breakup identi? d by Torza et al (1971), which remains 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 Sherwood’s 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 effort 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 emerging from a Taylor cone is dif? cult, but EHD processes are clearly involved. Observations of liquid drops emerging 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 sink onto the conical ? ow driven by a tangential electric stress varying 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 somewhat counterintuitive structure: Liquid is supplied to the jet from the surface of the cone, while a recirculating eddy moves ? uid down the axis of the cone toward the supply. Of course this analysis omits details of the jet, whose characteristics are submerged in the singularity at the cone tip. Accounting for charge convection on the surface of the cone, which clearly
Annu. Rev. Fluid Mech. 1997. 29:27-64. 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 Charged Cylinders (Free Jets) Annu. Rev. Fluid Mech. 1997. 29:27-64. 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 and increases growth rates. For a”o E 1 / = 1, the range is approximately 0 < ? < 1. 35 ( > 1. ? a). Interestingly, charge destabilizes non-axisymmetric deformations that are otherwise stable; the relation for these modes may 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 . 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 Huebner’s (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 axial ? elds, the topic is of considerable interest. Taylor (1964) observed that “induced charge has a very powerful effect in preventing the break up of jets into drops under certain circumstances and an equally powerful effect in causing violently unsteady movements ultimately
Shortly after Rayleigh’s 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 studying the dynamics of a charged, inviscid cylinder, Bassett showed that an axisymmetric disturbance of wavelength will grow if ? 0 2 ? a”o E 1 K 0 (? ) ? Io (? 2 1 ? 1+? o > 0. (43) Io (? ) K o (? ) 52 SAVILLE Annu. Rev. Fluid Mech. 1997. 29:27-64. 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. Raco’s (1968) experiments with poorly conducting liquids also show a strong stabilizing effect. These results produce a quandary of sorts. Axial ? elds promote 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 required ? 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 a””o 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 (? ) 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 made more unstable. Although axial ? elds are seen to promote 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 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

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