Dynamics of slender viscous dielectric liquid bridges subjected to axial AC fields

Journal of ELECTROSTATICS ELSEVIER Journal of Electrostatics 42 (1997) 259 278 Dynamics of slender viscous dielectric liquid bridges subjected to axial AC fields F.J. Garcia a'b'*, A. Castellanos a, H. Gonz/dez a'c Departamento de Electrbnica y Electromagnetismo, Facultad de Fisica, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain bDepartamento de Fisica Aplicada, E. U. 1. "i2 A., Universidad de Sevilla~ Ctra, Utrera km 1, 41013 Se~,illa, Spain Departamento de Fisica Aplicada, E. S. L, Universidad de Sevilla, At:da. Reina Mercedes sin 41012 Sevil/a, Spain Received 25 November 1996; received in revised form 14 May 1997; accepted 14 May 1997 Abstract An analysis of slender axisymmetric liquid bridges is performed on the basis of one-dimensional models recently derived, and generalized here to include the effect of dielectric forces at the interface. The natural frequencies and stability criteria in the absence of gravity are obtained. In the inviscid case, results are compared with the known exact linear solutions of the corresponding three-dimensional problem. ~ 1997 Elsevier Science B.V. Keywords: Electrodynamics; Liquid bridge; Stability criteria; Dielectric liquid; Natural frequency 1. Introduction Recently, there has been a renewed interest in the floating zone technique under microgravity conditions, owing to its employment in high-pure monocrystal manufacturing. In this technique, a liquid bridge is formed between two solid supports. The stability of these liquid bridges is governed by the surface tension, and many papers have been dedicated to study this effect (see Ref. [1] and references therein). More recently, the effect of an electric field upon the static stability of dielectric liquid bridges has been considered both experimentally and theoretically [2-5]. Also a linear dynamic analysis of inviscid liquid bridges subjected to electric fields has been performed [6]. However, when we try to solve the dynamics of viscous liquid bridges * Corresponding author. Tel.: -I34 95 4557910. 0304-3886/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S0304-3 886(97) 00 1 54-X 260 F.J. Garcia et aL/Journal of Electrostatics 42 (1997) 259-278 on the basis of the general three-dimensional (3-D) hydrodynamic equations, great difficulties arise. Even in the absence of electric fields, only partial results are known [7-11]. In the latter case, these difficulties have been successfully circumvented using one-dimensional (I-D) models [12]. In the absence of electric fields, the linear approach shows that liquid jets 1-13] as well as cylindrical liquid bridges [7] are unstable under axisymmetric perturbations whose wavelength or height is greater than the perimeter of the undisturbed column. Experimental observations also show that the breaking process is axisymmetric E5]. This justifies considering only axisymmetric motions. The smallness of the ratio of the radius to the initial wavelength or height of the column, allows obtaining 1-D models that greatly simplify the study of these columns [12, 14, 15]. Electrical forces acting upon the polarization charges present at the interface increase the minimum initial wavelength or height below which the column is stable. Thus, more slender columns are feasible. The derivation of 1-D models is here generalized to include the effect of dielectric forces at the interface. In this work, we study liquid bridges, either inviscid or viscous, on the basis of 1-D models. In Section 2 the 1-D models are generalized to include the electric-field effects. A linear stability analysis for these models is detailed in Section 3. In Section 4 some results are obtained and discussed. In order to test the validity of the models in the presence of an axial electric field E, a comparison is made with the exact 3-D results obtained by Gonz~tlez et al. [6] for inviscid liquids. Also, some results are presented for viscous liquid bridges, for which no 3-D results are known. Finally, the main conclusions are drawn in Section 5. 