aa r X i v : . [ phy s i c s . c l a ss - ph ] M a y Theory of water and charged liquid bridges
K. Morawetz , , M¨unster University of Applied Science, Stegerwaldstrasse 39, 48565 Steinfurt, Germany International Institute of Physics (IIP) Federal University of Rio Grandedo Norte Av. Odilon Gomes de Lima 1722, 59078-400 Natal, Brazil and Max-Planck-Institute for the Physics of Complex Systems, 01187 Dresden, Germany
The phenomena of liquid bridge formation due to an applied electric field is investigated. A newsolution of a charged catenary is presented which allows to determine the static and dynamicalstability conditions where charged liquid bridges are possible. The creeping height, the bridgeradius and length as well as the shape of the bridge is calculated showing an asymmetric profile inagreement with observations. The flow profile is calculated from the Navier Stokes equation leadingto a mean velocity which combines charge transport with neutral mass flow and which describesrecent experiments on water bridges.
PACS numbers: 05.60.Cd, 47.57.jd,47.65.-d, 83.80.Gv,
I. INTRODUCTIONA. The phenomenon
The formation of a water bridge between two beakersunder high voltage is a phenomenon known since over 100years [1]. When two vessels brought in close contact anda high electric field is applied between the vessels, thewater starts creeping up the beakers and forms a bridgewhich is maintained over a certain distance as schemat-ically illustrated in figure 1. Due to the voltage appliedby the vessels the electric field is longitudinally orientedinside the cylindrical bridge. It has remained attractiveto current experimental activities [2, 3]. On the one sidethe properties of water are such complex that a completemicroscopic theory of this effect is still lacking. On theother side the formation of water bridges on nanoscalesare of interest both for fundamental understanding ofelectrohydrodynamics and for applications ranging fromatomic force microscopy [4] to electrowetting problems[5]. Microscopically the nanoscale wetting is importantto confine chemical reactions [6] which reveals an interest-ing interplay between field-induced polarization, surfacetension, and condensation [7, 8].Molecular dynamical simulations have been performedin order to explore the mechanism of water bridges at Lz max xz Rf(x) FIG. 1: The schematic picture of water bridge between twobeakers. the molecular level leading to the formation of aligneddipolar filaments between the boundaries of nanoscaleconfinements [9]. A competition was found of orienta-tion of molecular dipoles and the electric field leadingto a threshold where the rise of a pillar overcomes thesurface tension [8]. In this respect the understanding ofthe microscopic structure is essential to explain such phe-nomena in micro-fluidics [10]. The problem is connectedwith the dynamics of charged liquids which is importantfor capillary jets [11], current applications in ink printersand electrosprays [12, 13]. Consequently the nonlineardynamics of breakup of free surfaces and flows has beenstudied intensively [14, 15].Much physical insight can be gained already on themacroscopic scale, where the phenomena of liquid bridg-ing is not restricted to water but can be observed inother liquids too [16] which shows that it has its originin electrohydrodynamics [17] rather then in molecular-specific structures. The traditional treatment is basedon the Maxwell pressure tensor where the electric fieldeffects comes from the ponderomotoric forces and dueto boundary conditions of electrodynamics [18]. Thisis based exclusively on the fact that bulk-charge statesdecay on a time scale of the dielectric constant dividedby the conductivity, ǫǫ /σ , which takes for pure water0 . ρ c = −∇ · j combinedwith Ohm’s law j = σ E = − σ ∇ φ where the source of theelectric field is given by the potential ∇ φ = − ρ c /ǫǫ .An overview about the different forces occurring in mi-croelectrode structures are discussed in [19].This simple Ohm picture leads to a problem in par-tially charged liquids. Following the Ohm picture onehas a constant velocity or current density of charged par-ticles caused by the external field and limited by friction.Contrary, for incompressible fluids the total mass fluxcannot be constant but is dependent on the area whereit is forced to flow through. Both pictures seem to be im-possible to reconcile. Here in this paper we will presenta discussion of this seemingly contradiction leading to adynamical stability criterion for the water bridge and acombined flow expression. This is in line with the ideaof [17] where the bulk charges have been assumed to