Full text of DOI:10.1063/1.870354

Stabilization of electrically conducting capillary bridges using feedback control of
radial electrostatic stresses and the shapes of extended bridges
Mark J. Marr-Lyon, David B. Thiessen, Florian J. Blonigen, and Philip L. Marston
Citation: Physics of Fluids 12, 986 (2000); doi: 10.1063/1.870354
View online: https://doi.org/10.1063/1.870354
View Table of Contents: http://aip.scitation.org/toc/phf/12/5
Published by the American Institute of Physics
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PHYSICS OF FLUIDS
VOLUME 12, NUMBER 5
MAY 2000
Stabilization of electrically conducting capillary bridges using feedback
control of radial electrostatic stresses and the shapes of extended bridges
Mark J. Marr-Lyon, David B. Thiessen, Florian J. Blonigen, and Philip L. Marston
Department of Physics, Washington State University, Pullman, Washington 99164-2814
͑Received 28 October 1999; accepted 12 January 2000͒
Electrically conducting, cylindrical liquid bridges in a density-matched, electrically insulating bath
were stabilized beyond the Rayleigh–Plateau ͑RP͒ limit using electrostatic stresses applied by
concentric ring electrodes. A circular liquid cylinder of length L and radius R in real or simulated
zero gravity becomes unstable when the slenderness SϭL/2R exceeds ␲. The initial instability
involves the growth of the so-called ͑2, 0͒ mode of the bridge in which one side becomes thin and
the other side rotund. A mode-sensing optical system detects the growth of the ͑2, 0͒ mode and an
analog feedback system applies the appropriate voltages to a pair of concentric ring electrodes
positioned near the ends of the bridge in order to counter the growth of the ͑2, 0͒ mode and prevent
breakup of the bridge. The conducting bridge is formed between metal disks which are grounded.
Three feedback algorithms were tested and each found capable of stabilizing a bridge well beyond
the RP limit. All three algorithms stabilized bridges having S as great as 4.3 and the extended
bridges broke immediately when feedback was terminated. One algorithm was suitable for
stabilization approaching Sϭ4.493... where the ͑3, 0͒ mode is predicted to become unstable for
cylindrical bridges. For that algorithm the equilibrium shapes of bridges that were slightly under or
over inflated corresponded to solutions of the Young–Laplace equation with negligible electrostatic
stresses. The electrical conductivity of the bridge liquid need not be large. The conductivity was
associated with salt added to the aqueous bridge liquid. © 2000 American Institute of Physics.
͓S1070-6631͑00͒00505-5͔
influence the RP instability.7 Of the various methods for sta-
I. INTRODUCTION
bilization which have been previously investigated, the ac-
tive control8 of acoustic radiation stresses9 has the greatest
The stability of a weightless column of liquid between
identical circular supports is ordinarily governed by capillary
forces. When the volume of the liquid column is constrained
to be that of a circular cylinder between coaxial supports, the
column first becomes unstable when its length L exceeds
2␲R where R is the radius of the supports.1–5 In the discus-
sion which follows, this condition is referred to as the
Rayleigh–Plateau ͑RP͒ limit and the corresponding limit for
the slenderness SϭL/2R is ␲. In this paper a novel method
for suppressing the RP instability is demonstrated based on
the active control of electrostatic stresses on a grounded liq-
uid bridge. While the experimental demonstration was lim-
ited to situations in which the electrical conductivity of the
bridge was enhanced by adding salt ͑NaCl͒ to the bridge
liquid, it is anticipated that a similar method may be useful
for certain situations in which the electrical conductivity of
the liquid is small. In the experiments, the weightless condi-
tion is established by the Plateau tank method where, for the
present application, the electrical conductivity is negligible
for the outer ͑density matched͒ liquid. The electric field and
associated stresses are nearly radial since the bridge is
grounded.
similarity with the approach considered here.
Some situations where the RP instability is relevant in-
clude the following: materials processing based on the float-
ing zone method,3,10,11 drop formation and the breakup of
laminar liquid jets,12,13 the coating of solid fibers,14 and the
release of air bubbles from an underwater nozzle.15 It is note-
worthy that the boundary conditions of fixed circular contact
curves of equal radius may be insufficient for describing
floating zone crystallization even in low gravity,16,17 and for
that application the liquid volume might not be constrained
to be at ͑or near͒ ␲R2L as in our experiments. Nevertheless,
feedback control strategies for suppressing capillary driven
instabilities are relevant.
II. ACTIVE STABILIZATION USING ELECTRIC FIELDS
As was previously demonstrated,8 the slenderness of a
liquid bridge in a Plateau tank may be extended significantly
beyond the RP limit by optically sensing the amplitude and
phase of the axisymmetric mode of interest and using the
information to rapidly control the spatial distribution of the
acoustic radiation stress. The first mode which naturally be-
comes unstable for a circular cylindrical bridge of fixed vol-
ume having pinned contact circles at the ends is denoted here
by (n,m)ϭ(2,0), where n is an axial index and m is an
azimuthal index. This mode displays a left–right ͑L–R͒
The present study differs in various aspects with prior
investigations of methods for suppressing the RP instability.
Electric-field based methods have emphasized liquids which
are good insulators and passive electric fields which are ap-
plied axially.6 Axisymmetric laminar flow has been shown to
1070-6631/2000/12(5)/986/10/$17.00
986
© 2000 American Institute of Physics

Phys. Fluids, Vol. 12, No. 5, May 2000
Stabilization of electrically conducting capillary bridges . . .
987
carried out in a Plateau tank in which the outer bath liquid is
a good insulator with a dielectric constant that is not large.
