Equilibrium of the floating water bridge: a theoretical and
experimental study
R M Namin1 and S A Lindi2
1
Department of Mechanical Engineering, Sharif University of Technology, Tehran,
Iran.
2
National Organization for Development of Exceptional Talents (NODET),
Farzanegan Highschool, Tehran, Iran.
E-mail: namin@mech.sharif.edu
Abstract. The floating water bridge is the phenomenon of suspension of a water thread
between two beakers with a high voltage applied in between. The equilibrium of this system is
investigated assuming both the surface tension force and the tension caused by the electric field
within the dielectric material. It was shown experimentally that the deflection of the bridge and
its diameter are not independent; this fact was not explained before and in the current paper we
show that the effect of surface tension in equilibrium of the bridge would explain it. The
deflection of the bridge was calculated theoretically and was experimented, and quantitative
agreement was shown while previous theories did not match with the experiments. Also the
equilibrium of the bridge in the longitudinal direction is studied, explaining the main effect of
the electric field and assuming water as a conductive to calculate the electric field. The nonuniform electric filed is suggested to be an important issue in this case. The conditions at the
edges have also been investigated and used to compute the overall shape of the bridge as a
function of the applied voltage and limitations of the voltage have been extracted. It is shown
that this phenomenon can be described by macro-scale investigations regardless of the microscale changes in the water as already suggested in other references.
PACS numbers: 47.65.-d, 47.55.nk, 77.84.Nh
Submission Date: 9 October 2012
Submitted to: Physical Review E
I. INTRODUCTION
A phenomenon known as the floating water bridge was first discovered in 1897 by Sir William George
Armstrong [1] and has caught attention recently. Rediscovered by Fuchs et al 2007 [2]. Two beakers field
with deionized water are subjected to a DC high voltage more than 10kV and a bridge is formed between
them (Fig. 1) which can last for hours and have a length exceeding 2cm [3]. The investigation of this
phenomenon seems interesting for it may reveal some hidden properties of water [4]. This experiment is
stable, easy to reproduce and leads to a special condition that the water in the bridge can be accessed and
experimented under high voltages and different atmospheric conditions [5]. This has lead to several
special experiments in this setup, including Neutron scattering [6, 7], visualization using optical
measurement techniques [3], Raman scattering [8], Brillouin scattering [9] and zero gravity experiments
[10]. Reviews on this topic have already been published such as [4] and [5] which the reader may refer to
for a comprehensive literature review.
Several papers have been published regarding the stability of such a bridge, and discussing the force
holding it against gravity [11-17]; although as we will explain in this paper, none are yet able to correctly
describe the observations. There have been several experiments regarding the density change [2] and fluid
motion in the beakers and also within the bridge [2, 3, 18 and 19].
Generally in explaining this phenomenon, two different approaches are considered [4]: a) treating it the
same as electro spray, assuming the dielectrophoresis (the motion of the dielectric fluid as a result of
polarization effects) responsible for the stability [11-15] and b) investigation of the structural changes
microscopically, which lead to a macroscopic influence on the behaviour of water [16]. For example
changes in the hydrogen bonds [17] and investigation of the coherent and non-coherent domains [16].
FIG. 1. The Floating Water Bridge.
In the current paper the theories regarding the dielectrophoresis and surface tension effects will be
combined to present a better theory concerning the shape and equilibrium of the bridge and comparing
them with experiments, it will be shown that macro-scale explanations will be sufficient in describing the
equilibrium of the bridge. Initially the experiment configuration will be explained in part II, then the
shape of the bridge’s centreline will be discussed in part III, equilibrium of the bridge in the longitudinal
direction will be discussed in part IV leading to equations about its diameter and its derivation in position.
The conditions at the sides of the bridge will be discussed in part V explaining the diameter of the bridge
as a function of other parameters, and calculations regarding the applied voltage will be discussed in part
VI leading to voltage limitations.
II. EXPERIMENT CONFIGURATION
Two 100ml glass beakers were filled with deionized water and the voltage was applied to the water by
two flat copper electrodes with a size of about 10cmx5cm which were placed inside the beakers. The
deionized water is 4 times distilled and initially has a resistivity of about 10MΩ/cm. At first the beakers
were placed closely that the edges were in touch and the electrodes were placed at the distant sides of the
beakers. The high voltage power supply was capable of providing a continuously adaptable potential
difference up to about 20kV. A digital oscilloscope was used for measuring the potential difference and
an ammeter for current intensity measurements. A high resolution camera was placed in front of the
beakers for measurements of the geometry. The captured frames of the bridge were analysed using a
MATLAB image processing code in order to obtain some properties of the bridge such as the average
diameter, amount of deflection in different sections etc.
