Drops with conical ends in electric and magnetic fields

Transcription

1 Drops with conical ends in electric and magnetic fields By Howard A. Stone 1, John R. Lister 2 and Michael P. Brenner 3 1 Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA 2 Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, UK 3 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 24 February 1997; accepted 7 May 1998 Slender-body theory is used to determine the approximate static shape of a conically ended dielectric drop in an electric field. The shape and the electric-field distribution follow from solution of a second-order nonlinear ordinary differential equation that can be integrated numerically or analytically. An analytic formula is given for the dependence of the equilibrium cone angle on the ratio, ɛ/ ɛ, of the dielectric constants of the drop and the surrounding fluid. A rescaling of the equations shows that the dimensionless shape depends only on a single combination of ɛ/ ɛ and the ratio of electric stresses and interfacial tension. In combination with numerical solution of the equations, the rescaling also establishes that, to within logarithmic factors, there is a critical field E min for cone formation proportional to (ɛ/ ɛ 1) 5/12, at which the aspect ratio of the drop is proportional to (ɛ/ ɛ 1) 1/2. Drop shapes are computed for E >E min.fore E min the aspect ratio of the drop is proportional to E 6/7. Analogous results apply to a ferrofluid in a magnetic field. Keywords: Taylor cones; drop deformation; conical ends; electric fields; dielectric liquids 1. Introduction When an electric field is applied to a dielectric liquid or when a magnetic field is applied to a ferrofluid, slender conical interfaces are frequently observed. For example, a small dielectric (magnetic) drop, when exposed to an electric (magnetic) field, first elongates into a prolate shape and then, if the field strength is sufficiently high, may develop what appear to be pointed or conical ends (see figure 1). In the absence of fluid motion, the shape of the interface is governed by a balance between interfacialtension stresses and Maxwell electric (magnetic) stresses, i.e. by a modified version of the familiar Young Laplace equation. The electric stresses depend on the details of the electric field at the interface, and, since calculation of the electric field requires knowledge of the interfacial shape, the determination of the interfacial shape requires solution of a non-local free-boundary problem. The coupling of the unknown shape of the drop and the unknown spatial variation of the electric field has inhibited the development of any exact description of even this static configuration. A variety of 455, Printed in Great Britain 329 c 1999 The Royal Society TEX Paper

2 330 H. A. Stone, J. R. Lister and M. P. Brenner Figure 1. Schematic illustration of the deformation of a dielectric-fluid drop exposed to a uniform electric field. Without an electric field the drop is held spherical with radius R by interfacial tension, γ. For small applied field strengths, E, the drop deforms into a prolate shape (l >a 0), while above a critical field strength, and for dielectric-constant ratios ɛ/ ɛ >17.6, the highly deformed shape forms conical ends. approximate theoretical and numerical studies have been developed, as summarized in table 1. We describe in two main steps an approximate solution for the slender shapes characteristic of deformed drops with conical, or nearly conical, ends: first, the normalstress balance is simplified for slender geometries; second, an integral equation for the electric field is approximated using the slenderness assumption to arrive at an ODE that couples the electric field to the shape of the drop. The approximations lead to scaling relationships between the dielectric-constant ratio, the cone angle, the aspect ratio of the drop, and the electric field, E min, at which a conical end is first observed. The conical-ended shapes of drops with an applied field E >E min are also determined. We begin with a brief historical account summarizing some important experimental contributions. The occurrence of conical tips at an interface exposed to an electric field was apparently first described by Zeleny (1917) for drops held at the end of a capillary. Wilson & Taylor (1925) examined this phenomenon further with soap films. Both papers observed that very fine threads of fluid may be emitted from the conical ends. Taylor (1964), in a seminal paper, demonstrated nearly conical equilibrium shapes of water drops at the end of a specially formed conical electrode. Similar investigations focusing on the nearly conical ends, and on the sprays of tiny droplets that typically accompany the formation of the conical ends, have now been made many times (see, for example, Fernández de la Mora 1992; Pantano et al. 1994); much of this interest stems from the uses of electroatomization as, for example, a method for spraying paint (Bailey 1988). Isolated drops have also been investigated in some detail. When a dielectric drop is exposed to a weak electric field, the drop deforms from a spherical to a prolate shape, which, with increasing field strength, deforms further and attains shapes with conical ends (e.g. Garton & Krasucki 1964); very small droplets are typically emitted from the pointed tips. Analogous observations have been made when magnetic-fluid drops, in a fluid with different magnetic properties, are exposed to magnetic fields (Bacri & Salin 1982, 1983; Boudouvis et al. 1988), though in this case the conical (or nearly conical) ends appear to be more stable. The description of the shape of a dielectric drop exposed to an electric field (figure 1) has also been considered both theoretically and numerically. In one theoretical