As a consequence, it may be shown ͑see the Appendix͒ that
the electrical conductivity of the bridge liquid is not required
to be large and it was sufficient to add 2% by weight of salt
to a mixture of water and methyl alcohol. The lower limit of
salt concentration was not determined experimentally. As ex-
plained in the Appendix, the dimensions of the electrodes
were selected in such a way that the electric-field distribution
outside the surface of the bridge was appropriate for apply-
ing a stress to the side of the bridge closest to the electrode
without significantly applying a stress to the other side of the
bridge. For the situation where the bridge liquid is main-
tained at ground potential, the electrostatic stress is radially
outward and in the region closest to the left ͑right͒ electrode
is proportional to V2L(VR2 ). The electric field in the portion of
the bridge near either electrode is only weakly affected by
the electrode potential on the opposite end. Stabilization of
the liquid bridge to a slenderness significantly beyond the
natural limit of ␲ was achieved by using three different al-
gorithms for adjusting the electrode voltages VL and VR . For
definiteness in the discussion which follows, let positive val-
ues of the error voltage Ve correspond to the situation where
the left side of the bridge is slender and the right side of the
bridge is rotund. In the simplified discussion of the control
method described below, the finite frequency response of the
control system is neglected so that any delays in adjusting
the potentials VL and VR are omitted from consideration. The
three methods for selecting the electrode potentials are sum-
marized as follows.
FIG. 1. The bridge is illuminated with an expanded laser beam, which is
focused onto the segmented photodiode by a two-element lens that projects
half of the bridge on each photodiode segment. The difference in signal
between the two segments is then a measure of the ͑2, 0͒ mode deformation.
The bridge is horizontal in the Plateau tank and this is the top view.
asymmetry when viewed from the side as shown, e.g., in Fig.
2͑b͒ of Morse et al.,9 where a similar notation is used. Ad-
ditionally, all other axisymmetric modes of a cylindrical
bridge ͑n, 0͒ for nϭ3,4,... also become unstable if S is suf-
ficiently large. The slenderness Sn at which the ͑n, 0͒ mode,
where nϭ2,3,..., becomes unstable, is given by the (n/2)th
lowest nonzero root of tan Sϭ0 if n is even, and by the (n
͓
Ϫ1)/2 th lowest nonzero root of tan SϭS if n is odd.18,19 For
͔
the two modes of interest—the ͑2, 0͒ and ͑3, 0͒ modes—S2
is simply the RP limit of ␲, and S3Ϸ4.493 41. The stability
limit of the ͑3, 0͒ mode is of interest since the stabilization
methods described herein are applied only to the ͑2, 0͒ mode
instability, thus the bridge stabilized by these methods will
break when the ͑3, 0͒ mode becomes unstable. This limit will
be discussed further in Sec. IV.
1. Method 1: Simple feedback
The potential of the electrode adjacent to the slender side
of the bridge is raised in proportion to Ve while the opposite
electrode remains at ground potential,
A. Stabilization methods
In the present work ͑as well as in Marr-Lyon et al.8͒, an
error voltage Ve is optically generated which is proportional
to the L–R asymmetry of the bridge. This signal is generated
by illuminating the bridge with a laser beam and by detecting
the light which passes by the bridge without being scattered
by the bridge using a photodiode having two segments as
shown in Fig. 1. An additional description of the apparatus is
given in Sec. III. The principle of operation is that if the left
side of the bridge becomes rotund and the right side thins,
the optical power detected by the photodiode on the left de-
creases while the power into the right diode increases and
this imbalance is used ͑with the aid of a differential ampli-
fier͒ to produce the error voltage Ve which may be either
positive or negative depending on which end of the bridge is
rotund.
VLϭKVe , VRϭ0
VLϭ0, VRϭKVe
V у0͒,
͑1a͒
͑1b͒
͑
e
V Ͻ0͒,
e
͑
where K is a positive gain constant incorporating the gain of
a high-voltage amplifier. This method has the property that
the difference in the stresses applied to the left and right side
of the bridge has a magnitude proportional to V2e, which
introduces certain complications as explained in Sec. IV.
2. Method 2: Square-root feedback
The simplest method to generate a stress difference
The error voltage is processed and used to adjust the
potentials of a pair of ring electrodes which are concentric
with the axis of the cylinder. Let VL and VR denote the
potentials of the electrodes closest to the left and right ends
of the cylinder, respectively. The circular metal disks on
each end of the bridge are held at ground potential. The
electrical conductivity of the bridge liquid is taken to be
sufficiently large that the bridge liquid ͑to a good approxi-
mation͒ is maintained at ground potential. The experiment is
which varies in magnitude proportional to
͉
Ve
͉ is to use a
circuit which takes the square root of Ve
͉
͉ prior to amplifi-
cation. As in method 1, the potential of the opposite elec-
trode remains at ground potential,
1/2
VLϭK
͉
Ve
͉
,
VRϭ0
V у0͒,
e
͑2a͒
͑2b͒
͑
1/2
VLϭ0, VRϭϪK
͉
Ve
͉
V Ͻ0͒,
͑
e
where K is a positive gain constant as in Eq. ͑1͒.

988
Phys. Fluids, Vol. 12, No. 5, May 2000
Marr-Lyon et al.
2
b
3. Method 3: Bias potential feedback
where kϭmb␻ and mb are the effective spring constant and
‘‘bare’’ mass for the ͑2, 0͒ mode of the corresponding invis-
cid system18 and ␥e is an effective damping constant in the
absence of feedback. For bridges longer than Sϭ␲, k be-
comes negative8 so the gain must be positive but not larger
than ␥emb /␶. The result that the maximum allowable gain is
proportional to 1/␶ agrees with standard control theory.20 As
S is increased significantly beyond ␲, k becomes increasingly
negative and it becomes impossible to control the ͑2, 0͒
mode if Ϫk exceeds ␥emb /␶ and ͑as assumed here͒ the con-
troller uses only displacement information. Our results pre-
sented in Sec. IV indicate that for the bridge system used in
these experiments, this crossover does not occur prior to the
onset of ͑3, 0͒ mode instability. For control method 1 the
feedback force is nonlinear in x and this model is not directly
applicable.