III. SHAPE OF THE BRIDGE
Initially in the paper the shape of the bridge will be discussed, explaining the forces opposing gravity and
supporting the bridge to float between the beakers.
Fuchs et al 2008 [18] suggests the force holding the water bridge is the Kelvin force because of the
vertical gradients of the electric displacement vector. Following this suggestion, Widom et al 2009[13]
suggest that a tension exists along the bridge caused by the electric field, supported by calculations in
terms of the Maxwell pressure tensor in a dielectric fluid medium. The tension equations have been
derived and shown to be in the same order of magnitude needed to hold the bridge in a catenary shape.
Marin and Lohse 2010 [11] use the same kind of calculations -the tension because of the electric field in
the dielectric material- and in a different controlled experiment setup show the agreement of the
predictions with experiments. Aerov 2011 [14] suggests a different mechanism for the stability of the
water bridge: “the force supporting it is the surface tension of water, while the role of the electric field is
to not allow the water bridge to reduce its surface energy by breaking into separate drops.” And uses
energy calculations to show that surface tension is strong enough to be responsible for this fact [14].
Morawetz 2012 [15] uses the electric tension and adds the effect of surface electric charges to better
describe the stability and also develops a model using the Navier-Stokes equations to find the velocity and
diameter profile along the bridge. The surface charges are suggested to be responsible for asymmetries in
the diameter and shape of the bridge [15]. In calculating the shape, Morawetz [15] states that "the surface
tension is negligible here", the same as [11] and [13]. So there are currently two kinds of calculations
regarding the shape: assuming the tension caused by electric field in the dielectric material to hold the
bridge as done by [11], [13] and [15]; and assuming the surface tension to be responsible for this fact as in
[14].
Both suggested mechanisms of the equilibrium (dielectric tension and surface tension) have lacks in
explaining the shape. Assuming the tension to be the electric one, the deflection of the bridge will not be a
function of its diameter and will only be a function of the applied electric field (as it can be seen in
calculations of [11], [13] and [15] for example that the diameter will not be involved in the final equation
for deflection). However according to our experiments the deflection of the bridge has same fluctuations
as its diameter as shown in Fig. 3, and the deflection and diameter do not seem to be independent. The
relation between the diameter and the parabola coefficient of the centreline has been extracted and shown
in Fig. 4.
On the other hand assuming the bridge to be held by the surface tension only, as suggested by Aerov [14]
will lead to the prediction that its curvature will not be a function of the electric field, and will only be
dependant of the surface tension coefficient and the diameter of the bridge. Experimental results by Marin
and Lohse [11] show this false as the deflection is shown to have an inverse relation with the applied
voltage.
The theory suggested here will be a combination of the two theories. It will be assumed that the tension
along the bridge is a sum of the tensions caused by electric field in dielectric and the surface tension. We
show that the surface tension and dielectric tensions have the same order of effects and none of the two
are negligible. To find the tension caused by the electric field, as done by Marin and Lohse [11] the
pressure jump across the interface is calculated according to balance of the normal stresses in the liquid
bridge as in [20], and the tension is assumed to be this pressure multiplied by the cross-sectional area.
This assumption has already been made by [11], [13] and [15]. The tension caused by the surface tension
is the surface tension coefficient multiplied by the cross-sectional perimeter.
T =(
πD 2 1
)[ ε 0 (ε − 1) E 2 ] + γπD
4 2
(1)
Where T is the overall tension along the bridge, D is the diameter, ε0 is the vacuum permittivity, ε is the
relative permittivity of the liquid, E is the electric field strength and γ is the surface tension coefficient.
Assuming a curved shape with this tension and writing the vertical equilibrium in the middle of the
bridge, the radius of curvature will be calculated and assuming the shape as a parabola, the following
equation for the parabola coefficient a will be derived:
a=
ρg
(2)
1
ε 0 (ε − 1) E 2 + 8γ / D
2
Where g is the gravity acceleration and ρ is the mass density of the liquid.
