Download Drops with conical ends in electric and magnetic fields

Drops with conical ends in electric
and magnetic fields
By Howard A. S t o n e1 , J o h n R. Lister2 a n d M i c h a e l P. B r e n n e r3
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, /¯
, 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 /¯
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 Emin for cone formation proportional to (/¯
− 1)−5/12 , at which the
1/2
aspect ratio of the drop is proportional to (/¯
− 1) . Drop shapes are computed
6/7
for E∞ > Emin . For E∞ Emin the aspect ratio of the drop is proportional to E∞ .
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
Proc. R. Soc. Lond. A (1999) 455, 329–347
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329
c 1999 The Royal Society
TEX Paper
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 (` > a0 ),
while above a critical field strength, and for dielectric-constant ratios /¯
> 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, Emin , at which a conical end is
first observed. The conical-ended shapes of drops with an applied field E∞ > Emin
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
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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
(magnetic liquids)
Séro-Guillaume et al . (1992)
numerical calculation
Brazier-Smith (1971), Miksis (1981),
Boudouvis et al . (1988), Basaran & Wohlhuter (1992)
numerical calculation with
asymptotics for a conical tip
Pantano et al . (1994)
slender-body theory
Sherwood (1991), Li et al . (1994)
slender-body theory for
field and shape utilizing
local conical structure of tip
this paper
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 /¯
> 18.08 (Miksis 1981), where and ¯ 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 /¯
> 20.8. 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, /¯
≈ 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
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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 < /¯
< 20, above which solutions were not
obtained. Recently, however, Pantano et al . (1994) reconsidered the case of a conducting drop (in air; /¯
→ ∞) 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 2θ0 between a fluid
with dielectric constant, , and a surrounding immiscible fluid of dielectric constant
¯. Li et 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 θ0 satisfies (0 < θ0 < π):
0
0
P1/2 (cos θ0 )P1/2
(− cos θ0 ) + (/¯
)P1/2 (− cos θ0 )P1/2
(cos θ0 ) = 0,
(1.1)
where P1/2 (x) is a Legendre function of the first kind. This equation has no solutions
for /¯
< 17.59 (corresponding to θ0 = 30◦ ), and allows two conical solutions for
/¯
> 17.59. Taylor (1964) had previously shown that a perfect conductor has a
cone angle such that P1/2 (− cos θ0 ) = 0, or θ0 = 49.3◦ . We note that in the case of
two conducting liquids, with conductivities σ and σ̄, a local conical solution is again
possible with /¯
replaced by σ/σ̄ 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 a0 /` 1,
where 2` is the length of the drop and a0 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)).
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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 and ¯ are constant. At the interface
between two dielectrics, the tangential component of the electric field, Et , is continuous, but the normal component, En , suffers a jump, given by the boundary condition
En = ¯Ēn , where overbars indicate values in the external fluid; the jump Ēn − En
gives the local induced surface charge density. The electric stress tensor in a medium
with dielectric constant is T E = 0 (EE − 12 |E|2 I), where 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]] = 12 0 ( − ¯)(Et2 − (/¯
)En2 )
(2.1)
[[n · T E · n]] ≈ 12 0 ( − ¯)E 2 ,
(2.2)
(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 /¯
→ ∞, though we shall think of /¯
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 (a0 /` 1), the electric field varies
primarily in the z-direction, so that E ≈ E(z)ez and Er = O(a0 E/`). Moreover,
the normal and tangential components of the electric field satisfy En = O(a0 Et /`).
