PHYSICAL REVIEW E 88, 033019 (2013)

Experimental investigation of the stability of the floating water bridge Reza Montazeri Namin* Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

Shiva Azizpour Lindi National Organization for Development of Exceptional Talents (NODET), Farzanegan Highschool, Tehran, Iran

Ahmad Amjadi, Nima Jafari, and Peyman Irajizad Physics Department, Sharif University of Technology, Tehran, Iran (Received 16 March 2013; published 24 September 2013) When a high voltage is applied between two beakers filled with deionized water, a floating bridge of water is formed in between exceeding the length of 2 cm when the beakers are pulled apart. Currently two theories regarding the stability of the floating water bridge exist, one suggesting that the tension caused by electric field in the dielectric medium is holding the bridge and the other suggesting surface tension to be responsible for the vertical equilibrium. We construct experiments in which the electric field and the geometry of the bridge are measured and compared with predictions of theories of the floating water bridge stability. We use a numerical simulation for estimation of the electric field. Our results indicate that the two forces of dielectric and surface tensions hold the bridge against gravity simultaneously and, having the same order of magnitude, neither of the two forces are negligible. In bridges with larger diameters, the effect of dielectric tension is slightly more in the vertical equilibrium than surface tension. Results show that the stability can be explained by macroscopic forces, regardless of the microscopic changes in the water structure. DOI: 10.1103/PhysRevE.88.033019

PACS number(s): 47.65.−d, 47.55.nk, 77.84.Nh

I. INTRODUCTION

The floating water bridge is an interesting phenomenon first reported by Armstrong in 1893 [1]. After more than a century Fuchs et al. [2] reported their investigation about this interesting phenomenon in 2007. They showed the different behaviors in it and suggested that it could reveal some hidden properties of water [3]. Two beakers filled with deionized water are subjected to a dc high voltage more than 10 kV and a bridge is formed between them (Fig. 1) which can last for hours and have a length exceeding 2 cm [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 [3,4], Raman scattering [8], Brillouin scattering [9], and zero gravity experiments [10], many of which have attempted to investigate the possible structural changes in the water bridge causing its formation, stability, and other properties observed. Some observations have been explained by quantum electrodynamic theories [11]. Aqueous solutions have been tested under the same conditions and the liquid bridging has been observed and conductivity and mass transfer differences have been investigated [12], as well as thermal differences in the behavior of the bridge [13]. Also the bridging has been observed in dielectric liquids other than water [14,15]. Midinfrared emission investigations of the water bridge suggest the existence of micro- and nanodroplets electrosprayed from the liquid-gas interface [16]. Transport

*

[email protected]; http://mech.sharif.edu/∼namin

1539-3755/2013/88(3)/033019(6)

and behavior of bacterial cells added to the water bridge have also been investigated [17]. Reviews on this topic have been published [5,15] which the reader may refer to for a comprehensive literature review. While many discussions have been published regarding the structural changes in water leading to the stability of the bridge and suggest the existence of anisotropic chains of molecules in the bridge, high-energy x-ray-diffraction experiments show no preferred orientation of the molecules in the water bridge, which is also approved in molecular dynamics simulations [18]. In the present investigation, we concentrate on the theories based on macroscopic forces explaining the stability of the bridge. In this case there are two different perspectives. In 2009 Widom et al. [19] suggested the existence of a tension along the bridge caused by the electric field within the dielectric material. They provide theoretical calculations based on the Maxwell pressure tensor within the dielectric to calculate this tension. The tension along the curved water bridge causes an upward force defying gravity. In 2010 Mar´ın and Lohse [14] apply a similar theory, while the tension is calculated as half the value derived by [19], and in a modified experimental setup compare the results with experiments. They also measure water flow along the bridge suggesting electrical charges responsible for that. In 2012 Morawetz [20,21] discusses the effect of electrical charges in a charged catenary and solves the flow and derives the stability criteria. On the other hand, Aerov [22] in 2011 performed calculations and stated that “It is proven that electrostatic field is not the origin of the tension holding the bridge”and the only force holding the bridge against gravity is surface tension. The effect of the electric field according to Aerov is to avoid the breakup of the bridge into small droplets and maintain stability.

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REZA MONTAZERI NAMIN et al.

PHYSICAL REVIEW E 88, 033019 (2013)

FIG. 1. (Color online) Floating water bridge from front and top views. Distance between the beaker tips: 14 mm.