berealized in a surface sheet. While there the migrationof charges to the surface has been considered forming acharged surface sheet, we adopt here the view point of ho-mogeneously distributed bulk charges which flow in fielddirection rather than forming a surface sheet.In the absence of bulk charges the forces on the waterstream are caused by the pressure due to the polarizabil-ity of water described by the high dielectric susceptibil-ity ǫ . This pressure leads to the catenary form of waterbridge like a hanging chain [20]. While already the sim-plified model of [16] employing a capacitor picture leadsto a critical field strength for the formation of the waterbridge, the catenary model [20] has not been reportedto yield such a critical field. In this paper we will showthat even the uncharged catenary provides indeed a min-imal critical field strength for the water bridge formationin dependence on the length of the bridge. This criti-cal field strength is modified if charges are present in thebridge which we will discuss here with the help of a newcharged catenary solution. This allows us to explain theasymmetry found in the bridge profile [3]. B. Overview about the paper
The scenario of water or other dielectric bridges isthought as follows. Applying an electric field parallelto two attached vessels the water creeps up the beakerand form a bridge as it is nicely observed and picturedin [2]. This bridge can be elongated up to a critical fieldstrength and it forms a catenary which becomes asym-metric for higher gravitation to electric field ratios [3].The critical value for stability is sensitively dependenton ion concentrations breaking off already at very lowconcentrations. The amount of mass flow through thebridge does not follow simple Ohmic transport as we willsee in this paper. The schematic picture of the waterbridge is given in figure 1.In this paper we want to advocate the following pic-ture. Imaging a snapshot of the charges flowing throughthe bridge we cannot decide whether the observed chargesare due to static bulk charges or due to the floating mo-tion of Ohmic bulk charges. This flow of charges withinthe liquid bridge we can associate with a dynamical bulkcharge in the mass motion which is not covered by thedecay of Ohmic bulk charges discussed above. Such apicture is supported by the experimental observation ofpossible copper ion motion [21] and by the observationthat the water bridge is highly sensitive to additional ex-ternal electric fields [22]. Strong fields even create smallcone jets [2]. This dynamical bulk charge will lead usto the necessity to solve the catenary problem includingbulk charges. Though charged membranes have been dis-cussed in the literature [23], a new analytical solution ofthe charged catenary is discussed in this paper.The picture of Ohmic resistors and capacitor as de- density ρ = 10 kg/m dielectric susceptibility ǫ = 81surface tension σ s = 7.27 × − N/mviscosity η = 1.5 × − Ns/m conductivity ofclean water σ = 5 × − A/Vmmolecular conductivityof NaCl λ = 12.6 × − Am /Vmolheat capacity c p = 4.187 J/gKTABLE I: Variables and parameters used within this paperfor water. scribed above is not sufficient, as one can see from theobservation that adding a small amount of electrolytes tothe clean water destroys the water bridge almost imme-diately. In other words good conducting liquids shouldnot form a water bridge. We will derive an upper boundfor charges possibly carried in water in order to remain instable liquid bridges. Though we present all calculationsfor water parameters summarized in table I, the theoryapplies as well to any dielectric liquid in electric fields.Four theoretical questions have to be answered: (i)How is the electric field influencing the height z max wa-ter can creep up? (ii) What is the radius R ( x ) along thebridge? (iii) What is the form z = f ( x ) of the waterbridge? What are the static constraints on the bridge?(iv) Which dynamical constraints can be found for pos-sible bridge formation?We will address all four questions with the help of fourparameters composed of the properties summarized intable I of water. The first one is the capillary height a = r σ s ρg = 3 . σ s , the particle density ρ andthe gravitational acceleration g . The second parameter isthe water column height balancing the dielectric pressurecalled creeping height in the following b ( E ) = ǫ ( ǫ − E ρg = 7 .
22 ¯ E cm (2)where the dimensionless electric field ¯ E is in units of10 V/cm. The third one is the dimensionless ratio ofthe force density on the charges by the field to the grav-itational force density c ( ρ c , E ) = ρ c Eρg = 15 .