Assuming, as will be verified later, that (a0 /`)2 (/¯
) 1 for slender drops, the
electric stress jump (2.1) can be approximated by
since Et ≈ 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 Et ≡ 0 and a non-zero En 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 0 (
− ¯)E 2 (z) + ∆p = γ/a(z),
2
(2.3)
2
where we have also neglected a contribution, d a/dz , to the curvature, since this is
O(a20 /`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
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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)



Z

/¯
−1
(x − y)
E∞ +
E (y) dSy = 12 [E(x) + Ē(x)], x ∈ S,
(2.4 b)
3 n

4π
|x
−
y|
S



 Ē(x),
x ∈ V̄ ,
(2.4 c)
where V , V̄ 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)ez . Then, from an integration of ∇ · E = 0, it follows that,
at the interface r = a(z), the small radial field is Er = − 12 a(z)∂E/∂z, with error
O(a20 /`2 ). The unit outward normal, n, has Cartesian coordinates proportional to
(cos θ, sin θ, −∂a/∂z), and so E · n = En ≈ −(1/2a)∂(a2 E)/∂z. Choosing a point
x = (0, 0, z) on the centreline, denoting the integration variable as
y = (a cos θ, a sin θ, s),
and taking the z-component of (2.4 a) then leads to
Z `
(z − s)(d/ds)[a2 (s)E(s)]
1 −1
ds = E∞ .
E(z) −
2 ¯
[a2 (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 2θ0 , the ratio of the radial to the axial scales of variation of E
is O(tan θ0 ), and hence, (2.5) is valid even near the tip for θ0 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| ` (Hinch 1991,
p. 43). Defining A 1 to be an effective length-to-width ratio, we obtain the differential equation
2
ln A d
E(z) −
−1
(a2 E) = E∞ ,
(2.6)
2
¯
dz 2
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 `/a0 . Asymptotic examination of
the integral in (2.5) near the conical ends suggests that A might alternatively be
taken to be 1/ tan θ0 . We note that either choice for A is asymptotically consistent
to O(1/ ln A) if `/a0 = O(1/ tan θ0 ) as A → ∞.
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 ,
Z `
π
a(z)2 dz = V,
(2.7)
−`
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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, (a2 /a20 )+
(z 2 /`2 ) = 1, the one-dimensional equation (2.6) is satisfied by a uniform electric field:
E∞
E1D =
,
(2.8)
1 + (/¯
− 1)(a0 /`)2 ln A
which may be compared with the exact result (Stratton 1941, pp. 211–214; eqn 45
has a misprint in a sign) for a spheroid (expanded for a0 /` 1):
E∞
.
(2.9)
Espheroid =
1 + (/¯
− 1)(a0 /`)2 [ln(`/a0 ) + ln(2/e)]
Taking A = `/a0 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 E1D is within 15% of
Espheroid for all dielectric-constant ratios at aspect ratios as small as `/a0 = 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 2θ0 . Since a → 0 as z → `, static
equilibrium requires that the local electric field diverges as E ∝ a−1/2 . Thus,
a(z) = (` − z) tan θ0
and E(z) ∝ (` − z)−1/2 ,
as z → `.
(3.1)
Substitution into (2.6) with A = 1/ tan θ0 , as appropriate for this local analysis,
yields
8/3
=1−
,
(3.2)
2
¯
tan θ0 ln(tan θ0 )
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 θ0 < 20◦ (i.e.
/¯
> 21), the approximate result differs by less than 5% from the exact result.
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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 /¯
→ ∞ in (3.2) corresponds to θ0 → 45◦ (or θ0 → 0), which
is in surprisingly good agreement with Taylor’s (1964) exact result of θ0 = 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),
−1 −1/2
= 1 + 16
(e
) ≈ 31.2◦ ,
(3.3)
3 e ≈ 15.5 at θ0 = tan
¯
compares well with the numerically determined minimum, /¯
= 17.6, at θ0 = 30◦
predicted by the exact formula (1.1). These agreements are encouraging, though
somewhat surprising since (3.2) was derived for small cone angles θ0 .
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
, changes the local strength of the
that a decrease in the cone angle, θ0 , at fixed /¯
electric field according to E ∝ (` − 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 puzProc. R. Soc. Lond. A (1999)
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, `cap , and ‘electric Bond
number’, B:
`cap =
2γ
2
0 ¯E∞
and B =
R
`cap
=
2
0 ¯E∞
R
,
2γ
(3.4)
where 43 πR3 = V , in order to scale the stress balance between the electric field and
−
interfacial tension. If we now non-dimensionalize radial dimensions by `cap /(/¯
− 1)]1/2 , the internal excess pressure by
1), axial dimensions by `cap [ 12 ln A/(/¯
γ(/¯
− 1)/`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 − (ã2 Ẽ)00 = 1,
2
Z
−1
Ẽ + p̃ = ã ,
1/2 5/2
2
2
4 3
−1
ã dz̃ = 3 B
,
ln A
¯
−`˜
`˜
(3.5 a)
(3.5 b)
(3.5 c)
where primes denote derivatives with respect to z̃. For a given value of p̃, 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
˜
as z̃ → `.