Another method in which the generalized force for the
mode of interest is linear in the error signal Ve is to introduce
a bias voltage and to consider the difference between the
stresses on the left and right sides of the bridge. The elec-
trode voltages are taken to be
VLϭVbϩKVe , VRϭϪVbϩKVe ,
͑3͒
for all Ve where Vb is a bias voltage. It follows that the
generalized force for the mode of interest is proportional to
V2LϪV2Rϭ4KVbVe ,
͑4͒
which is linear in Ve . The effective gain of the system is
now proportional to the bias voltage Vb . As explained in
Sec. IV, this method has the complication that for a symmet-
ric bridge giving Veϭ0, each electrode has a potential of
magnitude Vb so that there is a radially outward stress on
each end of the bridge.
III. DESCRIPTION OF EXPERIMENTAL APPARATUS
A. Plateau tank
B. Feedback delay time
A horizontal liquid cylinder consisting of a mixture of
62.0 wt% water, 36.0 wt% methanol, and 2.0 wt% salt
͑NaCl͒ is formed between two 0.432 cm diameter stainless-
steel disks in a density-matched bath of 20 cS silicone oil
͑Dow Corning 200 fluid, ␳ϭ0.951 g/cm3͒. The salt provides
significantly more than the minimum conductivity required,
as explained in the Appendix. Two ring electrodes which are
concentric with the bridge are spaced by 1.16 cm. The rings
are 1.23 cm in diameter and are made of 0.12 cm diameter
copper wire. The ring diameter was chosen to give optimal
coupling of the electrostatic stress profile to the ͑2, 0͒ mode
of the bridge, as explained in the Appendix. The voltages on
the ring electrodes are varied as needed in order to stabilize a
liquid bridge while the disks are held at ground potential.
An automated system injects the bridge liquid through a
hole in one of the end disks and simultaneously retracts the
disks at such a rate that the liquid bridge remains cylindrical
during extension. The disk separation and bridge volume
may also be adjusted independently.
In the analysis presented in the Appendix to Marr-Lyon
et al.,8 one of the requirements for feedback control to stabi-
lize the bridge is that the generalized force Ffeedback for the
mode of interest must not lag significantly behind the modal
amplitude detected by the optical sensor. In practice, some
delay is necessary because the frequency response of the
amplifier circuitry is diminished at high frequencies. The
simplest causal approximation of the impulse response of the
control circuitry is the one-sided exponential h(t)
ϭH(t)eϪt/␶, where ␶ is a time constant and H(t) is a step
function which vanishes for tϽ0. For control methods 2 and
3, Ffeedback is proportional to the convolution of the error
voltage Ve(t) with h(t). The characteristic equation of the
bridge mode with feedback follows from the analysis in
Marr-Lyon et al., as will now be demonstrated. Let x(t)
ϭx0ei⍀t denote the amplitude of the ͑2, 0͒ bridge mode
where ⍀ is a complex frequency and Ve is proportional to
the ͑real part of͒ x(t). Assuming that
convolution with h(t), the feedback force becomes
͉⍀␶͉Ӷ1, from the
B. Optics
F
feedbackϷϪGx0eϪi⍀␶ei⍀t
,
͑5͒
The deformation of the bridge is detected optically, as
shown in Fig. 1. The beam of a semiconductor laser is ex-
panded and illuminates the bridge, and a two-element lens
focuses the laser beam passing by each half of the bridge
onto a separate photodiode element. A spatial filter placed at
the focal plane of the lenses prevents light scattered by the
bridge from entering the photodiode. The photodiode is po-
sitioned as close to the focal plane of the lenses as possible
so all of the light collected by the lenses falls on the appro-
priate segment of the photodiode. This method allows for
increased sensitivity and easier alignment than the previously
described method using a single lens.8
The two-element lens is composed of two circular plano-
convex lenses of focal length f modified as shown in Fig. 2.
Both lenses are cut along a chord such that the lens centers
are separated by a distance d, determined by the size and
configuration of the photodiode elements. The radius r of the
where G is a gain constant. The force in Eq. ͑5͒ is of the
form considered by Marr-Lyon et al. The characteristic
equation for the natural frequency ⍀ of the controlled mode
is reducible to8
␻2Ϫ⍀2ϩ␣ i 1ϩi͒⍀3/2ϩi␥n⍀ϭ0,
͑6͒
͑
n
n
where ␣n and ␥n are normalized modal damping constants
which depend on the viscosities of the inner and outer liquids
2
n
and ␻ is a normalized effective spring constant for the
mode. For the bridge to be stable, it is necessary to have both
␻2nϾ0 and ␥nϾ0. ͑The damping parameter ␣n is positive
and is only weakly affected by the choice for ␶.͒ The varia-
tion of ␻ and ␥n with ␶ is such that for stability the gain G
n
must be in the range8
ϪkϽGϽ␥emb /␶,
͑7͒

Phys. Fluids, Vol. 12, No. 5, May 2000
Stabilization of electrically conducting capillary bridges . . .
989
FIG. 4. High-voltage circuit using diodes.