To check this experimentally, the bridge was detected and averaging the upper and lower curves of the
bridge the centreline was estimated. Fitting a parabola to this curve, the coefficient was extracted. Also
the average of diameter in the length of the bridge was measured. This was done for several frames and
results were obtained in terms of time. Fig. 3 shows the fluctuations of both parameters. To compare with
the theory, the voltage drop along the bridge was estimated (as explained later in section 6) and using the
length, electric field was calculated. Results are presented in Fig. 4.
FIG. 2. An Analyzed Bridge.
FIG. 3. Average Diameter of the Bridge and its Parabola Coefficient a in terms of Time. Similar
fluctuations show that they are not independent.
FIG. 4.Measurements and Different Theories regarding the Parabola Coefficient (a) as a Function of the
Bridge Diameter. Agreement of the measurements with the present theory is observed, and theories
suggested before are also plotted. Length of the bridge was 1cm and the voltage difference was estimated
to be 8.5kV according to section 6.
Curves for the prediction of the theory of electric tension and surface tension have been plotted
individually in Fig. 4. It can be deduced that surface tension and electric tensions have the same order of
effects on the vertical equilibrium in the common diameter ranges of the experiments, and non of the two
are negligible. The results show an acceptable agreement between the experiments and the theory
suggested here. It can be seen that as the diameter increases, the experiments will no longer follow the
theoretical trend. This may be an experimental error caused by the transient fluctuations, since it takes
time for the diameter to increase as the deflection increases. It may also be caused by the fluid velocity
within the bridge. With the increase of diameter, the velocity magnitude increases in the bridge which
causes a vertical acceleration in the curved shape.
This theory also applies to the axisymmetric configuration by Marin and Lohse [11] and our theory results
match their experiments (Fig. 8 in [11]). So we suggest that the effect of surface tension is not negligible,
however [11] has matched the theory with experiments neglecting surface tension. We suggest this is
caused by a calculation error as described in appendix A.
IV. LONGITUDINAL EQUILIBRIUM
According to the previous section, the bridge will be curved downwards so that the tensions will exert an
upward force opposing gravity. This curvature will make the bridge sloped to the sides, so the gravity will
have a longitudinal component towards the middle. We will derive equations regarding the equilibrium in
the longitudinal direction of the bridge, leading to equations of the diameter.
The force exerted to a dielectric in a non-uniform electric field has already been explained e.g. [20, 21].
There is a body force known as the Kelvin force density to the direction with higher electric field
strength.
1
FD = ε 0 (ε − 1)∇E 2 (4)
2
To calculate the electric field, the bridge can be assumed as an Ohmic conductive with a high resistivity
(ρs). In this case, the electric field is proportional to the electric current density. So assuming a current
intensity of I along the bridge in a section of area A; the electric field would be:
I
E = ρs
(5)
A
Showing that the electric field will be stronger in a point with a smaller cross-sectional area. This can lead
to an explanation of the longitudinal equilibrium: The sides of the bridge have a smaller diameter
compared to the diameter in the middle. This causes a body force on the water toward the sides where a
higher electric field exists; cancelling the gravity component.
Note that this effect seems to be the most important act of the electric potential. Not only it avoids the
bridge to break into droplets (as suggested by Aerov[14]), but it also is directly responsible for the
equilibrium in the longitudinal direction. With the same mechanism, the electric field is also responsible
for the rise of the water columns at the sides in the beakers (Section 5).
This explanation also describes the stability by the fact that; if a section has an area lower than the one
predicted by equilibrium equations in a moment, it will have higher electric field strength, attracting water
to this section and this will lead to an increase in its area. And if it has a larger area, water will be repelled
reducing the area. This stability mechanism explains the fact that the bridge can gain a large slenderness
ratio (ratio between length and diameter) and not break into droplets.
Three body forces are present in the longitudinal direction. The dielectric force which is the Kelvin force
density (FD) as described, the gravity component (FG), and the capillary force (FC) caused by the
derivation of the surface tension pressure along the bridge. Using equations (4) and (5) the forces will be
derived:
∂A
FD = ρ s 2 I 2ε 0 (ε − 1) A −3
(6)
∂x
FG = − ρg sin(θ )
π
(7)
∂A
(8)
2
∂x
Assuming the sum of these forces to be zero and using the shape described in part 3 to calculate the slope,
a differential equation will be extracted regarding A and its first derivation with respect to x. Solving that
numerically, we will obtain the area as a function of position in a known current intensity and boundary
conditions. Some results are presented in Fig. 5.