97 ¯ E ¯ ρ c (3)where the charge density ¯ ρ c is in units of ng/l . For dy-namical consideration the characteristic velocity u = ρga η ≈ . / s (4)will be useful as the fourth parameter.The outline of the paper is as follows. In the next sec-tion we repeat shortly the standard treatment of creepingheight and bubble radius of a liquid but add the pressureby the external electric field on the dielectric liquid. Thenwe present the form of the bridge in terms of a new so-lution of the catenary equation due to bulk charges insection IV. In section V we present the flow calculationproposing the picture of moving charged particles due tothe field which drag the neutral particles. This will leadto a dynamical stability criterion. Then we compare withthe experimental data and show the superiority of thepresent treatment. Summary and conclusion ends up thediscussion in section VI. II. ANSWER TO QUESTION (I): CREEPINGHEIGHT AND (II): RADIUS OF BRIDGE
We start to calculate the possible creeping height anduse the pressure tensor for dielectric media [18] σ ik = − pδ ik − σ s (cid:18) R + 1 R (cid:19) + ǫǫ E i E k −
12 ˜ ǫǫ E δ ik (5)where p is the pressure in the system, R , R the principalradii of curvature such that the second term on the righthand side describe the contribution due to surface tensionand the last terms are the parts due to the forces in thedielectric medium. We assume a density-homogeneousliquid such that for the dielectric susceptibility ˜ ǫ = ǫ − ρ ( dǫ/dρ ) T ≈ ǫ . Further we consider first the stationaryproblem which means that viscous forces can be neglectedin (5).Denoting the components of the normal vector by e k ,the stability condition between water (W) and air (A) isgiven by σ ik ( A ) e k ( A ) = − σ ik ( W ) e k ( W ) = − σ ik ( A ) e k ( W ) . (6)Since the principal curvature of the tube is much largerradially than parallel, we have R ∼ ∞ and denotingthe coordinate in the direction of the height with z , thepressure difference between water and air is p W − p L = ρgz . We employ the boundary conditions for the normal E n and tangential E t components of the electric field E n ( A ) = ǫE n ( W ) = ǫE n , E t ( A ) = E t ( W ) = E t . (7)and the balance (6) with (5) reads ρgz + σ s R = 12 ǫ ( ǫ − ǫE n + E t ) . (8)Please note that due to the migration of charges to thesurface one should consider a surface charge here in prin-ciple. We adopt thorough the paper the simplified pic-ture that the charges remain bulk-like due to the pre-ferred motion along the field and no surface charges areformed. The influence of such surface charges is con-sidered as marginal since the curvature of the bridge isminimal leading to preferential tangential components ofelectric fields. E z θ α z max x FIG. 2: The schematic picture of water bridge creeping upthe vessel due to the applied electric field.
We assume the electric field in x -direction such that E t = − E cos α , E n = E sin α where z ′ ( x ) = tan α is theincrease of the surface line of the water as illustrated infigure 2. Using the parameters (1) and (2) we obtainfrom the stability condition (8) the differential equation2 z − a z ′′ (1 + z ′ ) / = ǫ ( ǫ − ρg ( ǫE n + E t ) ≈ b (9)where we used the approximation of small normal electricfields justified if there are no surface charges. This showsthe modification of the standard treatment of capillaryheight by the applied field condensed on the right handside. The first integral of (9) is z a + 1 √ z ′ − bza = 1 (10)and we have used the condition that for x → ∞ the sur-face is z = z ′ = 0. The explicit solution of the surfacecurve z ( x ) is quite lengthy and not necessary here. In-stead we can give directly the maximally reachable heightin dependence on the electric field. Therefore we use theangle θ = 90 − α of the liquid surface with the wall suchthat z ′ ( x ) = − cot θ and from (10) we obtain z = b r b a (1 − sin θ ) ≤ b r b a = z max (11)which shows that without electric field the maximalcreeping height is just the capillary length (1) as it iswell known. The other extreme of very high fields leadsto the field-dependent length (2) which