FIG. 2. Front view of the dual-lens system.
tem could also be used. From the frequency response of the
Trek amplifier a lower limit for ␶ is estimated to be 0.03 ms.
lenses must be large enough so that vignetting of the bridge
by the lenses does not occur, and f must be long enough so
that all of the parallel rays entering the lens fall on the pho-
todiode element, which is positioned behind the focal plane
of the lenses. In the system used for stabilization, the param-
eters are as follows: fϭ175, dϭ6, rϭ12.5 mm, and the pho-
todiode is approximately 20 mm behind the focal plane of
the lenses.
Since the expanded semiconductor laser beam is not spa-
tially uniform, a horizontal masking bar is placed in the
beam path at the same height as the bridge ͑see Fig. 1͒. The
bar has a diameter slightly less than that of the end disks, but
greater than that of the supporting rods. This results in a
much decreased sensitivity of the optical signal difference to
the bridge length.
1. Method 1: Simple feedback
The simplest feedback algorithm is VfϭK1Ve , where
K1 is an adjustable positive gain constant. Using rectifiers as
shown in Fig. 4, the high-voltage output Va is split such that
VL and VR are given by Eq. ͑1͒. As previously noted in Sec.
II, the net force on the bridge is proportional to the square of
the electrode potential, so this linear feedback algorithm re-
sults in a nonlinear feedback force.
2. Method 2: Square-root feedback
In this method, VfϭK2Ve͉Ve͉
Ϫ1/2, where K2 is again an
adjustable positive gain constant. Using the high-voltage cir-
cuit in Fig. 4, the electrode potentials are given by Eq. ͑2͒.
Calculation of the square-root is accomplished using loga-
rithmic amplifiers, and therefore the circuit does not behave
ideally near Veϭ0. However, this had no observable effect
on the stabilized bridge.
C. Feedback algorithms
As described in Sec. II, three feedback methods were
used for stabilization. The general form of the implementa-
tion of all three methods is outlined in Fig. 3. The photodi-
ode signals are amplified, and the difference is taken. An
adjustable offset voltage Vo is then added to the difference
signal to compensate for laser beam irregularities and other
optical imperfections, giving the error signal Ve . The error
signal is input to the feedback circuit which generates a
single output voltage Vf that is then input to a high voltage
amplifier. The output of the amplifier Va is then input to a
high-voltage circuit which produces two output voltages VR
and VL on the electrodes.
3. Method 3: Bias potential feedback
In this method, a bias potential is introduced into the
high-voltage circuit as in Fig. 5 such that VLϭVaϩVb and
VRϭVaϪVb . The linear feedback algorithm VfϭK1Ve of
method 1 is used to produce the electrode potentials given by
Eq. ͑3͒.
IV. RESULTS AND DISCUSSION
The high-voltage amplifier ͑Trek model 677B͒ has a
maximum output voltage of Ϯ2 kV and a maximum slew
rate of 15 V/␮s, and has an adjustable current limit in the
range 0–5 mA to avoid catastrophic failure when the bridge
breaks and shorts an electrode to ground. All circuits in the
feedback loop use analog components to keep the delay time
␶ as small as possible, though in principle a fast digital sys-
As discussed in previous sections, bridge stabilization is
accomplished by sensing when the bridge is becoming de-
formed with a L–R asymmetry and then applying voltages to
the ring electrodes to counter the growth of the deformation.
The three different algorithms for determining the electrode
potentials which were discussed previously were each tested
and found capable of stabilizing a bridge well beyond the RP
FIG. 3. Feedback block diagram.
FIG. 5. High-voltage circuit using bias potentials.

990
Phys. Fluids, Vol. 12, No. 5, May 2000
Marr-Lyon et al.
FIG. 6. The dashed curve shows the quadratic dependence of surface free
energy with the ͑2, 0͒ mode amplitude x, while the dotted curve shows the
cubic dependence on x of the potential energy associated with the feedback
force. The solid line is the sum of the two, with an unstable equilibrium at
xϭ0, and two stable equilibria having xÞ0, as shown by the images of a
bridge of Sϭ3.5 near each stable equilibrium.
limit. The results of the tests on each of the three methods
are described here with a discussion of the relative advan-
tages at the end. Unless noted otherwise, the volume of the
bridge is VϭVcylϭ␲R2L.
FIG. 7. A time sequence showing a bridge with slenderness 4.48 actively
stabilized using the square-root feedback algorithm. Frames ͑a͒–͑c͒ show
the bridge being stabilized for about 5 min. In frame ͑c͒ the feedback control
is turned off. Frames ͑d͒ through ͑f͒ show the subsequent growth of the ͑2,
0͒ mode and bridge breakage which occurs less than a second after control
is turned off.
A. Method 1: Simple feedback
In this method, when the bridge begins to deform by one
end getting thin and the other end rotund, a voltage propor-
tional to the amplitude of the deformation is applied to the
electrode near the thin end of the bridge. It was found to be
possible to stabilize a bridge to a slenderness value of 4.3
with this method, although it was nearly impossible to main-
tain a cylindrical shape. As shown in the images of Fig. 6,
the static bridge shape was deformed, with one end rotund
and the other thin. By changing the offset voltage in the
feedback algorithm it was possible to cause the rotund part
of the bridge to move to the opposite side but it was very
difficult if not impossible to tune the offset voltage in such a
way that the bridge was cylindrical. This can be understood
by considering the potential energy of the system. For
bridges longer than Sϭ␲, the surface free energy contribu-
tion of the ͑2, 0͒ bridge mode grows negative in proportion
to the square of the deformation amplitude. The work done
by the electrostatic stress on the bridge as it goes from a
cylindrical to a deformed shape is proportional to the cube of
the absolute deformation amplitude. The potential energy of
the system is then the sum of the surface free energy and the
work done by the feedback force which, as shown in Fig. 6,
gives a double-well potential. Thus one would expect two
stable equilibrium shapes which have mirror-opposite defor-
mations while the cylindrical state is an unstable equilibrium.