2. Equations Let us consider an axisymmetric liquid bridge of height L anchored to two parallel coaxial disks, the anchors being of radius R. The liquid is assumed to be incompressible, with uniform density p and viscosity kt (see Fig. 1). The liquid bridge, confined by the surface tension a, is supposed to be in a zero-gravity environment. Quantities have been made dimensionless taking the scales: R for both the radial and axial lengths, 'Z ?" ~e ~30 Fig. 1. Schematic description of an axisymmetric liquid bridge subjected to an AC axial electric field. F.J. Garcia et al./Journal of Electrostatics 42 (1997) 259-278 261 r and z; (pR3/ff) 1/2 for the time t; [~/(pR)] 1/2 for both the radial and axial velocities, V and W; and a/R for the pressure. The bulk equations and boundary conditions for the 3-D axisymmetric problem were given by Meseguer [14]. The difficulty of such equations can be circumvented through the use of the so-called one-dimensional models. Garcia and Castellanos have recently derived a set of 1-D models in the context of liquid jets [ 15], which has been extended to liquid bridges as well [12]. These models are a good approximation to the 3-D equations as long as the liquid column is slender. The relative error introduced in neglecting small terms in the referred derivation can be expressed as a negative power of the nondimensional axial length scale 2, defined as the typical axial length of the system divided by R. Therefore, the results given by the 1-D models are expected to be good for a large value of 2. Notice that 2 does not coincide necessarily with the slenderness of the bridge, A = L/(2R), a nondimensional number which measures the length of the bridge. Instead, A is a maximum value for 2 [12]. Two kinds of 1-D models can be considered, according to their dependent variables: the mean-velocity models and the parabolic model. 2.1. Mean-velocity models The mean-velocity models have the shape of the interface r = F(z, t) and the mean axial velocity on a slice l,V(z, t) as variables. All of them must satisfy the kinematic condition (F2), + (FZJ,~)z = 0, (1) where the subscripts t and z indicate derivatives with respect to time and the axial coordinate, respectively. No approximations have been made to obtain this equation. The other equation, which is derived from the Navier-Stokes equations, is characteristic of each model. The simplest and best known I-D model for inviscid liquids is the inviscid slice model, derived by Lee [16] for jets, and extended to inviscid liquid bridges by Meseguer [14]. The relative error of this model, defined as the order of the neglected terms divided by the conserved ones of the same nature, is 22. This model has been recently generalized to viscous liquid jets [15, 17] and bridges [12], having the same relative error in the viscous terms. The viscous Lee model is given by F2(I/Vt qWWz) = - F2(g.n)z q3C(FZW~)~, (2) where C = p/(paR) 1/2 is the Ohnesorge number, which shows the ratio of viscous to capillary forces; n is the unitary vector normal to the interface, and the mean curvature V. n, which gives the capillary pressure jump, takes the form v.. - (I + v:~) 1/: 1 +/~ " 262 F.J, Garcia et al./Journal of Electrostatics 42 (1997) 259 278 A more sophisticated approach is the Cosserat model, derived by Green [18], which reads F2(I~, + WWO [1Fa(l~,z ½ -2 - - w~ + w w=)]= F2(V.n)~ + C[3(F2I~z), 1 4 - = - . -- ~ F W .... - F3Ffl~ -- ½(F3F~ + 3F2F2)W=]. (4) The relative error of this model is 2-4 in the inertial terms, but 2-2 in the viscous ones. Although it is more accurate than the Lee model for small viscosities, the error increases for moderate or large values of the Ohnesorge number. A new model, which may be called averaged model [12], corrects the mentioned deficiency, by estimating the viscous terms that were inconsistently neglected in the Cosserat model. It yields Fz(I~, + WW~) [~F4(~tz ½ -2 - - w~ + w w=)L = _ F2(V.n)z + C{3(F2I~)z + 3 [(F3F~z - 3FZF~)l~z]z}. (5) This model can be obtained by averaging the equations for the parabolic model (see below). Its relative error is 2 -4 in both the inertial and viscous terms. Thus, it improves the Lee model for any Ohnesorge number. To these equations, which are common to liquid jets, some appropriate boundary conditions must be added, accounting for the presence of rigid walls [12, 14]. The anchoring to the disks yields F(z = _A, t) = 1. (6) The impenetrability of the disks gives ff(z = +A, t) = 0. (7) Finally, the anchoring to the disks (6) can be put in terms of 1~ by means of the kinematic condition (1): ~z(Z = +A, t) = 0. (8) The