3 Drops with conical ends in electric and magnetic fields 331 Table 1. Summary of different theoretical and numerical approaches to the calculation of the static shape of a drop in an electric (magnetic) field method of solution reference Taylor two-point method Taylor (1964), Miksis (1981) method of moments Rosenkilde (1969) energy minimization Bacri & Salin (1982), Sherwood (1988) perturbation of a spheroid Dodgson & Sozou (1987) variational approach Séro-Guillaume et al. (1992) (magnetic liquids) numerical calculation Brazier-Smith (1971), Miksis (1981), Boudouvis et al. (1988), Basaran & Wohlhuter (1992) numerical calculation with Pantano et al. (1994) asymptotics for a conical tip slender-body theory Sherwood (1991), Li et al. (1994) slender-body theory for this paper field and shape utilizing local conical structure of tip approach it is assumed a priori that the shape is spheroidal (an exact expression for the uniform electric field inside such a drop is available), and that the normal-stress balance is satisfied at both the pole and the equator (Taylor 1964). This calculation predicts that the shape of the drop is a multiple-valued function of the electric field if ɛ/ ɛ > (Miksis 1981), where ɛ and ɛ are the dielectric constants inside and outside the drop, respectively. Rosenkilde (1969) also assumed the shape was spheroidal and developed a method-of-moments approach for approximately satisfying the momentum equation augmented with electric stresses, and so predicted multiple equilibria for ɛ/ ɛ > Because experimentally measured equilibrium shapes often do appear to be nearly spheroidal, Dodgson & Sozou (1987) studied the equilibrium by considering small perturbations from a prolate spheroid and obtained a critical dielectric-constant ratio in good agreement with that of Rosenkilde (1969). Alternatively, the shape may still be taken to be spheroidal, but the equilibrium aspect ratio is now determined by minimization of the sum of the electrostatic energy and the surface energy of the drop (Bacri & Salin 1982). Above a critical dielectricconstant ratio, ɛ/ ɛ 20.8, the relationship between aspect ratio and field strength becomes non-monotonic so that there are three aspect ratios (two stable and one unstable) for an intermediate range of field strengths, though only one long slender solution for large field strengths (see, for example, Sherwood 1988). The theoretical predictions of multiple equilibrium are in qualitative agreement with experiments (Bacri & Salin 1982, 1983) and the drop deformation displays hysteresis as a function of applied field, which is also observed in experiments. Nevertheless, the larger of the predicted aspect ratios (or the largest equilibrium configuration) in the approximate theories described above, is commonly associated with the experimental observation of highly distorted drops with pointed ends. It is clear that these experimental shapes are not prolate spheroids and, therefore, agreement between theory and experiment may be considered only qualitatively successful. Numerical investigations have been reported by many authors (e.g. Brazier-Smith

4 332 H. A. Stone, J. R. Lister and M. P. Brenner 1971; Miksis 1981; Wohlhuter & Basaran 1992). Both boundary-integral and finiteelement solutions have been considered. A numerical approach is capable of handling the coupling between the unknown shape and the electric field, though all the numerical studies either assume a rounded end and/or cannot resolve the structure in the neighbourhood of a nearly pointed end. The numerical studies have all reported a critical dielectric-constant ratio, 19 < ɛ/ ɛ < 20, above which solutions were not obtained. Recently, however, Pantano et al. (1994) reconsidered the case of a conducting drop (in air; ɛ/ ɛ ) at the end of a narrow capillary, and combined a numerical integral-equation solution for the electric potential with the local field to be expected of a conical end (as determined by Taylor (1964)). This work naturally leads to the question of a purely local, slender-body analysis for a conical tip. In order to focus on the pointed shapes, a slender-body approach was developed by Sherwood (1991). Recently, Li et al. (1994) used a similar slender-body analysis, which was combined with the local structure to be expected of a conical solution, to re-examine the appearance of conical ends on drops. Motivated by the work of Li et al. (1994), we have also studied the slender-body limit and have obtained numerical solutions to a one-dimensional description of the drop shape, as well as scaling relationships between the cone angle, the dielectric-constant ratio, the drop aspect ratio and the minimum electric field necessary to produce conical tips. By way of introduction to the intriguing problem of the occurrence of a conical shape, we note an analytical result for a conical tip of angle 2θ 0 between a fluid with dielectric constant, ɛ, and a surrounding immiscible fluid of dielectric constant ɛ. Liet al. (1994) and Ramos & Castellanos (1994) showed that the equations of electrostatics and the interfacial stress balance are consistent with the existence of a locally conical shape provided that θ 0 satisfies (0 <θ 0 <π): P 1/2 (cos θ 0 )P 1/2 ( cos θ 0)+(ɛ/ ɛ)p 1/2 ( cos θ 0 )P 1/2 (cos θ 0)=0, (1.1) where P 1/2 (x) is a Legendre function of the first kind. This equation has no solutions for ɛ/ ɛ <17.59 (corresponding to θ 0 =30 ), and allows two conical solutions for ɛ/ ɛ > Taylor (1964) had previously shown that a perfect conductor has a cone angle such that P 1/2 ( cos θ 0 )=0,orθ 0 =49.3. We note that in the case of two conducting liquids, with conductivities σ and σ, a local conical solution is again possible with ɛ/ ɛ replaced by σ/ σ in equation (1.1) (Ramos & Castellanos 1994). It remains to ask how such a local conical solution may be joined with the rest of the shape of the drop. To address this question, we present the governing equations for the electrostatics and the equilibrium of the drop in 2 and simplify them using the assumption of slenderness. An ODE coupled to an algebraic equation is obtained, which is studied in detail in 3. Several analytic characteristics of the drop shape and of the minimum electric field necessary to obtain conical drops are natural byproducts of the analysis. 2. Governing equations for slender shapes The analysis presented below for slender axisymmetric shapes requires a 0 /l 1, where 2l is the length of the drop and a 0 is a representative radial dimension, which we shall take as the equatorial radius (figure 1). After developing an integral equation for the electric field interior to the slender shape using several of the intermediate steps of Li et al. (1994), we take further advantage of the slenderness approximation to deduce an ODE for the axial field (as first suggested by Sherwood (1991)).