ã ∼ ( 4 )1/2 (`˜ − z̃) and Ẽ ∼ [ 4 (`˜ − z̃)2 ]−1/4
3
3
The scaled cone angle, tan−1 [( 43 )1/2 ], corresponds to an unscaled cone angle given
by (3.2) if A is taken as 1/ tan θ0 . However, for a description of the global shape of
the drop, it is more consistent to take A = `/a0 and we do this below.
˜ 0 , is
From the numerically determined shape, the dimensionless aspect ratio, `/ã
known, and so the actual aspect ratio, `/a0 , follows from
1/2 1/2 ˜
`
`
ln A
−1
=
.
(3.6)
a0
2
¯
ã0
In the derivation of (2.2) we assumed that (a0 /`)2 (/¯
) 1. We now note from (3.6)
) = O(1/ ln A) 1, which is consistent with the level of approximathat (a0 /`)2 (/¯
tion 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
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H. A. Stone, J. R. Lister and M. P. Brenner
electric field, Emin , 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
6
−1
= const.,
(3.7)
Emin
¯
(conical tips only form for /¯
1). Furthermore, from (3.6), we observe that the
aspect ratio of the conical-ended drop varies with the dielectric properties according
− 1)1/2 . Thus, the aspect ratio is related to the field necessary to
to `/a0 ∝ (/¯
−6/5
produce the conical tips by `/a0 ∝ Emin .
(ii) Reduction to a first-order ODE
In order to determine the drop shape numerically, we first let (ã2 Ẽ)0 = u and
obtain
(Ẽ − 1)(p̃ − 3Ẽ 2 )
du
=
.
dẼ
u(Ẽ 2 + p̃)3
(3.8)
It is clear that p̃ 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
u2 =
3Ẽ 2 − 2Ẽ + p̃
+ c,
(Ẽ 2 + p̃)2
(3.9)
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(z̃), which is symmetric about z̃ = 0, may then
be determined by integrating the first-order differential equation
(3Ẽ 2 − 2Ẽ + p̃)1/2 (Ẽ 2 + p̃)2
dẼ
,
(3.10)
=
dz̃
3Ẽ 2 − p̃
from the asymptotic behaviour, Ẽ ∼ [ 43 (`˜ − z̃)2 ]−1/4 at the tip towards the middle of
the drop, where Ẽ 0 = 0 by symmetry. If Ẽ 0 = 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 = 0 (corresponding to the
middle of a finite drop) if −1 6 p̃ 6 13 . As p̃ → 13 , the dimensionless drop volume
tends to a finite minimum, and as p̃ → −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 z̃(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 p̃ = 13 is of particular interest since it corresponds to the drop of minivolume; this dimensionless volume is found, from the analytic solution, to be
Rmum
`˜
˜ 0 = 1.275 . . . . It can be verified that the solution has
ã2 dz̃ ≈ 11.10 . . . , with `/ã
−`˜
an unbounded second derivative at z̃ = 0; this feature corresponds to the loss of
solution for p̃ > 13 .
Proc. R. Soc. Lond. A (1999)
Drops with conical ends in electric and magnetic fields
339
Figure 3. Drop shapes for different p̃ from the numerical solution of (3.10) and (3.5 b).
R `˜
Figure 4. Numerically determined drop volume 2 0 ã2 dz̃ for different values of p̃. The solid
circles denote the volume calculated analytically for some simple cases: p̃ = 13 , 0, − 14 and − 49 .