mean-velocity models approximate the radial and axial velocities, within a relative error of 2-2, as follows: V(r, z, t) = - ½rl~(z, t), (9) W(r, z, t) = ff'(z, t). (10) Notice that the condition of no-slip on the disks is then fulfilled by Eq. (8). Therefore, the number of independent boundary conditions at the disks is four, the same as the differential order in z of all these models. F.J. Garcia et al./Journal of Electrostatics 42 (1997) 259-278 263 2.2. Parabolic model The parabolic model [-12] provides an improved, two-term radial approximation of the velocity field, whose relative error is 2-4: V(r, z, t) = - ½rWo~(Z, t) - ~r3W2~(z, t), ll 1) W(F, 2, t) = Wo(z , t) -}- ½rZWz(z, t). (12) Its variables are F, Wo, and W2, which do not depend on r. In terms of these, # can be readily found by averaging (12) on a slice. Three equations define this model, the first of which is the corresponding version of the kinematic condition: F, + ½FWoz + F~Wo + ~F3W2~ + ½F2F~W: = 0, (13) the other two coming from the momentum equations: (Wo, + WoWo~) - [¼ V2(Wotz -½ WL + WoWozz)L = - (V. n)~ + C [2 Wozz + 2 W2 1 2 -- gF W0~zzz FF~Wo.z~ 1 z - _ -~(F~ + FFz~)Woz~ + ¼F2W2=z + {FF~W2~ + (FZ~ + FFz~)W2] (14) and !W F4(W2t + WoW2z + 2 otzz + ½ WoWozzz) 1 4. = C(4FZWo~z + 24FF~Woz - 8F2W2 + ~F Wo .... - 4FEF~Wo~z + 3F4Wzz~ + 14FSF~W2~ + 8FZFZW2). (15) The relative error of the parabolic model is 2-4 in both the inertial and viscous terms. Some appropriate boundary conditions at the disks are necessary to solve the problem. The anchoring conditions at the edge of the rigid disks apply again: F(z = ___A, t) = 1. (16) The impenetrability of the disks implies Wo(z = +_A, t) = O, W2(z = ±A, t) = 0. (17) If the liquid is viscous, the no-slip condition on the disks must be satisfied, which gives Woz(Z = ±A, t) = O, Wz~(Z = ±A, t) = 0. (18) Owing to the kinematic condition, only eight of the conditions (16)-(18) are independent, the same as the differential order in z of the model when C 4: 0. 264 F.J. Garcia et al./Journal of Electrostatics 42 (1997) 259-278 For inviscid liquids, W2 can be decoupled from the problem, which is formulated in terms of F and Wo, and its differential order reduces to six. In this limit the liquid is allowed to slip on the disks, and the four conditions given by Eq. (18) are no longer valid. Instead, the kinematic condition (13) along with (16) give us the other two conditions necessary to complete the problem. The six conditions are then Wo(z = +_A, t) = O, Wozz(Z = +_A, t) = O, and Wo~(Z = +_A, t) -- ~ Wozzz(Z = + A, t) = O. (19) (20) Contrary to the mean-velocity models, the parabolic model conserves the sensitivity of the 3-D equations and boundary conditions to the viscous or inviscid character of the problem. In particular, the boundary conditions as well as the differential order in z are different depending on whether C = 0 or not. 2.3. Presence of an axial AC electric field Let us consider now that an AC potential difference, with an effective value ~b o, is applied to the bounding plates (electrodes) where anchors are welded on (see Fig. 1). For frequencies much higher than the inverse of the charge relaxation time, the only forces of electrical origin that are of importance are the dielectric ones. We also assume that the period of the AC field is much smaller than the typical capillary time. Therefore, no parametric resonance is expected [19]. The exact formulation of the problem for static conditions has been given in Ref. [2] and for dynamic conditions in Ref. [6]. In both cases, the electric field enters the formulation of the hydrodynamic equations only in the normal stress boundary condition. The electrical pressure has to be added to the capillary one following the replacement rule V n~V.n 4A2zA a 2 1 z • - [e(:q~, - ~ "2 - Fz~,~Oz)], (21) where ~ is the ratio of the electric permittivity to the one of the inner liquid Cln, i.e., e = 1 inside the liquid bridge and/~ = eo/~in outside; the electric field has been made dimensionless using the scale 4~o/L (so q)oR/L for the electric potential q,); A denotes the jump of a quantity through the interface; and g = e~,~ZR/(aL z) is the electric Bond number, which shows the ratio between the electrical pressure and the capillary pressure. The Maxwell equations reduce to both the divergence and curl of the electric field being