5 Drops with conical ends in electric and magnetic fields 333 This latter step is crucial for obtaining the analytic results described in 3. For completeness, we outline all the major steps in the slender-body analysis. Throughout the discussion it should be kept in mind that a conical tip is assumed, and so the cone angle should be small for the slenderness approximation to be valid in the neighbourhood of the cone tip. (a) Normal-stress balance We assume that the material properties ɛ and ɛ are constant. At the interface between two dielectrics, the tangential component of the electric field, E t, is continuous, but the normal component, E n, suffers a jump, given by the boundary condition ɛe n = ɛēn, where overbars indicate values in the external fluid; the jump Ēn E n gives the local induced surface charge density. The electric stress tensor in a medium with dielectric constant ɛ is T E = ɛɛ 0 (EE 1 2 E 2 I), where ɛ 0 is the permittivity of free space. It follows that the jump in the normal electric stress across the interface separating two distinct dielectric liquids is [n T E n ] = 1 2 ɛ 0(ɛ ɛ)(et 2 (ɛ/ ɛ)en) 2 (2.1) (see, for example, Sherwood 1988), where [ ] denotes outside minus inside. We note that the limit of a perfectly conducting drop in a vacuum, studied by Taylor (1964), corresponds to ɛ/ ɛ, though we shall think of ɛ/ ɛ as a fixed parameter within the context of the slender-body analysis below. Let a(z) be the radius of the drop in cylindrical coordinates (r, θ, z) aligned with the applied electric field E, which is assumed to be uniform at large distances from the interface (figure 1). For slender shapes (a 0 /l 1), the electric field varies primarily in the z-direction, so that E E(z)e z and E r = O(a 0 E/l). Moreover, the normal and tangential components of the electric field satisfy E n = O(a 0 E t /l). Assuming, as will be verified later, that (a 0 /l) 2 (ɛ/ ɛ) 1 for slender drops, the electric stress jump (2.1) can be approximated by [n T E n ] 1 2 ɛ 0(ɛ ɛ)e 2, (2.2) since E t E(z). We note that this approximation neglects the effect of the normal component of the electric field relative to that of the tangential component and so it is not appropriate at a rounded end, where E t 0 and a non-zero E n is responsible for deformation of the drop. We assume that the drop is static. Thus the shape of the drop and the electric field are coupled through the normal-stress equation for the interface, which balances the electric stress, fluid pressure and interfacial tension according to [n T E n]+ p = γκ, where p is the constant pressure excess inside the drop, γ is the coefficient of interfacial tension and κ is the curvature. Thus, using (2.2), the normal-stress balance is approximated by 1 2 ɛ 0(ɛ ɛ)e 2 (z)+ p=γ/a(z), (2.3) where we have also neglected a contribution, d 2 a/dz 2, to the curvature, since this is O(a 2 0/l 2 ) relative to a 1. (b) Electric field The electrostatics problem requires solution of Laplace s equation, 2 φ = 0, inside and outside the drop, from which the electric field is given by E = φ. An integral

6 334 H. A. Stone, J. R. Lister and M. P. Brenner representation of the solution to Laplace s equation serves as a convenient starting point and, since only the normal component of the electric field has a jump, the electric field on either side of the interface or along the interface may be represented as a surface distribution of dipoles: E(x), x V, (2.4 a) E + ɛ/ ɛ 1 4π S (x y) x y 3 E 1 n(y)ds y = 2 [E(x)+Ē(x)], x S, (2.4 b) Ē(x), x V, (2.4 c) where V, V and S are the interior, exterior and surface of the drop, respectively. The case of a magnetic field applied to two distinct magnetic liquids (ferrofluids) is obtained by replacing the ratio of dielectric constants by the ratio of permeabilities and the electric field by the magnetic field. For slender shapes, we have noted that the internal electric field is primarily axial and given by E(z)e z. Then, from an integration of E= 0, it follows that, at the interface r = a(z), the small radial field is E r = 1 2a(z) E/ z, with error O(a 2 0/l 2 ). The unit outward normal, n, has Cartesian coordinates proportional to (cos θ, sin θ, a/ z), and so E n = E n (1/2a) (a 2 E)/ z. Choosing a point x =(0,0,z) on the centreline, denoting the integration variable as y =(acos θ, a sin θ, s), and taking the z-component of (2.4 a) then leads to ( ) E(z) 1 ɛ l 2 ɛ 1 (z s)(d/ds)[a 2 (s)e(s)] ds=e l [a 2 (s)+(z s) 2 ] 3/2. (2.5) This equation is equivalent to the integral equation given by Li et al. (1994) integrate by parts their eqn 6 and represents the electric field inside the slender drop using an area-averaged line distribution of dipoles. We note that near a conical tip of interior angle 2θ 0, the ratio of the radial to the axial scales of variation of E is O(tan θ 0 ), and hence, (2.5) is valid even near the tip for θ 0 1. For slender shapes, (2.5) may be approximated further since the integral is dominated by a logarithmically large contribution from a(s) z s l(hinch 1991, p. 43). Defining A 1 to be an effective length-to-width ratio, we obtain the differential equation E(z) ln A ( ) ɛ d 2 2 ɛ 1 dz 2 (a2 E)=E, (2.6) with error O(1/ ln A), which expresses a purely local relation between E(z) and a(z). In slender-body theory, A is usually taken to be l/a 0. Asymptotic examination of the integral in (2.5) near the conical ends suggests that A might alternatively be taken to be 1/ tan θ 0. We note that either choice for A is asymptotically consistent to O(1/ ln A) ifl/a 0 = O(1/ tan θ 0 )asa. The ODE (2.6) and the algebraic equation (2.3) are coupled equations for the equilibrium static drop shape, which must be combined with the equation for the drop volume V, l π a(z) 2 dz = V, (2.7) l