FC =
γA −3 / 2
FIG. 5.Theoretical Results A) Diameter along the bridge in a constant current intensity of I = 0.5mA,
there is a maximum diameter for the equations to have finite results. B) Different current intensities in
mA. As the current increases diameter changes decrease.
One of the predictions of the theory was that a maximum exists for the diameter at the boundaries in a
given voltage. If the diameter exceeds this minimum, equations will not have a finite solution.
V. The Sides of the Bridge
According to the theories described above, the shape of the bridge and its diameter as a function of
position could be calculated knowing the diameter at an end. Therefore we will discuss the conditions at
the ends. At the beaker walls it is observed that a column of water is formed vertically ending to the
bridge at the top. To find the conditions of the bridge at the sides, we will describe this rising mechanism
and the diameter at its end.
There have been some discussions in the literature about the diameter of the bridge. The theory developed
by Morawetz 2012 [15] leads to the result that “The radius of the bridge at the beaker is nearly
independent on the applied electric field but only dependent on the surface tension and gravitational
force” [15]. However the experiments show an evident increase of the diameter at the edges by an
increase in the applied voltage as shown in Fig. 6.
FIG. 6.The shape of the bridge in different voltages. The diameter at the sides seems to be a function of
the applied voltage.
According to the theory suggested in this paper, the forces acting here are the same as the forces
described for the bridge its self. There is a gravitational body force which is opposed by the dielectric
force. The dielectric force again is originated by the non-uniform field caused by the cross-sectional area
changes. The bridge is thinner at the top so higher current density and electric field exists at the top. To
find the area of this vertical column, we assume the Kelvin force density to be equal to ρg:
ρg = − ρ s 2 I 2ε 0 (ε − 1) A −3
∂A
∂y
(9)
The solution to this differential equation for the boundary of infinite area at zero elevation is:
1
(10)
A=
2 Ky
With
K=
ρg
ρ s I ε 0 (ε − 1)
2 2
(11)
This shows an inverse relation between the area and the level from the free surface. It also suggests a
direct linear relation between the area and the current intensity which is also observed in the experiments
(Fig. 7). The disagreement in low current intensities (small diameters) may be caused by the neglected
capillary effects. Also the assumption that the current is passing uniformly in the cross-section may not be
accurate enough in the lower parts of the vertical bridge.
FIG. 7.Linear Relation between the Diameter and Current Intensity. The line shows a linear fit with zero
intercept.
VI. VOLTAGE LIMITATION FOR EQUILIBRIUM
Until here all the calculation was based on the electric current intensity I. Now we may use I to find the
bridge shape, and calculate its resistance to find the voltage drop within the bridge.
dx
(12)
A
And the voltage difference between the two electrodes can be calculated:
R=∫
l/2
−l / 2
ρs
V = I ( R + Rb ) (13)
Where Rb is the resistance of the water between the electrodes and the bridge sides (inside the beakers). It
is evident from equations (10) and (11) that the area of the bridge at the sides is proportional to I in a
given y. If we assume the bridge as a cylinder to calculate its resistance, we will see that the voltage
difference across the bridge is not a function of the current intensity. When increasing the current
intensity, the bridge gets thicker, resistivity drops and voltage remains constant. This means that by
changing the voltage applied between the electrodes, we’re actually changing the voltage drop in the
beaker and not along the bridge. This explains the relation between I and V, the voltage when current
intensity approaches to zero is the constant voltage drop along the bridge, and the linear increase is for the
resistance in the beakers as shown experimentally in Fig. 8. So there is a minimum voltage between the
electrodes, and that is when the current intensity approaches to zero and is equal to the voltage drop in the
bridge. This minimum voltage is a function of the distance between the beakers and the height of the
vertical column at the beaker walls:
V = l ⋅[
2 ρgy 1/ 2
]
ε 0 (ε − 1)
(14)
FIG. 8. The Relation between the Current Intensity and the Applied Voltage between the Electrodes. The
line shows a linear fit.
This fact may be explained differently. One may assume the voltage to be constant, and this equation
leads to a maximum length of the bridge as a function of the applied voltage. Note that according to (14),
there must be a linear relation between the minimum voltage and the length of the bridge. This limitation
has been measured experimentally in Fig. 9 and drawn theoretically for the height of the beaker side as
5mm. y was assumed to be the height plus the radius of the bridge at the side, and an equation has been
solved to find theoretical results. Note that the theoretical limitation is only for having equilibrium and
this is not enough for the phenomenon to be observed since stability may demand further conditions. This
might be the reason for the mismatch seen in Fig. 9.