We try to examine the theoretical perspectives experimentally by designing quantitative experiments which are comparable with the two theories. The experimental setup is explained in Sec. II. We precisely measure the geometry of the bridge by image processing, and estimate the electric field with use of a numerical simulation explained in Sec. III for all data in experiments. Experimental results and the theories are compared in Sec. IV leading to the conclusion in Sec. V. II. EXPERIMENT

The experimental setup consists of two 50 ml beakers filled with deionized water. The water was produced with a Millipore SimPak1 purification pack kit and initially had a resistivity of 18.2 M cm. Resistivity of pure DI water decreases rapidly by contamination of impurities, e.g., the CO2 gas from air. The resistivity during our experiments was 1.8 M cm. The resistivity also varies by temperature changes; it decreases from 1.8 M cm at 25 ◦ C to 1 M cm at 45 ◦ C. A high voltage power supply was used which could provide a voltage up to 25 kV with 20 mA current intensity (Plastic Capacitors, Inc. HV250-103M). A resistance of 50 M was placed after the power supply as a ballast resistor to control the current in the circuit which had a great effect on the stability of the bridge, as shown in Fig. 2. Also a resistance of 100  was placed so that the voltage difference along it demonstrates

FIG. 2. Schematic of the experimental setup.

the current intensity. Two aluminum plates were placed as electrodes at the far ends of the beakers connected to the power supply. The voltage difference between the electrodes and the current intensity was measured. An infrared thermometer (TES 1326S) was used to measure the surface temperature of water. The current intensity and experimenting time were kept small enough so that the temperature rise of the surface of water did not exceed 2 ◦ C and was kept between 24 ◦ C and 26 ◦ C during the experiments. So there might be about 5% change of resistivity in different points in water. Two cameras were recording the bridge from the top view and front view. A third camera was recording the current intensity and voltage. For the extraction of quantitative data, we developed an image processing code using MATLAB to read the three movies and extract the desired data which includes the average diameter from top and front views Dt and Df , respectively, the curvature of the center line of the bridge at its center ξ , the electrode voltage Ve , and the current intensity I at every frame. To estimate the curvature of the center line, initially the center line was calculated by averaging the top and bottom boundaries of the bridge from the front view; then a parabola was fitted to the line and using the coefficient of the parabola the curvature was estimated. The amount of voltage was used to estimate the electric field in the center of the bridge for every experiment data as explained in Sec. III. III. EVALUATION OF THE ELECTRIC FIELD

An important parameter which should be evaluated to compare the theories with experiments is the electric field inside the bridge. We apply the leaky-dielectric model by Taylor and Melcher [23] as reviewed in [24] which has been applied in liquid bridging before [25]. The electric relaxation time in a dielectric is ε0 ε/σ , where σ is electrical conductivity, ε0 is vacuum permittivity, and ε is relative permittivity of the liquid. Thus it is in order of 10−5 s for our system and, since the maximum velocities are in the order of tens of centimeters per second [4] and minimum lengths are millimeters, the characteristic time is of the order of 10−2 s and so the system can be assumed electrostatic, i.e., electrical charges are at the surfaces, while inside water there is no bulk charge density. Thus the electric potential follows the solution of Laplace’s equation: ∇ 2 φ = 0,

(1)

with the electric field defined by  E = −∇φ.

(2)

Using a three-dimensional numerical solution to the Laplace equation (1) we find the electric field in the bridge as a function of the electric potential between the electrodes (Ve ) and the diameter (D), length (l), and curvature (ξ ) of the bridge. Later in experiments, we measure these values and use the results here to evaluate the electric field in experiments. For this purpose the volume of water was meshed as shown in Fig. 3 in different bridge diameters, lengths, and curvatures. The boundary conditions were Dirichlet boundaries with known voltages at the electrodes. Other boundaries at surfaces of water were the Neumann boundary condition with zero flux because no electric current flows from those boundaries. The equation was solved in the software COMSOL Multiphysics (see Fig. 4).

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PHYSICAL REVIEW E 88, 033019 (2013) 90 l = 10mm l = 20mm l = 30mm

E (V/m)

80 70 60 50 40 30

0

1

2 D (mm)

3

4

0.1

0.2 D/l

0.3

0.4

2.5

E*/E

2

1.5

FIG. 3. Geometry and grid of the numerical model.