7 Drops with conical ends in electric and magnetic fields 335 in order to determine the shape of the drop completely. Symmetry requires that E(z) =E( z) and a(z) =a( z). The reduction of the non-local integral equation (2.5) to the local differential equation (2.6) is a significant simplification that allows for analytic progress in 3. An alternative derivation of (2.6) is given in Appendix A. Before proceeding further, we note that (2.6) may also be used to predict the electric field inside a prolate spheroid despite the rounded ends. It is well known (see, for example, Stratton 1941) that the electric field induced inside a spheroid is uniform. For the spheroid, (a 2 /a 2 0)+ (z 2 /l 2 ) = 1, the one-dimensional equation (2.6) is satisfied by a uniform electric field: E E 1D = 1+(ɛ/ ɛ 1)(a 0 /l) 2 ln A, (2.8) which may be compared with the exact result (Stratton 1941, pp ; eqn 45 has a misprint in a sign) for a spheroid (expanded for a 0 /l 1): E E spheroid = 1+(ɛ/ ɛ 1)(a 0 /l) 2 [ln(l/a 0 ) + ln(2/e)]. (2.9) Taking A = l/a 0 as appropriate for the spheroidal shape, there is agreement to within terms of O(1), which is the expected error since only the leading-order logarithmically large term was retained in (2.6). We note, for example, that E 1D is within 15% of E spheroid for all dielectric-constant ratios at aspect ratios as small as l/a 0 =4. 3. Drops with conical ends We now illustrate how equations (2.3), (2.6) and (2.7) can be used to analyse the static shape of a drop in an electric field. We first show in 3 a that a local solution near a conical tip yields analytical results in excellent agreement with the known exact analytical expression given in terms of Legendre functions in equation (1.1). Second, in 3 b we show that the coupled nonlinear equations can be non-dimensionalized and reduced to a first-order nonlinear ODE, which can be solved numerically or analytically as a function of the single remaining parameter. A combination of the numerical solutions in 3 c with the non-dimensionalization establishes a number of scaling results for drop shapes with conical ends. (a) Local solution near a conical end Consider a drop with conical ends of cone angle 2θ 0. Since a 0asz l, static equilibrium requires that the local electric field diverges as E a 1/2. Thus, a(z) =(l z) tan θ 0 and E(z) (l z) 1/2, as z l. (3.1) Substitution into (2.6) with A =1/tan θ 0, as appropriate for this local analysis, yields ɛ ɛ =1 8/3 tan 2 θ 0 ln(tan θ 0 ), (3.2) which relates the cone angle and dielectric-constant ratio necessary to have a conical tip. A comparison of this approximate expression, the exact result (1.1), and the integral approximation reported by Li et al. (1994) is shown in figure 2; the approximate formula is an excellent approximation for all cone angles and for θ 0 < 20 (i.e. ɛ/ ɛ >21), the approximate result differs by less than 5% from the exact result.

8 336 H. A. Stone, J. R. Lister and M. P. Brenner Figure 2. Dielectric-constant ratio versus cone angle. The solid curve is the prediction of equation (3.2), the long-dashed curve is the slender-body approximation developed by Li et al. (1994), and the short-dashed curve is the exact result given by equation (1.1). Equation (3.2) also gives reasonable agreement with two results for larger cone angles. First, the limit ɛ/ ɛ in (3.2) corresponds to θ 0 45 (or θ 0 0), which is in surprisingly good agreement with Taylor s (1964) exact result of θ 0 =49.3 for perfectly conducting drops. Second, as is clear in figure 2, there is a minimum dielectric-constant ratio, above which conical drops are capable of forming. The minimum predicted by (3.2), ɛ ɛ = e 15.5 at θ 0 = tan 1 (e 1/2 ) 31.2, (3.3) compares well with the numerically determined minimum, ɛ/ ɛ =17.6, at θ 0 =30 predicted by the exact formula (1.1). These agreements are encouraging, though somewhat surprising since (3.2) was derived for small cone angles θ 0. Each of the calculations shown in figure 2 predicts that above the minimum dielectric-constant ratio there are two equilibrium cone angles. Li et al. (1994) pointed out that, for a given dielectric-constant ratio, the solution with the smaller angle is expected to be stable while the larger angle is unstable. Their argument is that a decrease in the cone angle, θ 0, at fixed ɛ/ ɛ, changes the local strength of the electric field according to E (l z) ν 1. For the smaller of the two cone angles, ν increases with decreasing angle that implies an increased pressure at the tip and so restores the tip to the original angle. The interpretation of this result is rather puz-