Proc. R. Soc. Lond. A (1999)
340
H. A. Stone, J. R. Lister and M. P. Brenner
Figure 5. (a) Drop shapes at the first appearance of conical tips for /¯
= 20, 40, 100. (b) Increasing elongation of the drop shape with increasing electric-field strength above Emin for /¯
= 40.
AtR large field strengths the shapes are approximately ellipsoidal with conical tips. In all cases,
`
2 0 ã2 dz̃ 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
p̃ are shown in figure 3. The dimensionless volume of the drop is shown in figure 4 as
a function of p̃. 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 p̃ = 13 may be combined with equations (3.5 c) and (3.6) to obtain
explicit formulae for the variation of the minimum electric field, Emin , necessary to
obtain conical ends, and the actual aspect ratio at this critical value as functions of
/¯
. 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 /¯
in figure 6. The termini of the curves
correspond to the critical fields, Emin , 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 < Emin 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 `/a0 > 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
Proc. R. Soc. Lond. A (1999)
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
/¯
= 20, 40, 100 (solid). The curves terminate at the minimum Bond number required to
produce a conical tip (long dashes). At large Bond number, `/a0 ∝ B3/7 , as is also true of the
spheroidal approximation of Sherwood (1988) (short dashes).
proportional to (/¯
)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 p̃ → −1. In this limit, the scales of ã and z̃ become large (figure 3), and
hence, Ẽ 2 ∼ −p̃ ∼ 1. Consideration of the balance of terms in (3.5 a) and (3.5 b)
suggests that the rescalings
p̃ = −1 + δ,
Ẽ = 1 + δF,
ã = δ −1 α,
z̃ = δ −3/2 ζ,
(3.11)
which give
F − (α2 )00 = δ(α2 F )00 ,
(3.12 a)
(3.12 b)
2F + 1 − α−1 = −δF 2 ,
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 α0 = O(δ −1/2 ).
A first integral of (3.12), as δ → 0, gives
(α − 1)2 + (2αα0 )2 = [α(0) − 1]2 ,
(3.13)
Proc. R. Soc. Lond. A (1999)
342
H. A. Stone, J. R. Lister and M. P. Brenner
where we have imposed the boundary condition α0 = 0 at ζ = 0. If the solution of
(3.13) is to be matched to a conical tip with α = O(δ), and α0 = 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.
˜ 0 = O(δ −1/2 ) and `ã
˜ 2 = O(δ −7/2 ). It
We note from the scalings (3.11) that `/ã
0
3/7
˜
follows from (3.5 c) that `/ã0 = O(B ) as B → ∞. 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 /¯
≈ 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 `/a0 ≈ 7, which is in reasonable agreement with the theoretical predictions of § 3 that a drop with /¯
= 39 forms
conical ends at `/a0 = 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, `/a0 , as a function of the electric Bond number Bel .
For /¯
= 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
Proc. R. Soc. Lond. A (1999)
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, 0 /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.
, between the drop fluid
(i) The cone angle 2θ0 and dielectric constant ratio, /¯
() and the surrounding fluid (¯
) are related by
8/3
=1−
.
2
¯
tan θ0 ln(tan θ0 )
(5.1)
Since this function has a minimum, there is a minimum dielectric-constant
ratio, /¯
= 15.5, at which conical tips may form. At the minimum, the cone
half-angle is θ0 = tan−1 (e−1/2 ). Both of these approximate results are in good
agreement with the exact formula (equation (1.1)) relating /¯
to θ0 .
(ii) The minimum critical field, Emin , necessary to produce a drop with a conical
− 1)−5/12 , and the aspect ratio of the
end, approximately satisfies Emin ∝ (/¯
drop at which the conical end exists, increases proportional to (/¯
− 1)1/2 .
(iii) Large electric fields (E∞ Emin ) produce highly distorted drops with aspect
6/7
ratios proportional to E∞ . 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.
Proc. R. Soc. Lond. A (1999)
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 55-97-0114) 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
(πa2 E).
dz
(A 1)
The potential, φlc , generated at a point (r, z) external to the drop by this line charge
is
Z `
q(s)
p
φlc (r, z) =
ds.