The results show that the electric field decreases with the increase of the bridge diameter. Defining E ∗ = Ve / l, the ratio E ∗ /E is one when the diameter is very small compared to length and decreases with an increase in diameter. In our range of variables this ratio is only a function of l/D, and not independently a function of l and D. According to Fig. 5, there is a linear relation between E ∗ /E and D/ l. Note that, as a simplified estimation, one may assume the electrical resistance of the bridge to be proportional to D −2 and the resistance of water in the beakers to be independent of D, which results in the prediction that E ∗ /E would be linearly proportional to D 2 and not D as Fig. 5 shows. The point is that the resistance of water in the beakers is not independent of D because the

1

0

FIG. 5. (Color online) Above: evaluated electric field from the numerical simulation. Ve = 1 V and ξ = 0.001, for three values of l. Bottom: processed data. The line represents a linear fit: E ∗ /E = 2.9096(D/ l) + 1.0144 with the regression of R 2 = 0.98.

structure of electric current flows therein is affected by the diameter of the bridge, i.e., a bridge with a smaller section means that the electrical current in the beakers will be more dense near the bridge and thus the resistance will be higher. For this reason, this simplification is not valid and that is the reason we had to use this numerical solution. Based on our simulation results, the effect of curvature on the electric field was less than 1.5% in the range of experiments and was neglected. The shape of the bridge was assumed to be a bent cylinder of constant curvature, while theoretically it is closer to a hyperbolic cosine similar to a hanging rope. However because of the range of small curvatures, the difference was not distinguishable in our experiments. In the range of our experiments, this fact can cause a difference in the length of the bridge maximum of up to 1% which is negligible here. Elevation of water in the beakers was effective in this case, and since in our experiments water was up to 0.5 mm below the beaker edges, it caused less than 8% of error for l = 10 mm and 5% of error for l = 20 mm and was considered in calculating the error bars. IV. RESULTS AND DISCUSSION

FIG. 4. (Color online) Results of the numerical simulation. Above: contours of electric potential on a cut plane along the bridge. It shows that the equipotential surfaces are oriented normal to the bridge center line. Bottom: equipotential surfaces near the tip of one beaker. Units for the voltages are in volts when Ve = 1 V.

The extracted data from experiments are the average diameter from top view Dt , average diameter from front view Df , curvature of the bridge from the front view ξ , voltage difference between the electrodes Ve , and current intensity I . Every five extracted data (every 200 ms) has been averaged to present one data point. The distance between the beaker tips lb and the voltage difference across the high voltage power supply VH V were the independent parameters we could change during experiments. Our experimental data is presented in Fig. 6.

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PHYSICAL REVIEW E 88, 033019 (2013)

intensity and as a result the voltage drop across the bridge remains fairly constant in a specific bridge length. In the absence of the ballast resistor, the water in the beaker acts fairly similar to the ballast resistor, keeping the voltage difference across the bridge constant in different current intensities. We suggest this to be a reason for not achieving longer bridges in higher electric voltages; i.e., by increasing the electric voltage, the bridge increases its thickness passing a higher current intensity, while current density and electric field remain constant. To analyze the experimental data,the average diameter of the bridge was calculated as D = Dt Df . By having the length and diameter of the bridge and the voltage across the electrodes in every moment, the electric field was estimated using the results of Sec. III as a function of Ve and D and the length of the bridge l. To quantitatively compare our experiments with theoretical results of Widom et al. [19] and Aerov [22], their theory is used to find the fraction of their suggested forces to the force needed to hold the bridge in experiments. In the equilibrium condition, since the sum of vertical forces holding the bridge should be zero, the total fraction of the holding forces to the gravitational force must be equal to one. The tension because of the electric field in a dielectric medium calculated by Widom et al. [19] follows this relation: (3) TDE = ε0 (ε − 1)E 2 A,

−3

Diameter (m)

2.5

x 10

Dt

2

Df

1.5 1 0.5 290

300

310

320

330

340

320

330

340

(a)

−1

Curvature (m )

80 60 40 20 290

300

310

Current Intensity (mA)

(b) 0.4 0.3

where A is the cross-sectional area of the bridge, and is equal to π D 2 /4, ε is the relative permittivity of water, which was assumed to be 80, and ε0 is the vacuum permittivity. If a tension T is acting on a curved bridge with a curvature of ξ , the vertical force produced per unit length of the bridge is ξ T , while the gravitational force per unit length is ρAg. Thus the ratio of the dielectric force and gravitational force (RDE ) will be ε0 (ε − 1)E 2 ξ RDE = . (4) ρg

0.2 0.1 0 290

300

310

320

330

340

(c)

Voltage (kV)