bridge which begins to get thin; however, in this case, the
voltage is applied in proportion to the square-root of the
deformation amplitude instead of in direct proportion. This
overcomes the difficulty experienced with the first method,
which gave stable equilibria which were not a cylindrical
shape. With the proper feedback gain the potential energy of
the system in this case has a single stable equilibrium point
corresponding to a cylindrical shape. This is because both
the surface free energy and the work done by the electric
field are quadratic in the deformation amplitude. The experi-
mental tests of this method confirm that the cylindrical state
is stable. Using this method, bridges with near-cylindrical
volume could be extended to slenderness values near the
theoretical cylindrical-bridge limit S3Ϸ4.49. In approaching
this limit, the volume of liquid in the bridge became critical
to its stability. The bridge deployment system described in
Sec. III A was not capable of reproducibly creating bridges
of precisely cylindrical volume, with errors of 1%–3% being
typical. Accurate measurements of bridge volumes were not
made in real time. The equilibrium shapes of bridges with a
slenderness near S3 take on ͑3, 0͒ mode characteristics if the
volume is slightly greater or less than that of the cylinder.
Bridges with just a few percent greater volume than the cyl-
inder have a bulge in the center and could be stabilized even
beyond the S3 limit for the cylinder. A stabilized bridge with
a volume 2.8% greater than that of a cylinder and with a
slenderness of 4.48 is shown in Fig. 7͑a͒. Bridge volume was
measured by integrating ␲r2 over the length of the bridge
B. Method 2: Square-root feedback
The second method tried is similar to the first method in
that a voltage is applied to the electrode near the end of the

Phys. Fluids, Vol. 12, No. 5, May 2000
Stabilization of electrically conducting capillary bridges . . .
991
bridge volume using digital image analysis which gave a
volume 2% greater than that of a cylinder. The bias potential
on the ring electrodes is thus seen to make the bridge look
more cylindrical even though the volume is greater than the
cylindrical volume. On the other hand, when the volume is
that of a cylinder, the bias voltage tends to give a static shape
which bulges beneath the electrodes and is thin in the center.
This leads to pinch off of the bridge in the center before
reaching the slenderness limit of S3 .
D. Discussion of results
All three methods of stabilization which were tested
proved capable of stabilizing bridges having a volume ␲R2L
well beyond the Rayleigh–Plateau limit. The simplest feed-
back method ͑where the electrode voltages are proportional
to the error signal͒ has the disadvantage that it gives a static
͑2, 0͒ shape during stabilization. The ultimate slenderness
achieved with this method was significantly less than the
theoretical maximum of S3 .
The square-root method has the advantage over the other
two methods that it does not affect the static shape of the
bridge; this generally makes for a more stable situation. The
noncylindrical equilibrium shapes that are seen if the volume
is not exactly that of a cylinder are those expected from
theory as discussed below.
The main drawback of the bias-potential method is that
for a bridge with cylindrical volume, the bias potentials on
the rings cause a noncylindrical equilibrium shape which is
thin in the center, causing a reduction in stability. On the
other hand, for bridges with volumes a few percent greater
than cylindrical, the bias potentials on the rings cause the
bridge shape to be more cylindrical than it would otherwise
be. The problems caused by the bias potential could be re-
duced by reducing the magnitude of the bias potential; how-
ever, the minimum level of bias is dictated by the minimum
gain needed for stabilization.
FIG. 8. A time sequence showing a bridge with slenderness 4.4 actively
stabilized using the bias-potential method where Vbϭ300 V. Frames ͑a͒–͑c͒
show the bridge being stabilized for over 5 min. In frame ͑c͒ the feedback
control is turned off. Frames ͑d͒ through ͑f͒ show the subsequent growth of
the ͑2, 0͒ mode and bridge breakage which occurs less than 1 sec after
control is turned off.
where the bridge radius profile, r(z), was determined by
digital image analysis. The image sequence in Fig. 7 shows a
bridge that was stabilized for about 5 min, at which time the
feedback control was turned off. Control was turned off at
time tϭ0 s shown in Fig. 7͑c͒, and Figs. 7͑d͒–͑f͒ show the
subsequent growth of the ͑2, 0͒ mode and ultimate breakup
of the bridge which occurs less than a second after turning
off the control.
While it was not the emphasis of our observations, it is
noteworthy that the conical shapes visible in Figs. 7͑f͒ and
8͑f͒ are reminiscent of shapes associated with jet
breakup.12,13,21
C. Method 3: Bias potential feedback
Applying a bias voltage to the ring electrodes as de-
scribed in Sec. II is another way to avoid the problem of a
double-well potential experienced with method 1. Figure 8 is
a sequence showing a bridge of slenderness 4.4 being stabi-
lized with a bias voltage Vbϭ300 V. The bridge was stabi-
lized for more than 5 min before the control was turned off.
The control and bias voltages are turned off in frame ͑c͒ at
time tϭ0 s and frames ͑d͒–͑f͒ show the subsequent growth
of the ͑2, 0͒ mode and bridge breakage. It is interesting to
note that shortly after control was turned off, in frame ͑d͒ the
bridge relaxes to a near-equilibrium ͑3, 0͒ shape such as that
seen during active control with the square-root method. This
is because the relaxation time of the stable ͑3, 0͒ mode is
shorter than the growth time of the unstable ͑2, 0͒ mode
which ultimately leads to breakup. The ͑3, 0͒ shape with a
bulge in the center seen in Fig. 8͑d͒ indicates that the volume
of this bridge was somewhat greater than that of a cylinder.