justifies the namecreeping height. This answers the first question concern-ing creep heights.The second question, how large the radius of the bridgeis, one finds by equating the pressure due to surface ten-sion with the gravitational force density σ s R = ρgz ≈ ρg R (12)such that the radius of the water bridge is at the beaker R ≈ a/ . (13)Without using this approximation we could express thecurvature again by differential expressions in z ( x ) defin-ing a radial profile, as it can be found in text books [18].The radius of the bridge at the beaker is nearly indepen-dent on the applied electric field but only dependent onthe surface tension and gravitational force. Along thebridge the radius will change with the applied electricfield as we will see later in section IV.C. III. ANSWER TO QUESTION (III): LIQUIDBRIDGE SHAPEA. Charged catenary
Now we turn to the question which form the waterbridge will take. Therefore we consider the center ofmass line of the bridge being described by z = f ( x )with the ends at f (0) = f ( L ) = 0. The force densi-ties are multiplied with the area and the length element ds = p f ′ dx to form the free energy. We have thegravitational force density ρgf and the volume tension ρgb as well as the force density by dynamical charges ρ c Ex which contributes. The surface tension is negligi-ble here. The form of the bridge will be then determinedby the extreme value of the free energy L Z F ( x ) dx = ρg L Z ( f ( x ) + b − cx ) p f ′ dx → extr. (14)where c is given by (3) and b defined in (2).As shown in [24] and shortly outlines in appendix Athe solution can be represented parametrically as f ( t ) = 11+ c (cid:26) c t + ξ (cid:20) cosh (cid:18) tξ − Ld ξ (cid:19) − cosh (cid:18) Ld ξ (cid:19)(cid:21)(cid:27) x ( t ) = t − cf ( t ) , t ∈ (0 , L ) . (15)with d = 2 ξL arcosh bξ (16)and ξ to be the solution of the equation c = c m ( ξ, b ) c m ( ξ, b ) = − ξL sinh L ξ bξ sinh L ξ − s b ξ − L ξ ! . (17) B. Static stability criteria
Without dynamical bulk charges, c = 0 , d = 1, thesolution (15) is just the well known catenary [20]. Theboundary condition (17) reads in this case2 bL = 2 ξL cosh L ξ ≥ ξ c = 1 . ... (18)which means that without bulk charges the condition fora stable bridge is b > Lξ c . (19)Together with (2) this condition provides a lower boundfor the electric field in order to enable a bridge of length L . This lower bound for an applied field appears obvi-ously already for the standard catenary and has been notdiscussed so far.Lets now return to the more involved case of bulkcharges and the new solution of charged catenary (15).The field-dependent lower bound condition (17) is plot-ted in figure 3. One see that in order to complete (17)the bulk charge parameter c has to be lower than themaximal value of c m which reads c ≤ c m ( ξ , b ) (20)and which is plotted in the inset of figure 3. Remember-ing the definition of the bulk charge parameter (3) wesee that (20) sets an upper bound for the bulk charge independence on the electric field. The lower bound (19) ofthe electric field for the case of no bulk charges is obeyedas well since the curve in the inset of figure 3 starts at b > Lξ c / IV. ANSWER TO QUESTION (IV):DYNAMICAL CONSIDERATIONA. Mass flow of the bridge
We consider now the actual motion of the liquid inthe bridge. Here we want to propose the picture thatpossible charges in the water will move according to theapplied electric field and will drag water particles suchthat a mean mass motion starts. Due to the low Reynoldsnumbers (40-100) for water we can consider the motionas laminar and we can neglect the convection term u ∇ u in the Navier Stokes equation [25] which reads then forthe stationary case η ∇ u − ∇ p + ρ c E = 0 . (21) b[L] c m ( ξ , b ) ξ [L] -10-8-6-4-202 c m ( ξ , b ) ξ c ξ c +22b/L = ξ c +4 FIG. 3: The upper critical bound for the parameter c accord-ing to (17). The inset shows the maximum in dependence onthe creeping parameter b . The gradient of the electric pressure (8) can be given inthe direction of the bridge by − ∇ p = ǫ ( ǫ − E L = b L ρg. (22)Here we can adopt the stationary