20 15

Aerov [22] states that the electric tension along the bridge is zero, and the tension holding the bridge is surface tension. The electric field causes stability of the bridge and avoids it breaking to droplets. The tension caused by surface tension is the sum of the tension on the sides (γ P ) and the repulsing tension caused by the pressure jump at the surface (−γ P /2): (5) TST = 12 γ P ,

10 VHV

5 0 290

300

Ve

310 320 time (s)

330

340

(d)

FIG. 6. (Color online) Data extracted directly from experiments in means of time. (a) Diameters from top and side view. (b) Curvature of the bridge center line. (c) Current intensity passing through the bridge. (d) Voltage differences across the electrodes Ve and the power supply VH V . Time is started with the bridge formation.

where P is the perimeter of the cross section of the bridge and is equal to π D. According to this assumption, the ratio between the upwards surface-tension force and gravitational force (RST ) can be calculated as 2γ ξ RST = . (6) ρgD

Figure 6 shows that by increasing the power supply voltage, the voltage difference between the electrodes does not change significantly. Instead, current intensity increases and the residual voltage will be dropped at the ballast resistor. The reason for this fact seems to be that the diameter of the bridge increases with the increase of current intensity, causing a fairly linear relation between current intensity and the cross-sectional area of the bridge. This causes the electric resistance of the bridge to drop with an increase in current

To compare the results with experiments, we plotted in Fig. 7 RDE and RST as found in experiments. We see that neither of the theories predict a force high enough for stability since the forces they predict are significantly lower than the force needed to hold the bridge. We have also added the two forces of theories and plotted in Fig. 7 the ratio of the resultant force to the force needed to hold the bridge. Our experimental results show that the sum of the two forces of surface tension and electric tension are the same as the force needed to hold the bridge. With this experimental evidence

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PHYSICAL REVIEW E 88, 033019 (2013)

Force fraction to the total force

1.6 Dielectric Tension

1.4

Surface Tension

Total

1.2 1 0.8 0.6 0.4 0.2 0.6

0.8

1

1.2 1.4 Average Diameter (m)

1.6

1.8

2 −3

x 10

FIG. 7. (Color online) Ratio of the calculated forces of surface tension (light shade) and dielectric tension (midshade) to the force needed to hold the bridge. This result suggests that a sum of the surface tension and dielectric tension (dark shade) is convenient to explain the vertical equilibrium of the bridge. l = 14 mm.

we suggest that both mechanisms of surface tension and electric tension are holding the bridge simultaneously and each hold about half of the bridge’s weight. Note that the system is not actually in the equilibrium state since water is flowing in the bridge and the motion of water in the curved bridge causes a vertical acceleration; thus the sum of the forces per unit mass should not be zero, but equal to the acceleration. We estimated the velocity in our experiments to be less than 10 cm s−1 for the maximum diameter and decrease in less diameters. The curvature is 80 m−1 so the velocity causes a maximum acceleration of a = v 2 ξ = 0.8 m s−2 in the maximum diameter. This causes 8% of increased force and is a reason for the increased force in large diameters in Fig. 7. V. CONCLUSION

We have experimentally investigated the forces holding the floating water bridge against gravity. By analyzing the shape of the bridge from top and side views and evaluating the electric field using a numerical solution of the Laplace equation for every point in the experiment range, we have estimated the forces of dielectric tension and surface tension and compared them to the weight of the bridge. Based on our experimental results, we suggest that the vertical components of the two forces of dielectric tension and surface tension simultaneously hold the bridge against gravity. Our data shows that in smaller diameters of the bridge the effect of surface tension gets more important, while in thick bridges the dielectric tension is more important in holding the bridge. Neither of the two forces are negligible according to our data, each being responsible for holding about half of the [1] W. Armstrong, Electr. Eng. 10, 153 (1893). [2] E. Fuchs, J. Woisetschl¨ager, K. Gatterer, E. Maier, R. Pecnik, G. Holler, and H. Eisenk¨olbl, J. Phys. D: Appl. Phys. 40, 6112 (2007).

weight of the bridge and the sum of them is equal to the weight of the bridge. We have shown that increasing the electric voltage of the power supply does not necessarily increase the electric field along the bridge, because the cross-sectional area of the bridge varies fairly linearly with current intensity. We suggest this to be a reason for not achieving bridges longer than 25 mm in high electric voltages in experiments, and also an explanation for Aerov’s claim [22] about the dielectric tension hypothesis: “The electrostatic field hypothesis of the bridge tension (τ ∼ E 2 ) is not really consistent with experiments, because it allows the existence of bridges longer than 4 cm in stronger fields, which seems to be not the case.” We have shown that the stability of the floating water bridge can be fully explained with the two forces of dielectric tension and surface tension. Changes in the structure of water are not needed for explaining this fact. ACKNOWLEDGMENTS