(A 2)
r2 + (z − s)2
−` 4π¯
This integral may be evaluated asymptotically for r = a(z) ` to give φlc (z) ∼
−2 ln A(q(z)/4π¯
), where A is the aspect ratio, and the induced axial field Elc =
−∇φlc . Therefore, in the presence of a uniform external field E∞ , the total electric
field inside the dielectric is given by
ln A d2 2
E(z) = E∞ −
(a E),
(A 3)
¯ 2 dz 2
which, for /¯
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 3Ẽ 2 − 2Ẽ + p̃ = 0, so that
Ẽ(0) = 13 (1 + (1 − 3p̃)1/2 ),
ã(0) =
9/2
.
1 + 3p̃ + (1 − 3p̃)1/2
The length and volume are given by the integrals
Z ∞
(3Ẽ 2 − p̃) dẼ
`˜ =
,
2
2
2
1/2
Ẽ(0) (Ẽ + p̃) (3Ẽ − 2Ẽ + p̃)
Z `˜
Z ∞
(3Ẽ 2 − p̃) dẼ
2
ã dz̃ =
,
2
4
2
1/2
Ẽ(0) (Ẽ + p̃) (3Ẽ − 2Ẽ + p̃)
0
(B 1)
(B 2)
(B 3)
which can be evaluated from § 2.25 of Gradshteyn & Ryzhik (1980). The two simplest
such results are given below.
Proc. R. Soc. Lond. A (1999)
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 p̃. The four cases shown
are typical of the topology for p̃ < −1, −1 < p̃ < 0, 0 < p̃ < 13 and 13 < p̃, respectively. Trajectories corresponding to a conical tip are shown bold. Trajectories cannot cross the short-dashed
barrier lines (p̃ < 0) and trajectories crossing the long-dashed lines (p̃ > 0) give multi-valued
functions.
(i) p̃ = 0
Ẽ(z̃) is given by
(3Ẽ)3/2 − (1 + 3Ẽ)(3Ẽ − 2)1/2
= `˜ − z̃,
Ẽ 3/2
(B 4)
where the drop shape follows from ã = Ẽ −2 and
Ẽ(0) = 23 ,
ã0 = 94 ,
√
`˜ = 3 3,
4
`˜
= √ ≈ 2.31.
ã0
3
(B 5)
The scaled volume enclosed by the drop is given by
Z
2
Proc. R. Soc. Lond. A (1999)
0
`˜
ã2 dz̃ =
1296 √
3
77
≈ 29.15.
(B 6)
346
H. A. Stone, J. R. Lister and M. P. Brenner
(ii) p̃ =
1
3
Ẽ(z̃) is given by
√
3(1 − Ẽ)
1
−1
√
− tan
= 23 (`˜ − z̃).
3Ẽ 2 + 1
3Ẽ
It follows that
Ẽ(0) = 13 ,
√
`˜ = 34 3 + 12 π ≈ 2.87,
ã0 = 94 ,
and
Z
2
0
`˜
ã2 dz̃ =
45
8 π
−
243 √
3
64
˜ 0 ≈ 1.275,
`/ã
≈ 11.10.
(B 7)
(B 8)
(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 = ±(−p̃)1/2 when p̃ < 0 and any singular points are
centres or saddles. Inspection of (3.8) shows that the singular points are (1, 0) and,
when p̃ > 0, (±( 13 p̃)1/2 , 0). As shown in figure 7, the topology of the phase plane
changes at p̃ = −1 when the barrier line Ẽ = +(−p̃)1/2 crosses the singular point
(1, 0), and at p̃ = 0 when the barrier lines Ẽ = ±(−p̃)1/2 give way to the singular
points (±( 13 p̃)1/2 , 0). A further significant change in topology occurs at p̃ = 13 when
the separatrix from the saddle at (+( 13 p̃)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 = ±( 13 p̃)1/2 are not physically allowable solutions, since Ẽ and hence, from (3.5 b),
ã, are multiple-valued functions of z̃.