pressure since the vis-cous pressure is accounted for by the Navier-Stoke equa-tion. Assuming that the flow in the bridge has only atransverse component which is radial dependent, u ( r ),we can write the Navier Stokes equation (21) as ηρg ddr (cid:18) r dudr (cid:19) + r (cid:18) b L + c (cid:19) = 0 (23)with the resulting velocity profile in the direction of thebridge u ( r ) − u ( R ) = 2 u (cid:18) b L + c (cid:19) (cid:18) − r R (cid:19) (24)where R is the radius of the bridge and we have intro-duced the characteristic velocity (4). Please note thatwe keep the undetermined velocity at the surface of thebridge u ( R ). We will assume in the follwoing that it isnegligible. The resulting profile (24) has the form of aPoiseullie flow but with an interplay between forces dueto bulk charges and dielectric pressure in relation to grav-ity.The mean current relative to the surface motion is eas-ily calculated I = 2 πρ R Z drr [ u ( r ) − u ( R )] ≡ ρvπR (25) I [ m l / s ] ρ c =0 ng/l ρ c =1 ng/l L=1cm L=2cm FIG. 4: The mean mass current through the bridge in depen-dence on the electric field and for two different bulk chargedensities. The thick lines are for a bridge length of 1cm andthe thin lines for the corresponding length of 2cm. The min-imal field strength for stability (19) are indicated by corre-sponding vertical lines. providing the mean velocity of the bridge from (24) as v = u (cid:18) b L + c (cid:19) . (26)One sees that the ratio of the field-dependent creepingheight (2) to the bridge length determines the mean ve-locity together with possible dynamical bulk charges de-scribed by (3). Since we have presently no good controlover the surface velocity u ( R ) we approximate it in thefollowing as zero.Please note that the bulk charge transport describedby (3) leads to Ohmic behavior and the neutral particletransport due to dielectric pressure leads to a quadraticfield dependence condensed in (2). The formula (26)combines the effect of charge transport and neutral par-ticle mass transport. It answers the problem raised inthe introduction how the two pictures can be broughttogether, the one of an incompressible fluids where thevelocity is dependent of the area and the one of Ohmictransport where the velocity is only dependent on theelectric field.The resulting total mass current is given in figure 4.The current increases basically with the square of theapplied field scaled by the bridge length. For additionalbulk densities the mass flow is higher. B. Comparison with the experiment
To convince the reader about the validity of the veloc-ity formula (26) we compare now with the mass flow andthe charge flow measurements. The experimental valuesof Figure 4 in [2] are reported to be 40mg/s for a bridge Ρ c @ ng (cid:144) l D @ kV D I= (cid:144) s H Ohmic 10 Ρ c L FIG. 5: The necessary applied voltage versus bulk chargedensities in order to maintain a mass current of 40ml/s. Fol-lowing [2] the length of the bridge was L = 1cm and thediameter 2 . of 1cm length, a diameter of 2 . σ = λ ρ c eN A + σ (27)where for clean water the conductivity is σ , λ is themolecular conductivity of the solved charge (electrolyte),and N A the Avogadro constant, see table I. We see thatour formula (26) leads to a realistic necessary voltage -which was 12 . . × which il-lustrates the difference between our model and the Ohmictransport.While the difference in charge transport is not verysignificant provided the fact that the conductivity of wa-ter varies in the order of 3 magnitudes, the mass flow offigure 5 has shown that our result here with (26) is su-perior since it considers the drag of neutral particles dueto dielectric pressure together with the charge transport.Having the current at hand one estimates the Joule @ mm D @ kV D I= FIG. 6: The necessary applied voltage versus bridge lengthin order to maintain a charge current of 0 . . × . Thesame offset of U = 8 kV is used as in the experiments. heating easily as ∆ T ∆ t = jEρc p . (28)From figure 5 of [2] one sees that the reported increaseof 10K in 30min would translate into field strengths of0.7kV/cm in our calculation. This is much lower thanour result. We would obtain here 2-3 orders of magni-tude higher heating rates. Please note that the coolingmechanisms like evaporating and cooling due to waterflow is beyond the present consideration. Since these areprobably the major cooling effects in the experiments [26]we cannot compare seriously the theoretical heating ratewith the experimentally observed ones. C. Profile of bridge