FIG. 9. Minimum voltage observed in different lengths and the equilibrium limitation predicted by the
theory for h = 5mm.
Also individual effects of resistivity and current intensity are of interest. It is evident in all the formulas
that ρ and I are never involved independently but it’s the product of ρ and I that has always been involved
i.e. in a distinct applied voltage changing the resistivity causes a change in the current intensity and as a
result their product remains constant. This fact yields to the questionable result that the resistivity does
not influence the stability and shape of the bridge; however it is observed and reported that this
phenomenon cannot be reproduced in ionized water which has a low resistivity [15].
Two explanations are suggested here. One may be the limitation of the power supply, as the experiments
have been performed with high-voltage power supplies supporting low current intensities. This means that
by decreasing resistivity (as in ionized water) current intensity rises and voltage drops. So although the
phenomenon is not a function of the resistivity in a constant voltage, resistivity affects the voltage.
Another explanation might be the electric heat production. When the resistivity drops, current intensity
increases causing an increase in the heating power. It was observed that by decreasing the resistivity the
temperature of the bridge rises, and sometimes large bubbles were observed which seemed to be caused
by the water boiling in the areas with higher current densities (Fig. 10). This increase in temperature and
possible boiling may have effects such as decreasing the surface tension and dynamical effects which
might reduce the stability.
FIG. 10.Large Bubbles in the Bridge.1/30 Second time intervals.
VII. CONCLUDING REMARKS
A consistent theory has been suggested which can describe the equilibrium of the floating water bridge
regarding the macro-scale electrohydrodynamical and surface tension forces. The curvature of the bridge
was suggested to be a function of the surface tension and the tension caused by electric field within the
dielectric material, and this suggestion was compared with experiments showing acceptable results. The
effect of the electric field was most importantly in the longitudinal equilibrium along the bridge and in the
vertical sides at the beaker. The increase of diameter with voltage was explained and a voltage limitation
was extracted theoretically, however practically a higher voltage limit was observed. This may be for the
approximations in calculating the vertical column.
Further calculations regarding the stability of the bridge are suggested to be done. It may also be possible
to simulate the phenomenon knowing the electro-chemical and electrical equations and equations of fluid
dynamics in terms of computational fluid dynamics methods. This may lead to the explanation of the
complicated motion of water in the bridge and beakers during and after the formation phase which may be
influenced by the advection and diffusion of charges in the flow and the fluid motion derived by the
charges.
It has been shown that the theory regarding the electrohydrodynamics and surface tension forces is able to
explain the equilibrium of the bridge in this phenomenon and the structural changes in water are not
necessary in explaining the equilibrium.
ACKNOWLEDGEMENTS
The authors would like to thank the International Young Physicists’ Tournament community, participants
and jurors for the introduction to this phenomenon and presentations and discussions during IYPT 2012,
Bad Saulgau - Germany.
APPENDIX A
Marin and Lohse 2010 [11] have neglected the surface tension in calculating the shape of the bridge and
matched experiments with their theory. However according to our study, tension caused by surface
tension along the bridge is not negligible in calculating its shape. We suggest that there was a calculation
error in [11].
According to equation (3) in [11] and by the simplifications explained there, they would come to this
equation:
1
πd2
T = ε 0ε r E 2 (
)
2
4
(A1)
Where T is the tension along the bridge. And equation (4) in [11] states:
ρVg = 2T sin(θ )
(A2)
Where V is the volume of the bridge, which will be:
V = (π d 2 / 4) L
(A3)
Placing (A3) and (A1) in (A2) will lead to:
sin(θ ) =
Lρ g
ε 0ε r E 2
(A4)
They claim in [11] that rearranging the terms using the dimensionless numbers leads to the following
expression:
sin(θ ) =
GE Λ 2 1/ 3
( )
2 2π
(A5)
Where Ge and Λ are defined as follows:
GE =
ρ gV 1/ 3
(A6)
ε 0ε r E 2
Λ = L/d
(A7)
However placing (A7) and (A6) in (A5) leads to this result:
sin(θ ) =
Lρ g
4ε 0ε r E 2
(A8)
Which is four times smaller than what should have been resulted as (A4). So it seems that equation (5) in
[11] is not correct, and the comparison made with experiments is not valid. Adding the surface tension
term will make an acceptable comparison there.
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