9 Drops with conical ends in electric and magnetic fields 337 zling since it predicts, for example, that the static Taylor cone is unstable, although away from the cone tip most experiments with conducting fluids (see, for example, Pantano et al. 1994) are consistent with an angle close to that predicted by Taylor. It should be noted, however, that very close to the cone tip an emitted jet is almost always observed in experiments with dielectric liquids and so these configurations are not entirely static. (b) Scaling of the equations To proceed further, it is convenient to non-dimensionalize equations (2.3), (2.6) and (2.7). We introduce an electro-capillary length-scale, l cap, and electric Bond number, B: l cap = 2γ ɛ 0 ɛe 2 and B = R = ɛ 0 ɛe R 2, (3.4) l cap 2γ where 4 3 πr3 = V, in order to scale the stress balance between the electric field and interfacial tension. If we now non-dimensionalize radial dimensions by l cap /(ɛ/ ɛ 1), axial dimensions by l cap [ 1 2 ln A/(ɛ/ ɛ 1)]1/2, the internal excess pressure by γ(ɛ/ ɛ 1)/l cap and the electric field by E, and denote dimensionless variables with tildes, then the governing equations may be reduced to the simpler form Ẽ (ã 2 Ẽ) =1, (3.5 a) Ẽ 2 + p=ã 1, (3.5 b) l ( ) 1/2 ( ) 5/2 2 ɛ ã 2 d z = 4 3 B3 l ln A ɛ 1, (3.5 c) where primes denote derivatives with respect to z. For a given value of p, equations (3.5 a) (3.5 b) determine the scaled shape of the drop. The corresponding electric field (or, equivalently, the electric Bond number) necessary to produce a drop with this shape, is obtained from equation (3.5 c). Solution of (3.5 a) (3.5 b) near the tip of a drop shows that ã ( 4 3 )1/2 ( l z) and Ẽ [ 4 3 ( l z) 2 ] 1/4 as z l. The scaled cone angle, tan 1 [( 4 3 )1/2 ], corresponds to an unscaled cone angle given by (3.2) if A is taken as 1/ tan θ 0. However, for a description of the global shape of the drop, it is more consistent to take A = l/a 0 and we do this below. From the numerically determined shape, the dimensionless aspect ratio, l/ã 0,is known, and so the actual aspect ratio, l/a 0, follows from ( ) 1/2 ( ) 1/2 l ln A ɛ = a 0 2 ɛ 1 l. (3.6) ã 0 In the derivation of (2.2) we assumed that (a 0 /l) 2 (ɛ/ ɛ) 1. We now note from (3.6) that (a 0 /l) 2 (ɛ/ ɛ) =O(1/ ln A) 1, which is consistent with the level of approximation used in deriving (2.6). (i) Scaling relationships As we shall see via numerical solution of (3.5 a) (3.5 c), there is a minimum dimensionless drop volume for which a solution is possible and hence there is a minimum

10 338 H. A. Stone, J. R. Lister and M. P. Brenner electric field, E min, necessary to produce a drop with conical ends. The scaling of equation (3.5 c) reveals the fact that, to within a logarithmic factor, the minimum field depends on the dielectric-constant ratio according to ( ) 5/2 ɛ Emin 6 ɛ 1 = const., (3.7) (conical tips only form for ɛ/ ɛ 1). Furthermore, from (3.6), we observe that the aspect ratio of the conical-ended drop varies with the dielectric properties according to l/a 0 (ɛ/ ɛ 1) 1/2. Thus, the aspect ratio is related to the field necessary to produce the conical tips by l/a 0 E 6/5 min. (ii) Reduction to a first-order ODE In order to determine the drop shape numerically, we first let (ã 2 Ẽ) = u and obtain du dẽ = (Ẽ 1)( p 3Ẽ2 ). (3.8) u(ẽ2 + p) 3 It is clear that p determines the structure of the solution in the (u, Ẽ) phase plane and it is useful to note that a first integral of (3.8) gives u 2 = 3Ẽ2 2Ẽ + p +c, (3.9) (Ẽ2 + p) 2 where c is a constant of integration. At a conical end u 0 (see equation (3.1)), which gives c = 0. The electric field Ẽ( z), which is symmetric about z = 0, may then be determined by integrating the first-order differential equation dẽ d z = (3Ẽ2 2Ẽ + p)1/2 (Ẽ2 + p) 2, (3.10) 3Ẽ2 p from the asymptotic behaviour, Ẽ [ 4 3 ( l z) 2 ] 1/4 at the tip towards the middle of the drop, where Ẽ = 0 by symmetry. If Ẽ = 0 is attained, then the shape of the drop follows from (3.5 b). (Alternatively, (3.5 b) may be used to eliminate Ẽ from (3.5 a) and the resulting second-order differential equation for ã integrated numerically.) Consideration of the phase plane of (3.8) (Appendix C) shows that the asymptotic behaviour of the tip only connects to a point where Ẽ = 0 (corresponding to the middle of a finite drop) if 1 p 1 3.As p 1 3, the dimensionless drop volume tends to a finite minimum, and as p 1, the dimensionless drop volume tends to infinity. We also note with a fair amount of algebra that it is possible to integrate (3.10) analytically (Gradshteyn & Ryzhik 1980, 2.25) and determine the electric field in implicit form z(ẽ), from which all the geometric properties of the solution can be determined. Some of the relevant manipulations are summarized in Appendix B. The case p = 1 3 is of particular interest since it corresponds to the drop of minimum volume; this dimensionless volume is found, from the analytic solution, to be l l ã2 d z , with l/ã 0 = It can be verified that the solution has an unbounded second derivative at z = 0; this feature corresponds to the loss of solution for p > 1 3.

11 Drops with conical ends in electric and magnetic fields 339 Figure 3. Drop shapes for different p from the numerical solution of (3.10) and (3.5 b). Figure 4. Numerically determined drop volume 2 l 0 ã2 d z for different values of p. The solid circles denote the volume calculated analytically for some simple cases: p = 1 3,0, 1 4 and 4 9.