Let us now calculate the profile of the bridge alongthe length. We consider to this end the total mass flowof the bridge and neglect the viscous term compared tothe kinetic energy (which includes part of the convectionterm), u ∇ u = ∇ u + curl u × u ≈ ∇ u . Then onearrives at the Bernoulli equation ρ v ( x ) ρgf ( x )+ σ s ( 1 R ( x ) ) − ρ c Ex = ρ v σ s R . (29)Here we have neglected the curvature of the bridge com-pared to the curvature due to the radius and have com-pared the position-dependent radius R ( x ) and veloc-ity v ( x ) in the bridge with the situation at the beaker( x = 0). The Bernoulli equation (29) can be rewritten interms of the capillary height (1) and the velocity (26) as f ( x ) − cx = v − v ( x )2 g + a − a R ( x ) (30)which determines the radius R ( x ) from the profile of thebridge (15) and the velocity v ( x ) if we observe the currentconservation through an area R ( x ) v ( x ) = R v. (31)The results are presented figures 7 and 8. We plot theshape of the bridge, the radius and the velocity togetherwith a 3D plot. The case of no bulk charges which leadsto the standard catenary can be found in figure 7 andfigure 7 shows the situation for extreme bulk charges al-most at the stability edge (20). We see a deformation ofthe catenary due to the applied field. This deformationis observed, e.g. if an additional field is brought nearthe bridge [2, 22]. One sees that the radius is becom-ing smaller at one end of the bridge accompanied withhigher velocities as it is known from falling water pipes[27]. The bulk charge leads to deformations of this pro-file which are exaggerated in the plot due to the choiceof unequal scales.Interestingly such asymmetry is experimentally ob-served [2], where after 3 min of operation the asymmetryfor the bridge of 0 . . . D. Dynamical stability
We turn now to the question of dynamical stabilityof the flow and consider the motion of water togetherwith the motion of charged particles characterized by themass m i and charge e i . This charge current is given byOhm’s law σE and the corresponding mass current canbe written j i = m i e i j = x i ρρ c σE (32)where we introduced the mass ratio of the number ofcharged particles (e.g. NaCl) to the water particles x i = i m NaCl w m H = ρ c m i ρe i . (33)The mass current of the neutral (water) particles are then j n = ρ n v n = ( ρ − m i e i ρ c ) v n = (1 − x i ) ρv n (34)such that the total mass current reads ρv = j i + j n = x i ρρ c σE + (1 − x i ) ρv n . (35)The total current (left side) should be larger than thecurrent only from the charged particles (last term on theright side). However the velocity of charged particles, σE/ρ c should be larger than the velocity of the dragged -0.08-0.06-0.04-0.020 f [ L ] r [ L ] v [ m / s ] FIG. 7: The center of mass coordinate (above), the radius(middle) and the velocity (bottom) together with the 3D plotof water bridge (in cm) for no bulk charges c = 0. The pa-rameter are b = 1 . x and y, z direction. water molecules v n and therefore larger than the meanvelocity v of the mass motion. Together with (26) this isexpressed by the inequalities σEρ c > u (cid:18) bL + c (cid:19) > x i σEρ c (36) -0,2-0,15-0,1-0,050 f [ L ] c=1c=0 r [ L ] v [ m / s ] FIG. 8: The center of mass coordinate (above), the radius(middle) and the velocity (bottom) together with the 3D plotof water bridge (in cm) with bulk charges c = 1. The param-eter are b = 1cm and according to table I. which gives an upper and lower bound on the possiblemass motion created by the drag of particles due to theforce on charged particles.If we now take into account the dependence of the con-ductivity on the density of the solved ions in water wecan