We wish to thank Professor K. Morawetz and Professor A. Aerov for helpful discussions in different stages of our investigation. We also thank the help of M. Habibpour and Z. Karimi in experiments. We acknowledge the International Young Physicists’ Tournament (IYPT) society for introducing this phenomenon and for discussions at IYPT Bad Saulgau Germany. We thank Sharif Applied Physics Research Center for its financial support at Medical Laser Physics Lab. All the experiments were performed at the Medical Physics Laboratory at Department of Physics, Sharif University of Technology. [3] E. Fuchs, K. Gatterer, G. Holler, and J. Woisetschl¨ager, J. Phys. D: Appl. Phys. 41, 185502 (2008). [4] J. Woisetschl¨ager, K. Gatterer, and E. Fuchs, Exp. Fluids 48, 121 (2010).

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[5] E. Fuchs, Water 2, 381 (2010). [6] E. Fuchs, B. Bitschnau, J. Woisetschl¨ager, E. Maier, B. Beuneu, and J. Teixeira, J. Phys. D: Appl. Phys. 42, 065502 (2009). [7] E. Fuchs, P. Baroni, B. Bitschnau, and L. Noirez, J. Phys. D: Appl. Phys. 43, 105502 (2010). [8] R. Ponterio, M. Pochylski, F. Aliotta, C. Vasi, M. Fontanella, and F. Saija, J. Phys. D: Appl. Phys. 43, 175405 (2010). [9] E. Fuchs, B. Bitschnau, S. Di Fonzo, A. Gessini, J. Woisetschl¨ager, and F. Bencivenga, J. Phys. Sci. Appl. 1, 135 (2011). [10] E. Fuchs, L. Agostinho, A. Wexler, R. Wagterveld, J. Tuinstra, and J. Woisetschl¨ager, J. Phys. D: Appl. Phys. 44, 025501 (2010). [11] E. Del Giudice, E. Fuchs, and G. Vitiello, Water 2, 69 (2010). [12] M. Eisenhut, X. Guo, A. Paulitsch-Fuchs, and E. Fuchs, Cent. Eur. J. Chem. 9, 391 (2011). [13] E. C. Fuchs, A. D. Wexler, L. L. F. Agostinho, M. Ramek, and J. Woisetschl¨ager, J. Phys.: Conf. Ser. 329, 012003 (2011). ´ G. Mar´ın and D. Lohse, Phys. Fluids 22, 122104 (2010). [14] A.

[15] J. Woisetschl¨ager, A. Wexler, G. Holler, M. Eisenhut, K. Gatterer, and E. Fuchs, Exp. Fluids 52, 193 (2012). [16] E. Fuchs, A. Cherukupally, A. Paulitsch-Fuchs, L. Agostinho, A. Wexler, J. Woisetschl¨ager, and F. Freund, J. Phys. D: Appl. Phys. 45, 475401 (2012). [17] A. Paulitsch-Fuchs, E. Fuchs, A. Wexler, F. Freund, L. Rothschild, A. Cherukupally, and G. Euverink, Phys. Biol. 9, 026006 (2012). [18] L. B. Skinner, C. J. Benmore, B. Shyam, J. K. R. Weber, and J. B. Parise, Proc. Natl. Acad. Sci. USA 109, 16463 (2012). [19] A. Widom, J. Swain, J. Silverberg, S. Sivasubramanian, and Y. N. Srivastava, Phys. Rev. E 80, 016301 (2009). [20] K. Morawetz, Phys. Rev. E 86, 026302 (2012). [21] K. Morawetz, AIP Adv. 2, 022146 (2012). [22] A. A. Aerov, Phys. Rev. E 84, 036314 (2011). [23] J. Melcher and G. Taylor, Annu. Rev. Fluid Mech. 1, 111 (1969). [24] D. Saville, Annu. Rev. Fluid Mech. 29, 27 (1997). [25] C. Burcham and D. Saville, J. Fluid Mech. 452, 163 (2002).

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Experimental investigation of the stability of the floating ...

Sep 24, 2013 - ... zero flux because no electric current flows from those boundaries. The equation was solved in the software COMSOL Multiphysics (see Fig.

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