Equilibrium of the floating water bridge: a theoretical and experimental study

R M Namin1 and S A Lindi2 1Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran. 2National 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 non-uniform 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 micro-scale 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.

DEDT γπεεπ +−= ])1(21)[

4( 2

0

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:

DE

ga/8)1(

21 2

0 γεε

ρ

+−= (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.

20 )1(

21 EFD ∇−= εε (4)

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:

AIE sρ= (5)

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:

xAAIF sD ∂

∂−= −30

22 )1(εερ (6)

)sin(θρgFG −= (7)

xAAFC ∂

∂= − 2/3

2γπ (8)

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.

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:

yAAIg s ∂

∂−−= −30

22 )1(εερρ (9)

The solution to this differential equation for the boundary of infinite area at zero elevation is:

KyA

21= (10)

With

)1(022 −

=εερ

ρI

gKs

(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.

∫−= 2/2/

ll s A

dxR ρ (12)

And the voltage difference between the two electrodes can be calculated:

)( bRRIV += (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:

2/1

0]

)1(2[

−⋅=

εερgylV (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:

22

01 ( )2 4r

dT E πε ε= (A1)

Where T is the tension along the bridge. And equation (4) in [11] states:

2 sin( )Vg Tρ θ= (A2) Where V is the volume of the bridge, which will be:

2( / 4)V d Lπ= (A3) Placing (A3) and (A1) in (A2) will lead to:

20

sin( )r

L gE

ρθε ε

= (A4)

They claim in [11] that rearranging the terms using the dimensionless numbers leads to the following expression:

21/3sin( ) ( )

2 2EGθ

πΛ= (A5)

Where Ge and Λ are defined as follows:

1/3

20

Er

gVGE

ρε ε

= (A6)

/L dΛ = (A7)

However placing (A7) and (A6) in (A5) leads to this result:

20

sin( )4 r

L gE

ρθε ε

= (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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