find a condition on possible bulk charges in water tomaintain a stable bridge. To this aim we consider very FIG. 9: The range of possible water bridges for an electricfield of E = 0 . small charge densities solved in water which allows toconsider the lowest order dependence of the conductivityon the bulk charge concentration (27).Noting the charge-density dependencies of x i , b and c via (33), (2) and (3) one obtains from (36) the dynamicalrestriction on possible bulk charges ρ c ∈ ρ − ρ ± q ( ρ − ρ ) + ρ ρ c (1 − ρ /ρ i ) > ρ /ρ i − ρ (37)with the auxiliary densities ρ = ǫ ( ǫ − E L , ρ = 16 ηλeN A a ρ = 32 ησ a , ρ i = e i ρm i . (38)The results for NaCl in water (table I) are plotted infigures 9-10. The static stability condition (19) gives theupper and charge-density-independent limit in figure 10.The static condition (20) with bulk charges leads to theborder of maximal densities on the right side which agreeswith (19) at zero densities, of course. The lower minimallength of the bridge at a given field strength and bulkcharge is provided by the dynamical condition (37). Forno bulk charge the possible range of lengths of the bridgestarts at zero and is limited by the upper length (19).If there are charges present, there is a minimal lengthrequired to have a stable bridge.From the 3D plot in figure 10 one can see that forfinite charges and for fixed bridge lengths there is a lower FIG. 10: The range of possible water bridges in dependenceon the bridge length, the electric field and the electrolyte bulkcharges. and an upper critical field where bridges can only bestable. From the experiments [2] it is seen that the bridgeforms jets for fields higher than 15kV/cm and thereforebecomes unstable. With a bridge length of 0 . cm thistranslates into a bulk charge of 4ng/l according to ourfound boundary conditions. This is in agreement withthe value needed to reproduce the flow measurementsdescribed in section IV.B. V. SUMMARY
The formation of water bridges between two vesselswhen an electric field is applied has been investigatedmacroscopically. Electrohydrodynamics is sufficient todescribe the phenomena in agreement with the experi-mental data. The four necessary parameters which arebuild up from microscopic properties of the charged liq-uid are the capillary height (1), the creeping height (2),the dimensionless ratio between field and gravitationalforce density (3), and the characteristic velocity (4).As new contribution to the discussion, an exact solu-tion has been found of a charged catenary. This leadsto a static stability criterion for possible charges in theliquid dependent on the applied field strengths and onthe length of the bridge. With no bulk charges presentthe maximal bridge length is determined and no minimallength occurs. This changes if bulk charges are present.Then also a minimal length is required. However, onlyvery small concentrations of bulk charges are possible andthe bridge is easily destroyed when bulk charges exceed 50 ng/l. As a further an asymmetric profile in the diam-eter along the bridge is obtained which was observed byasymmetric heating.For the dynamical consideration a picture is proposedof dragged liquid particles due to the motion of thecharged ones besides the ponderomotoric forces due tothe dielectric character of the liquid. The resulting con-sideration of dynamical stability restricts the possible pa-rameter range of bridge formation. The resulting massflow combines the charge transport and the neutral massflow dragged by dielectric pressure and is in agreementwith the experimental data.The presented simple classical theory applies forcharged liquids as long as the Reynolds number is suchlow that laminar flow can be assumed.
Acknowledgments
The discussions with Bernd Kutschan who pointed outthis interesting effect to me and the clarifying commentsof Jacob Woisetschl¨ager are gratefully mentioned. Thiswork was supported by DFG-CNPq project 444BRA-113/57/0-1 and the DAAD-PPP (BMBF) program. Thefinancial support by the Brazilian Ministry of Science andTechnology is acknowledged.
Appendix A: Solution of charged catenary