12 340 H. A. Stone, J. R. Lister and M. P. Brenner Figure 5. (a) Drop shapes at the first appearance of conical tips for ɛ/ ɛ = 20, 40, 100. (b) Increasing elongation of the drop shape with increasing electric-field strength above E min for ɛ/ ɛ = 40. At large field strengths the shapes are approximately ellipsoidal with conical tips. In all cases, 2 l 0 ã2 d z has been scaled to unity. (c) Solutions and drop shapes (i) Numerical results Numerical solutions for the dimensionless shape of the drop for different values of p are shown in figure 3. The dimensionless volume of the drop is shown in figure 4 as a function of p. The shapes with increasing dimensionless volume correspond to an increasing electric field on a drop with a given dielectric-constant ratio (3.5 c). The fact that there is a minimum volume sets the lowest electric field at which conically tipped drops form, while the existence of solutions with volume tending to infinity (in the rescaled equations) simply gives a drop with conical ends for each electric-field value larger than the critical minimum value. The numerically determined minimum dimensionless drop volume and aspect ratio corresponding to p = 1 3 may be combined with equations (3.5 c) and (3.6) to obtain explicit formulae for the variation of the minimum electric field, E min, necessary to obtain conical ends, and the actual aspect ratio at this critical value as functions of ɛ/ ɛ. Some representative drop shapes, all scaled to have the same drop volume, are shown in figure 5. Drops become progressively elongated as the dielectric-constant ratio is increased (figure 5a) and as the electric field is increased (figure 5b). These trends are in accordance with the qualitative prediction of energy arguments that approximate the shape of the drop a priori as a prolate spheroid. Each of the shapes shown in figure 5 has a conical tip, though the near-tip region, in which the drop is approximately conical, decreases in size with increasing E (figure 5b). The actual aspect ratio of the drop is shown as a function of electric Bond number, B, for some representative values of ɛ/ ɛ in figure 6. The termini of the curves correspond to the critical fields, E min, for cone formation. The form of the curves is similar to the upper solution branches of fig. 2 of Sherwood (1988), and we interpret the loss of solution for E < E min as a transition to the lower solution branches, which correspond to rounded rather than conical ends. We note that all solutions shown in figure 6 have aspect ratios l/a 0 5. The assumption that the aspect ratio is large, which forms the basis of the theory described here, is an increasingly good approximation at large dielectric-constant ratios, since the aspect ratio increases

13 Drops with conical ends in electric and magnetic fields 341 Figure 6. The actual aspect ratio of the drop as a function of electric Bond number B for ɛ/ ɛ = 20, 40, 100 (solid). The curves terminate at the minimum Bond number required to produce a conical tip (long dashes). At large Bond number, l/a 0 B 3/7, as is also true of the spheroidal approximation of Sherwood (1988) (short dashes). proportional to (ɛ/ ɛ) 1/2, and at large fields since the aspect ratio is shown below to increase proportional to B 3/7. (ii) Asymptotics for strong electric fields The behaviour for strong electric fields (B ) is given by solution of (3.5 a) (3.5 c) as p 1. In this limit, the scales of ã and z become large (figure 3), and hence, Ẽ 2 p 1. Consideration of the balance of terms in (3.5 a) and (3.5 b) suggests that the rescalings p = 1+δ, Ẽ =1+δF, ã = δ 1 α, z = δ 3/2 ζ, (3.11) which give F (α 2 ) = δ(α 2 F ), (3.12 a) 2F +1 α 1 = δf 2, (3.12 b) where δ 1, and primes denote differentiation with respect to ζ. This rescaling is inappropriate near the conical tip where F = O(δ 1 ), α = O(δ) and α = O(δ 1/2 ). A first integral of (3.12), as δ 0, gives (α 1) 2 +(2αα ) 2 =[α(0) 1] 2, (3.13)

14 342 H. A. Stone, J. R. Lister and M. P. Brenner where we have imposed the boundary condition α =0atζ= 0. If the solution of (3.13) is to be matched to a conical tip with α = O(δ), and α = O(δ 1/2 ) then α(0) = 2. We can thus integrate again to get the scaled asymptotic shape ζ = ±2(cos 1 (α 1)+[α(2 α)] 1/2 ), (3.14) which must be matched to tip regions near ζ = ±2π. We note from the scalings (3.11) that l/ã 0 = O(δ 1/2 ) and lã 2 0 = O(δ 7/2 ). It follows from (3.5 c) that l/ã 0 = O(B 3/7 )asb. The same scaling can be derived from energy estimates based on a spheroidal shape (Bacri & Salin 1982; Sherwood 1988), since it reflects a physical balance of terms, though we note that the asymptotic shape is not spheroidal as assumed, even far from the tip, and hence the numerical factors are somewhat different (figure 6). The conical tip occupies a small proportion, O(δ 1/2 )=O(B 3/7 ), of the total drop length, and so measurement of this local cone angle becomes difficult as the field strength increases. 4. Discussion The theory outlined above provides a self-consistent description of the shapes of drops with conical ends. Four points of comparison with previous work can be made. 1. Bacri & Salin (1982) performed experiments with ferrofluid drops corresponding to ɛ/ ɛ 39 and found that modestly deformed prolate drops jumped to much more elongated shapes when the magnetic field was increased above a critical value. These large elongations then persisted even when the magnetic field was decreased below the critical value and so the drop deformation followed a hysteresis loop. The smallest experimentally observed deformation for the more highly extended shapes had l/a 0 7, which is in reasonable agreement with the theoretical predictions of 3 that a drop with ɛ/ ɛ = 39 forms conical ends at l/a 0 =8.0 (choosing the larger of the two aspect ratios allowed by equation (3.6)). 2. The common theoretical explanation for the hysteresis loop in the drop deformation versus applied electric (or magnetic) field is based on an energy analysis. It is assumed that the shape of the drop is a prolate spheroid and that the total electric energy and surface energy of the drop is then calculated. Minimization of the total energy, while maintaining the drop volume constant, leads to a prediction of the aspect ratio, l/a 0, as a function of the electric Bond number B el. For ɛ/ ɛ = 20.8, the predicted equilibrium aspect ratio is multiple valued over a range of electric fields. Nevertheless, this theoretical argument is predicated on the shape being prolate spheroidal, which is a shape that cannot satisfy the normal-stress balance and excludes the possibility of pointed ends. By contrast, the present analysis applies to the case of large deformations and pointed ends, but is inappropriate for the case of rounded ends and small deformations. 3. Our analysis has been based on the assumption of constant material properties. Basaran & Wohlhuter (1992) used finite-element simulations to argue that the experimental observation of hysteresis could be explained on the basis of nonlinear polarization effects owing to the strong fields at the nearly pointed ends. While it is unclear from their paper what the quantitative agreement