This was confirmed by an independent measurement of
E. Theoretical bridge equilibria and comparison with
observation
The equilibrium shapes seen for stabilized bridges near
S3 can be better understood by looking at the stability dia-
gram in Fig. 9 calculated using the methods of Lowry and
Steen.22–24 In this graph, the stability of bridges is plotted as
a function of the normalized bridge volume and slenderness.
In the shaded areas, bridges are only unstable to the ͑2, 0͒
mode, and in principle may be stabilized by the methods
described. In the unshaded region, the ͑3, 0͒ mode is also
unstable and the bridge will break. This graph is only valid
for the square-root feedback method, since the bias potential
method introduces additional forces which alter the equilib-
rium bridge shapes and would be expected to shift stability
boundaries.
Representative bridge shapes are shown for each of the
stabilized regions. In the small darker-shaded area with V

992
Phys. Fluids, Vol. 12, No. 5, May 2000
Marr-Lyon et al.
FIG. 9. A stability plot of normalized volume V/Vcyl vs slenderness S for
near-cylindrical capillary bridges near the critical slenderness S3Ϸ4.4934.
In the shaded regions, only the ͑2, 0͒ mode is unstable, and the bridge may
be stabilized by the method described. In the unshaded region, the ͑3, 0͒
mode is unstable, and the stabilized bridge will break.
FIG. 11. ͑a͒ Photograph of a bridge with Sϭ4.1 which is actively stabilized
using the square-root algorithm. The volume was measured to be 2.5% less
than that of a cylinder. ͑b͒ Computer generated image of a bridge shape
calculated from the Young–Laplace equation assuming no external forces.
The calculation was performed for a bridge with Sϭ4.1 and a volume which
is 3.2% less than that of a cylinder.
ϽVcyl and SϽS3 , the shape is rotund at the ends, and thin in
the middle. Along the VϭVcyl line for SϽS3 , the bridge
shape is a cylinder. In the lighter-shaded area, the bridge is
rotund in the middle, and thin on the ends. Note that on the
VϭVcyl line for SϾS3 the stabilized shape is not a cylinder,
even though the bridge has cylindrical volume.
Near S3 , the slenderness limit is strongly dependent on
bridge volume. A deficiency in bridge volume of less than
0.4% from cylindrical causes the maximum slenderness to
decrease by 2%, while it is predicted that a slightly overin-
flated bridge can be stabilized to SϾ4.55.
Comparisons between the theoretical shapes obtained as
above and the observed shapes of bridges stabilized using the
square-root method are shown in Figs. 10 and 11. The vol-
umes of the bridges for the theoretical shape calculations
were chosen to give a good qualitative shape comparison
with the experimental images. Independent measurement of
the bridge volumes from the corresponding experimental im-
ages gave volumes very close to those chosen for the theo-
retical calculations. The experimentally observed shapes
seen in Figs. 10͑a͒ and 11͑a͒ are compared to theoretical
shapes which are solutions to the Young–Laplace equation
with no external force fields as shown in Figs. 10͑b͒ and
11͑b͒. The theoretical shapes are shown as computer-
generated ray-traced images for a qualitative comparison
with the experimental images. This comparison suggests that
the static shape of the bridge is not influenced by the control
system. The control system simply prevents the growth of
the lowest-order unstable mode, allowing equilibrium shapes
which are normally unstable to be seen. As expected, bridges
with volumes less than that of the cylinder are found to be
more unstable, tending to pinch off in the center before
reaching a slenderness of S3 .
V. RELATED APPLICATIONS
While the research results described concern the active
stabilization of the ͑2, 0͒ mode of electrically conducting
bridges in ͑simulated͒ zero gravity, there are related note-
worthy applications. With a modified sensor and electrode
array, it is plausible that both the ͑2, 0͒ and ͑3, 0͒ modes
could be controlled. The control method described should
also be applicable to the suppression of the RP instability of
a dielectric liquid layer which coats a grounded conducting
cylinder. For the situation in which there are two electrodes,
the liquid dielectric will have the greatest attraction to the
electrode where the potential has the largest magnitude, as
with the electrically conducting bridges studied. For electri-
cally conducting bridges having SϽ␲, axisymmetric modes
of the bridge were excited with an appropriate modulation of
the electrode potentials. ͑That demonstration is analogous to
the excitation of modes with modulated acoustic radiation
pressure.9͒ Observations ͑not presented here͒ also show that
it is possible to modify the static profile of a vertical electri-
cally conducting bridge in air with radial electric fields even
for non-negligible Bond numbers. For this application, how-
ever, the fields required to counteract gravity may be large,
as is evident by the following estimate. Consider a vertical
FIG. 10. ͑a͒ Photograph of a bridge with Sϭ4.48 which is actively stabi-
lized using the square-root algorithm. The volume was measured to be 2.8%
greater than that of a cylinder. ͑b͒ Computer generated image of a bridge
shape calculated from the Young–Laplace equation assuming no external
forces. The calculation was performed for a bridge with Sϭ4.48 and a
volume which is 2.9% greater than that of a cylinder.

Phys. Fluids, Vol. 12, No. 5, May 2000
Stabilization of electrically conducting capillary bridges . . .
993
conducting bridge surrounded by a gas in an effective gravity
of ge . The magnitude of the hydrostatic pressure differential
for a bridge of length L and density ␳ is ␳geL. If a radial
field E is applied to the upper portion of the bridge to coun-
teract the bulge of the lower portion of the bridge, the Max-
well stress is radially outward with a magnitude of25
(1/2)⑀0E2, where ⑀0Ϸ8.85ϫ10Ϫ12 F/m is the permittivity of
free space. Though the magnitude of the applied stress re-
quired to suppress a bulge of a given size is affected by
surface tension, for the purpose of estimating the magnitude
of E, take (1/2)⑀0E2ϭ␳geL; with ␳ϭ1 g/cm3, Lϭ1 cm,
and a reduced gravity of geϭ29 cm/s2ϭ30 mg, this estimate
gives EϷ8.1ϫ105 V/m. The electrode potentials required
depend on the electrode dimensions. The gap between the
electrode and the bridge may not be small so as to avoid a
lateral instability of a liquid column described by Taylor.26
ACKNOWLEDGMENT
This work was supported by NASA.