The trajectory of interest is the one with u → 0 as Ẽ → ∞, 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 < p̃ < 13 , since, for
p̃ < −1, it asymptotes to the barrier line at Ẽ = +(−p̃)1/2 , and, for p̃ > 13 , it crosses
Ẽ = +( 13 p̃)1/2 and gives a multi-valued function.
References
Bacri, J. C. & Salin, D. 1982 Instability of ferrofluid magnetic drops under magnetic field. J.
Phys. Lett. 43, L649–L654.
Bacri, J. C. & Salin, D. 1983 Dynamics of the shape transition of a magnetic ferrofluid drop. J.
Phys. Lett. 44, L415–L420.
Bailey, A. G. 1988 Electrospraying of liquids. Wiley.
Basaran, O. A. & Wohlhuter, F. K. 1992 Effect of nonlinear polarization on shapes and stability
of pendant and sessile drops in an electric (magnetic) field. J. Fluid Mech. 244, 1–16.
Boudouvis, A. G., Puchalla, J. L. & Scriven, L. E. 1988 Magnetohydrostatic equilibria of ferrofluid drops in external magnetic fields. Chem. Eng. Comm. 67, 129–144.
Brazier-Smith, P. R. 1971 Stability and shape of isolated and pairs of water drops in an electric
field. Phys. Fluids 14, 1–6.
Dodgson, N. & Sozou, C. 1987 The deformation of a liquid drop by an electric field. ZAMP 38,
424–432.
Fernández de la Mora, J. 1992 The effect of charge emission from electrified liquid cones. J.
Fluid Mech. 243, 561–574.
Proc. R. Soc. Lond. A (1999)
Drops with conical ends in electric and magnetic fields
347
Garton, C. G. & Krasucki, Z. 1964 Bubbles in insulating liquids: stability in an electric field.
Proc. R. Soc. Lond. A 280, 211–226.
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Table of integrals, series and products. Academic.
Hinch, E. J. 1991 Perturbation methods. Cambridge University Press.
Li, H., Halsey, T. & Lobkovsky, A. 1994 Singular shape of a fluid drop in an electric or magnetic
field. Europhys. Lett. 27, 575–580.
Miksis, M. J. 1981 Shape of a drop in an electric field. Phys. Fluids 24, 1967–1972.
Pantano, C., Gañán-Calvo, A. M. & Barrero, A. 1994 Zeroth-order, electrohydrostatic solution
for electrospraying in cone-jet mode. J. Aerosol Sci. 25, 1065–1077.
Ramos, A. & Castellanos, A. 1994 Conical points in liquid–liquid interfaces subjected to electric
fields. Phys. Lett. A 184, 268–272.
Rosenkilde, C. E. 1969 A dielectric fluid drop in an electric field. Proc. R. Soc. Lond. A 312,
473–494.
Séro-Guillaume, O. E., Zouaoui, D., Bernardin, D. & Brancher, J. P. 1992 The shape of a
magnetic liquid drop. J. Fluid Mech. 241, 215–232.
Sherwood, J. D. 1988 Breakup of fluid droplets in electric and magnetic fields. J. Fluid Mech.
188, 133–146.
Sherwood, J. D. 1991 The deformation of a fluid drop in an electric field: a slender-body analysis.
J. Phys. A 24, 4047–4053.
Stratton, J. A. 1941 Electromagnetic theory. McGraw-Hill.
Taylor, G. I. 1964 Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280,
383–397.
Taylor, G. I. 1966 Studies in electrohydrodynamics. I. The circulation produced in a drop by an
electric field. Proc. R. Soc. Lond. A 291, 159–166.
Wilson, C. T. R. & Taylor, G. I. 1925 The bursting of soap-bubbles in a uniform electric field.
Proc. Camb. Phil. Soc. 22, 728–730.
Wohlhuter, F. K. & Basaran, O. A. 1992 Shapes and stability of pendant and sessile dielectric
drops in an electric field. J. Fluid Mech. 235, 481–510.
Zeleny, J. 1917 Instability of electrified liquid surfaces. Phys. Rev. 10, 1–6.
Proc. R. Soc. Lond. A (1999)