15 Drops with conical ends in electric and magnetic fields 343 (as compared to the qualitative hysteretic behaviour present in their numerical model) is with the experiments to which they refer, there is evidence for a significant analogous nonlinear effect in experiments on large deformations of ferrofluid drops (Boudovis et al. 1988). 4. Finally, everything discussed in this paper has assumed that the fluid is at rest so that a static analysis is appropriate. Nevertheless, even in a poorly conducting liquid there are still charge carriers present, generally owing to impurities (ions, for example (Ramos & Castellanos 1994)). Conduction processes give rise to fluid motion (as was demonstrated clearly by Taylor (1966)), thus destroying the static state, and the accompanying viscous stresses may not be negligible in the local analysis of the cone tip. Conduction processes have a typical timescale, ɛ 0 ɛ/σ, where σ is the conductivity of the liquid. Therefore, an experimental investigation using dielectric liquids of the static solutions described here, requires that the observation time at a given field strength be shorter than the typical conduction time in the liquid. As discussed by Ramos & Castellanos (1994), these charge relaxation times are typically smaller than 10 5 s, so experimental verification using dielectric liquids will be difficult without careful purification. 5. Conclusions When an electric field is applied to a dielectric liquid, the interface may form conical tips. In this paper, we used the approximations of slender-body theory to derive a second-order nonlinear ODE that describes the coupling between the electric field and the shape of a slender drop. This equation has solutions that correspond to drops with conical ends. Integration yields detailed drop shapes (figure 5), and several new analytic approximations are evident from the form of the equations, as follows. (i) The cone angle 2θ 0 and dielectric constant ratio, ɛ/ ɛ, between the drop fluid (ɛ) and the surrounding fluid ( ɛ) are related by ɛ ɛ =1 8/3 tan 2 θ 0 ln(tan θ 0 ). (5.1) Since this function has a minimum, there is a minimum dielectric-constant ratio, ɛ/ ɛ =15.5, at which conical tips may form. At the minimum, the cone half-angle is θ 0 = tan 1 (e 1/2 ). Both of these approximate results are in good agreement with the exact formula (equation (1.1)) relating ɛ/ ɛ to θ 0. (ii) The minimum critical field, E min, necessary to produce a drop with a conical end, approximately satisfies E min (ɛ/ ɛ 1) 5/12, and the aspect ratio of the drop at which the conical end exists, increases proportional to (ɛ/ ɛ 1) 1/2. (iii) Large electric fields (E E min ) produce highly distorted drops with aspect ratios proportional to E 6/7. The size of the nearly conical tip region decreases as E increases, which makes measurement of the local cone angle increasingly difficult. It would be interesting to test points (ii) and (iii) experimentally, which may be easiest to accomplish with ferrofluids.

16 344 H. A. Stone, J. R. Lister and M. P. Brenner We are grateful to E. J. Hinch and A. J. Mestel for constructive and valuable comments on an earlier version of this paper. M.P.B. thanks the Petroleum Research Fund for partial support of this research. H.A.S. thanks the Army Research Office (DAAG ) for partial support of this research. Appendix A. A physical argument for equation (2.6) An alternative way of deriving the differential equation for the electric field inside a slender dielectric drop was communicated to us by Dr E. J. Hinch (University of Cambridge, UK). A slender dielectric drop, when exposed to an electric field, appears from the outside to be a line distribution of charges. By Gauss s law, this charge per unit length, q(z), is the axial derivative of the cross-sectional flux, q(z) = d dz (πa2 ɛe). (A 1) The potential, φ lc, generated at a point (r, z) external to the drop by this line charge is l q(s) φ lc (r, z) = l 4π ɛ ds. (A 2) r 2 +(z s) 2 This integral may be evaluated asymptotically for r = a(z) l to give φ lc (z) 2lnA(q(z)/4π ɛ), where A is the aspect ratio, and the induced axial field E lc = φ lc. Therefore, in the presence of a uniform external field E, the total electric field inside the dielectric is given by ( ) ɛ ln A d 2 E(z) =E ɛ 2 dz 2 (a2 E), (A 3) which, for ɛ/ ɛ 1, is equation (2.6) in the text. Appendix B. Analytic solution By the symmetry of Ẽ about the centre of the drop, and consideration of (3.10), it can be shown that Ẽ(0) is the larger root of 3Ẽ2 2Ẽ + p= 0, so that Ẽ(0) = 1 3 (1+(1 3 p)1/2 ), ã(0) = The length and volume are given by the integrals l 0 l = ã 2 d z = Ẽ(0) Ẽ(0) 9/2 1+3 p+(1 3 p) 1/2. (B 1) (3Ẽ2 p)dẽ (B 2) (Ẽ2 + p) 2 (3Ẽ2 2Ẽ + p)1/2, (3Ẽ2 p)dẽ (B 3) (Ẽ2 + p) 4 (3Ẽ2 2Ẽ + p)1/2, which can be evaluated from 2.25 of Gradshteyn & Ryzhik (1980). The two simplest such results are given below.