APPENDIX: ELECTRODES AND BRIDGE ELECTRICAL
CONDUCTIVITY
FIG. 12. Equipotential contours of the axisymmetric electric field surround-
ing the bridge. ͑a͒ Analytical result for the simplified geometry of an infinite
grounded cylinder and concentric ring of charge. For ease of comparison
with ͑b͒, the plane of the electrode is offset to z/RϭϪ2.69. ͑b͒ Finite
element result for a more realistic geometry with a bridge of slenderness
Sϭ4.0 and with the dimensions of the electrodes, support disks, and posts
corresponding to the experimental situation. The center of the bridge corre-
sponds to zϭ0.
The selection of the radius and thickness of the conduct-
ing rings used as the electrodes was guided by the following
calculation. Consider the electrostatic potential ␾(z,r) for a
uniformly distributed ring of charge of radius aϾR which is
coaxial with an infinitely long grounded conducting cylinder
of radius R. The ring lies in the plane zϭ0. The solution for
this simplified potential distribution is useful for estimating
the field distribution near the bridge under conditions which
apply to the actual system under consideration. Some of the
simplifications involved include neglecting the perturbation
of the field introduced by the second electrode and the re-
quirement that an equipotential surface for the simplified
problem closely approximate the surface of the conducting
electrode. In addition, the perturbation of the field associated
with the wire which supports the electrode ͑and connects it
to the circuits described in Sec. III͒ is neglected, as are the
perturbations resulting from deformation of the bridge. The
potential for the simplified geometry is proportional to the
Green function
where rϾ(rϽ) is the greater ͑lesser͒ of r and a, B(kz)
ϭI0(kzR)/K0(kzR), and G is given by
ϱ
1
z
G r,z͒ϭ
͑
g r,k ͒eik zdkz .
͑A3b͒
͑
͵
z
2
4␲
Ϫϱ
The electric field at the surface of the circular cylinder is
proportional to ϪץG/ץr evaluated at rϭR and is given by
ϱ
E z͒ϭA
k cos k z B k ͒KЈ k R͒ϪIЈ k R͒
͓ ͑ ͔
͑
͑
͑
͵
r
z
z
z
z
z
0
0
0
ϫK0 k a͒dk ,
͑A4͒
͑
z
z
4␲
ٌ2G r͒ϭϪ
␦ rϪa͒␦ z͒,
͑A1͒
͑
͑
͑
where A is a constant. The relevance of these results to the
electrode configuration used is evident by inspection of Fig.
12͑a͒, which shows equipotential contours for the situation
where the radius a of the charged ring is 2.87R. Inspection
of the contours shows them to be nearly circular in the region
close to the ring. Consequently, the potential distribution
closely approximates that of a conducting torus provided the
cross-sectional area of the torus is sufficiently small. The
ratio a/RϷ2.87 was selected by evaluating the distribution
of the Maxwell stress on the surface of a cylindrical bridge
normalized to the peak value ͑which occurs in the plane of
the ring which is at zϭ0͒. Figure 13 shows this normalized
stress E2r (z)/E2r (0) which from Eq. ͑A4͒ may be expressed
as
r
where r denotes the radius in cylindrical coordinates. The
boundary condition is that Gϭ0 at rϭR for all z. This is
solved using standard potential theory27 by using a Fourier
transform representation for the z dependence of both G and
␦(z), which leads to the following condition on g(r,kz), the
Fourier transform of G:
d
dg
r
Ϫrk2gϭϪ4␲␦ rϪa͒.
͑A2͒
͑
ͩ ͪ
z
dr
dr
The solution is
g r,k ͒ϭϪ4␲ B k ͒K k r ͒ϪI k r ͒ K k r ͒,
͑
͓ ͑
͑
͑
͔
͑
z
z
0
z
Ͻ
0
z
Ͻ
0 z
Ͼ
͑A3a͒

994
Phys. Fluids, Vol. 12, No. 5, May 2000
Marr-Lyon et al.
given the same potential as the electrode potential in Fig.
12͑b͒. The FEM results indicate that the second electrode
being held at ground potential shields the part of the bridge
beneath it. Aside from this, the stress distribution computed
by the two methods is very similar in the region where the
stress is large.
While this analysis assumes that the bridge is sufficiently
electrically conducting to remain at zero potential, the re-
quirement on the conductivity of the liquid is not very re-
strictive. The requirement on the conductivity ␴ of the
e
bridge liquid may be estimated as follows. The resistance
from the closest end to the adjacent displacement antinode is
roughly ReϷL/(␴e4␲R2), while the actual resistance will
be somewhat lower because of current flow from the support
at the opposite side of the bridge. The electrical time con-
stant of the electrode-bridge system may be approximated as
␶eϭReCe , where Ce is the electrode capacitance with re-
spect to the bridge. Taking ␶eՇ␶, the time constant in Eq.
FIG. 13. A plot of the normalized stress distribution for the following nor-
malized ring radii a/R:2.0, 2.5, 2.87, 3.25, and 3.75.