Here the drivation of the charged catenary [24] isshortly sketched. We solve the variation problem (14) L Z F ( x ) dx → extr. (A1)with the functional F ( x ) = ρg [ f ( x ) + b − cx ] p f ′ ( x ) and the boundary conditions f (0) = f ( L ) = 0.It is useful to introduce t ( x ) = f ( x ) + b − cx (A2)such that F ( x ) = ρg t ( x ) p t ′ ( x ) + c ] . (A3)The corresponding Lagrange equation ddx δ F δt ′ ( x ) − δ F δt ( x ) = 0 (A4)possesses a first integral t ′ ( x ) δ F δt ′ ( x ) − F = const = − ξ p c (A5)where we introduced the first integration constant ξ in aconvenient way.0The resulting differential equation t (¯ x )[ ct ′ (¯ x ) + 1] = ξ p t ′ (¯ x ) + ( ct ′ (¯ x ) + 1) (A6)with ¯ x = x (1 + c ) is solved in an implicit way t (¯ x ) = ξ cosh (cid:26) ξ (cid:20) ¯ x + ct (¯ x ) − cb + L d (cid:21)(cid:27) (A7)with a second integration constants d . The profile istherefore given by the implicit equation f ( x ) = cx − b + ξ cosh (cid:26) ξ (cid:20) x + cf ( x ) + L d (cid:21)(cid:27) . (A8)The boundary condition f (0) = 0 leads to the deter-mination of the integration constant d = 2 ξL arcosh (cid:18) bξ (cid:19) (A9)in terms of the yet unknown ξ constant. The solution(A8) can be written with the help of (A9) as f ( x ) = cx + ξ (cid:26) cosh (cid:20) x + cf ( x ) ξ − Ld ξ (cid:21) − cosh (cid:18) Ld ξ (cid:19)(cid:27) . (A10) The boundary condition f ( L ) = 0 lead to the determi-nation of the remaining constant ξ to be the solution ofthe equation c = − ξL sinh L ξ bξ sinh L ξ − s b ξ − L ξ ! . (A11)Finally we can rewrite the implicit solution (A10) inparametric form. Therefore we choose as parameter t = x + cf ( x ) which runs obviously through the inter-val t ∈ (0 , L ) and we obtain the solution (15) f ( t ) = 11+ c (cid:26) c t + ξ (cid:20) cosh (cid:18) tξ − Ld ξ (cid:19) − cosh (cid:18) Ld ξ (cid:19)(cid:21)(cid:27) x ( t ) = t − cf ( t ) , t ∈ (0 , L ) . (A12) [1] W. G. Armstrong, The Electrical Engineer (The New-castle Literary and Philosophical Society, New Castle,1893), pp. 154–155, 18 February 1893.[2] J. Woisetschlager, K. Gatterer, and E. C. Fuchs, Exp. inFluids , 121 (2010).[3] A. G. Marin and D. Lohse, Phys. of Fluids , 122104(2010).[4] G. M. Sacha, A. Verdaguer, and M. Salmeron, J. Phys.Chem. B , 14870 (2006).[5] J. M. Oh, S. H. Ko, and K. H. Kang, Physics of Fluidsp. 032002 (2010).[6] A. Garcia-Martin and R. Garcia, Appl. Phys. Lett. ,123115 (2006).[7] S. Gomez-Monivas, J. J. Saenz, M. Calleja, and R. Gar-cia, Phys. Rev. Lett. (2003).[8] T. Cramer, F. Zerbetto, and R. Garcia, Langmuir ,6116 (2008).[9] S. Chen, X. Huang, N. F. A. van der Vegt, W. Wen, andP. Sheng, Phys. Rev. Lett. , 046001 (2010).[10] T. M. Squires and S. R. Quake, Rev. Mod. Phys. , 977(2005).[11] A. M. Ganancalvo, J. of Fluid Mechanics , 165(1997).[12] M. Gamero-Castano, J. of Fluid Mechanics , 493(2010).[13] F. Higuera, Physics of Fluids p. 112107 (9 pp.) (2010).[14] J. Eggers, Phys. Rev. Lett. , 3458 (1993).[15] J. Eggers, Rev. Mod. Phys. , 865 (1997). [16] F. Saija, F. Aliotta, M. E. Fontanella, M. Pochylski,G. Salvato, C. Vasi, and R. C. Ponterio, J. of Chem.Phys. , 081104 (2010).[17] J. R. Melcher and G. I. Taylor, Ann. Rev. of Fluid Mech. , 111 (1969).[18] L. D. Landau and E. M. Lifschitz, Lehrbuch der Theo-retischen Physik: Elektrodynamik der Kontinua , vol. VIII(Akademie-Verlag, Berlin, 1990).[19] A. Ramos, H. Morgan, N. G. Green, and A. Castellanos,J. Phys. D - Appl. Phys. , 2338 (1998).[20] A. Widom, J. Swain, J. Silverberg, S. Sivasubramanian,and Y. N. Srivastava, Phys. Rev. E , 016301 (2009).[21] L. Giuliani, E. D’emilia, A. Lisi, S. Grimaldi, A. Fo-letti, and E. Del Giudice, Neural Network World , 393(2009).[22] E. C. Fuchs, J. Woisetschlager, K. Gatterer, E. Maier,R. Pecnik, G. Holler, and H. Eisenkolbl, J. Phys. D -Appl. Phys. , 6112 (2007).[23] D. E. Moulton and J. A. Pelesko, Siam J. on Appl. Math. , 212 (2009).[24] K. Morawetz, AIP Advances (2012), in press.[25] D. S. Chandrasekharaiah, Continuum mechanics (Aca-demic Press, Boston, 1994).[26] J. Woisetschlager, priv. communication.[27] M. J. Hancock and J. W. Bush, J. Fluid Mech.466