17 Drops with conical ends in electric and magnetic fields 345 Figure 7. The (E,u) phase plane of equation (3.8) for various values of p. The four cases shown are typical of the topology for p < 1, 1 < p <0, 0 < p < 1 and 1 < p, respectively. Trajectories corresponding to a conical tip are shown bold. Trajectories cannot cross the short-dashed 3 3 barrier lines ( p <0) and trajectories crossing the long-dashed lines ( p >0) give multi-valued functions. (i) p =0 Ẽ( z) is given by (3Ẽ)3/2 (1+3Ẽ)(3Ẽ 2)1/2 Ẽ 3/2 = l z, (B 4) where the drop shape follows from ã = Ẽ 2 and Ẽ(0) = 2 3, ã 0 = 9 4, l =3 3, l ã 0 = (B 5) The scaled volume enclosed by the drop is given by 2 l 0 ã 2 d z = (B 6)

18 346 H. A. Stone, J. R. Lister and M. P. Brenner (ii) p = 1 3 Ẽ( z) is given by ( ) 3(1 Ẽ) 1 3Ẽ2 +1 tan 1 = Ẽ z). (B 7) It follows that Ẽ(0) = 1 3, ã 0 = 9 4, l = π 2.87, l/ã , (B 8) and 2 l 0 ã 2 d z = 45 8 π (B 9) Appendix C. The phase plane of equation (3.8) The trajectories of (3.8) are simply the contours of the first integral (3.9). Thus no trajectories can cross the lines Ẽ = ±( p)1/2 when p <0 and any singular points are centres or saddles. Inspection of (3.8) shows that the singular points are (1, 0) and, when p >0, (±( 1 3 p)1/2, 0). As shown in figure 7, the topology of the phase plane changes at p = 1 when the barrier line Ẽ =+( p)1/2 crosses the singular point (1, 0), and at p = 0 when the barrier lines Ẽ = ±( p)1/2 give way to the singular points (±( 1 3 p)1/2, 0). A further significant change in topology occurs at p = 1 3 when the separatrix from the saddle at (+( 1 3 p)1/2, 0) closes off at a finite positive value of Ẽ rather than extending to Ẽ =. We note from (3.10) that trajectories crossing Ẽ = ±( 1 3 p)1/2 are not physically allowable solutions, since Ẽ and hence, from (3.5 b), ã, are multiple-valued functions of z. The trajectory of interest is the one with u 0asẼ, shown bold in figure 7, which corresponds to the behaviour (3.1) at a conical end. This trajectory only corresponds to a finite drop with two conical ends if 1 < p < 1 3, since, for p < 1, it asymptotes to the barrier line at Ẽ =+( p)1/2, and, for p > 1 3, it crosses Ẽ =+( 1 3 p)1/2 and gives a multi-valued function. References Bacri, J. C. & Salin, D Instability of ferrofluid magnetic drops under magnetic field. J. Phys. Lett. 43, L649 L654. Bacri, J. C. & Salin, D Dynamics of the shape transition of a magnetic ferrofluid drop. J. Phys. Lett. 44, L415 L420. Bailey, A. G Electrospraying of liquids. Wiley. Basaran, O. A. & Wohlhuter, F. K Effect of nonlinear polarization on shapes and stability of pendant and sessile drops in an electric (magnetic) field. J. Fluid Mech. 244, Boudouvis, A. G., Puchalla, J. L. & Scriven, L. E Magnetohydrostatic equilibria of ferrofluid drops in external magnetic fields. Chem. Eng. Comm. 67, Brazier-Smith, P. R Stability and shape of isolated and pairs of water drops in an electric field. Phys. Fluids 14, 1 6. Dodgson, N. & Sozou, C The deformation of a liquid drop by an electric field. ZAMP 38, Fernández de la Mora, J The effect of charge emission from electrified liquid cones. J. Fluid Mech. 243,

19 Drops with conical ends in electric and magnetic fields 347 Garton, C. G. & Krasucki, Z Bubbles in insulating liquids: stability in an electric field. Proc. R. Soc. Lond. A 280, Gradshteyn, I. S. & Ryzhik, I. M Table of integrals, series and products. Academic. Hinch, E. J Perturbation methods. Cambridge University Press. Li, H., Halsey, T. & Lobkovsky, A Singular shape of a fluid drop in an electric or magnetic field. Europhys. Lett. 27, Miksis, M. J Shape of a drop in an electric field. Phys. Fluids 24, Pantano, C., Gañán-Calvo, A. M. & Barrero, A Zeroth-order, electrohydrostatic solution for electrospraying in cone-jet mode. J. Aerosol Sci. 25, Ramos, A. & Castellanos, A Conical points in liquid liquid interfaces subjected to electric fields. Phys. Lett. A 184, Rosenkilde, C. E A dielectric fluid drop in an electric field. Proc. R. Soc. Lond. A 312, Séro-Guillaume, O. E., Zouaoui, D., Bernardin, D. & Brancher, J. P The shape of a magnetic liquid drop. J. Fluid Mech. 241, Sherwood, J. D Breakup of fluid droplets in electric and magnetic fields. J. Fluid Mech. 188, Sherwood, J. D The deformation of a fluid drop in an electric field: a slender-body analysis. J. Phys. A 24, Stratton, J. A Electromagnetic theory. McGraw-Hill. Taylor, G. I Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280, Taylor, G. I Studies in electrohydrodynamics. I. The circulation produced in a drop by an electric field. Proc. R. Soc. Lond. A 291, Wilson, C. T. R. & Taylor, G. I The bursting of soap-bubbles in a uniform electric field. Proc. Camb. Phil. Soc. 22, Wohlhuter, F. K. & Basaran, O. A Shapes and stability of pendant and sessile dielectric drops in an electric field. J. Fluid Mech. 235, Zeleny, J Instability of electrified liquid surfaces. Phys. Rev. 10, 1 6.

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