E2r z͒
2
ϱ
͑
␤z K0 ␤a/R͒
͑
ϭ
cos
d␤
͵
͵
ͩ ͪ
ͫ
ͬ
Ͳ
E2r 0͒
ͭ
ͭ
ͮ
R
K0 ␤͒
͑
͑
0
͑5͒, and Lϭ2SR ͑where S is the slenderness͒ gives ␴
e
2
ϱ
տSCe/2␲R␶. For example, typically CeϷ200 pF,
R
K0 ␤a/R͒
͑
d␤
,
͑A5͒
ͫ
ͬ
ͮ
Ϸ2 mm, Sϭ4, and ␶Ϸ0.1 ms gives ␴eտ10 ␮S/cm, whereas
the salt solution used had ␴eϾ20 mS/cm. For comparison,
molten and hot solid silicon have28 ␴eտ10 000 S/cm.
K0 ␤͒
͑
0
where ␤ϭk R and the expression W K (␤),I (␤) ϭ1/␤
͓
͔
z
0
0
has been used where W denotes the Wronskian. The final
result for Er(z) was confirmed using a finite element method
͑FEM͒ to evaluate the potential and the associated field. As
the ratio a/R approaches unity, the ratio in Eq. ͑A5͒ becomes
narrowly peaked around zϭ0 and if a is taken to be much
larger than R, the stress distribution is overly broad so that
both the left and right sides of the bridge are stressed. To
optimize the ‘‘footprint’’ of the stress, the following Fourier
coefficient was evaluated;
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2
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͑1997͒.
3␭/4
1
2␲z E z͒
͑
r
Cfϭ
cos
dz,
͑A6͒
͵
ͩ ͪ
ͫ
ͬ
R
␭
E 0͒
͑
r
Ϫ␭/4
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for ␭ϭ2␲R with a range of a/R used in the evaluation of
Eq. ͑A5͒. This coefficient is a useful measure of the coupling
to the ͑2, 0͒ mode at the critical wavelength ␭ϭ2␲R which
naturally becomes unstable when Sϭ␲. This is because
cos(2␲z/␭) approximates the shape of the mode in the limit
of infinitesimal deformation.3 Numerical evaluation shows
the Fourier coefficient Cf has a broad maximum centered on
aϭ2.87R.
8M. J. Marr-Lyon, D. B. Thiessen, and P. L. Marston, ‘‘Stabilization of a
cylindrical capillary bridge far beyond the Rayleigh-Plateau limit using
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͑1997͒.
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͑1996͒.
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104, 47 ͑1990͒.
The stress distribution on the bridge predicted by the
analytical solution for the simplified geometry described
above was checked using a FEM to compute the field for a
more realistic geometry. A finite element solution was deter-
mined for a geometry which included two wire electrodes,
one of which has a nonzero potential and the second being
held at ground potential. The electrode dimensions, including
wire diameter and ring diameter, and the support-post geom-
etry corresponding to the experimental situation for a bridge
of slenderness Sϭ4.0, were used in the computation. Figure
12͑b͒ gives the equipotential contours from the FEM com-
putation for the experimental geometry. Figure 12͑a͒ shows
the analytical results for the potentials using the same ratio
of electrode ring radius to bridge radius. The potentials in
Fig. 12͑a͒ were scaled so that the equipotential contour
which is positioned approximately at the electrode surface is
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bridges,’’ J. Fluid Mech. 121, 25 ͑1990͒.
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Mod. Phys. 69, 865 ͑1997͒.
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͑1999͒.
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bubbles from an underwater nozzle,’’ J. Fluid Mech. 230, 365 ͑1991͒.
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crystals,’’ J. Cryst. Growth 37, 253 ͑1977͒.
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liquid bridges,’’ J. Fluid Mech. 156, 101 ͑1985͒.
19The estimate S3Ϸ3␲/2 noted in Marr-Lyon et al. ͑Ref. 8͒ exceeds the

Phys. Fluids, Vol. 12, No. 5, May 2000
Stabilization of electrically conducting capillary bridges . . .
995
correct value of S3 , although as n→ϱ, for odd n,Sn→n␲/2. See Sanz
͑Ref. 18͒ for the correct S3 .
liquid bridge equilibria—stability of long cylindrical liquid bridges,’’
Phys. Fluids 12, 996 ͑2000͒.
25D. A. Saville, ‘‘Electrohydrodynamics: The Taylor–Melcher leaky dielec-
tric model,’’ Annu. Rev. Phys. Chem. 29, 27 ͑1997͒.
26G. I. Taylor, ‘‘Electrically driven jets,’’ Proc. R. Soc. London, Ser. A 308,
453 ͑1969͒.
27J. D. Jackson, Classical Electrodynamics, 2nd ed. ͑Wiley, New York,
1975͒.
28Y. Masuda, T. Tsukada, M. Hozawa, N. Imaishi, and N. Ohnishi, ‘‘Nu-
merical simulation of heat transfer in floating zone crystal growth process
with radio-frequency induction heating,’’ Int. J. Heat Mass Transf. 39,
3035 ͑1996͒.
20C. R. Fuller, S. J. Elliott, and P. A. Nelson, Active Control of Vibration
͑Academic, San Diego, 1997͒.
21I. Cohen, M. P. Brenner, J. Eggers, and S. R. Nagel, ‘‘Two fluid drop
snap-off problem: Experiments and theory,’’ Phys. Rev. Lett. 83, 1147
͑1999͒.
22B. J. Lowry and P. H. Steen, ‘‘Capillary surfaces: Stability from families
of equilibria with application to the liquid bridge,’’ Proc. R. Soc. London,
Ser. A 449, 411 ͑1995͒.
23B. J. Lowry, ‘‘Shape stability of slender liquid bridges in axial flows,’’
Ph.D. dissertation, Cornell University, Ithaca, NY, 1994.
24B. J. Lowry, ‘‘The double-helical branch structure of fixed contact line