zero. Introducing the electrical potential ~, it must fulfill the Laplace equation 724 = 0, (22) as well as the appropriate boundary conditions in nondimensional form. These come from the imposition of the potential difference between the electrodes, • (r, z = A, t) = A, ~(r, z = -- A, t) = - A; (23) F.J. Garcia et al./Journal of Electrostatics 42 (1997) 259-278 265 the regularity of the potential along the axis of symmetry and its asymptotic value far from it, ~b(0, z, t) finite, lim ~b(r, z, t) = z; (24) the continuity of the tangential component of the electric field through the interface, Aq~ = 0; (25) and the continuity of the normal component of the displacement field in the absence of free surface charge, A[e(- 4)~ + F~q~=)] = 0. (26) 3. Linear dynamics Gonzb, lez et al. [6] have solved the linearized 3-D problem associated with the cylindrical inviscid liquid bridge subjected to an AC electric field. A similar treatment can be applied to the 1-D models shown above. This procedure allows us to check the error of the solutions provided by such models. Furthermore, the linear 1-D problem can be solved for viscous liquids in the same manner as for inviscid ones. The same is not true for the 3-D boundary problem, which is much more difficult to solve in the viscous case, because of the additional boundary conditions at the disks [9]. The static cylindrical solution is characterized by having zero velocity, constant values of the pressure and shape of the interface, and the electric potential of a plane capacitor. Let q be the greatest initial deviation of the shape of the interface from the cylindrical one. The following perturbative solution is proposed: F = 1 + q f, I~ = ~W, Wo = qWo, W2 = ~]W2, t~ = Z + qq~, (27) where the perturbative parameter r/is very small (r/~ 1). If this solution is introduced in the above equations and terms of order q2 or higher are neglected, the resulting problem is linear. Therefore, it is expected a time dependency of the form [~(z, t), Wo(Z, t), w2(z, t),f(z, t), c~(r, z, t)] = Re {em [w(z), ~o(Z), v~2(z), f(z), ~o(r, z)] }, (28) where w, ~o, ~2,f, and ~ are complex; and (2 = ~ + ko is a complex eigenvalue, whose real and imaginary parts represent the growth factor ~ and the oscillation frequency co, respectively. Eqs. (22)-(26) lead to a linear problem for the perturbation of the electric potential ~, given by q$,, + 1 q~r + ~zz = 0, (29) g q~(r, _ A) = 0, (30) 266 F.J. Garcia et al./Journal of Electrostatics 42 (1997) 259-278 ~(0, z) finite, lim ~(r, z) = 0, r~oo Aq] = 0, a(~&) = (zx~)f', where the primes stand for total derivatives with respect to z. The most general solution of the system of Eqs. (29)-(33) is ~)(r, z) = ~ A,~,(r)sin[x,(z + A)], n-1 where ~n(r) = ~Ko(xnr)/Ko(xn) (Io(xnr)/Io(Xn) (31) (32) (33) (34) and (35) Notice that Eq. (34) couples the electric problem to the mechanical one. 3.1. Mean-velocity models Proceeding similarly, the application of Eqs. (27) and (28)to the kinematic condition (1) allows us to obtain f=-~', (38) which is valid for the Lee, Cosserat, and averaged models. The linear counterparts of Eqs. (2), (4), and (5) can be obtained by applying the same procedure to these equations, and to the definition of the capillary pressure (3) as well. Eq. (38) allows eliminating the variable f, which leads to the following linear 1-D momentum equation: e2~ Iv + c1~" + Co~ = - 2Zs'2A(~z), (39) where the coefficients Co, ca, and c2 depend on f2 and C. Their values for each model are as follows. Lee: Co = 2f22, cl = 1 - 6Cf2, c2 = 1; (40) n~ x, - 2A (37) if r~>l, if r .%< 1; (36) F.J. Garcia et al./Journal of Electrostatics 42 ¢1997) 259-278 267 Cosserat: co = 2g? 2, cl = 1 -- 6Cf2 - ¼f22, C 2 = 1 + ¼Cf2; (41) averaged: 1 2 C0 = 2(2 2, c1 1 - 6Cf2 - J2 , c,_ = 1. ~42) Finally, the four boundary conditions at the disks (7) and (8) become ~'(_+ A) = 0, w'(+A) = 0. (43) Although the evolution of a liquid bridge is an initial-value problem with respect to time, we do not consider initial conditions here. Instead, we address our interest to a modal analysis, from which a countable infinite set of eigenvalues f2m as well as their corresponding eigenfunctions are obtained. The final solution of a particular problem, characterized by its initial conditions, would be an appropriate superposition of these eigenmodes. The linear problem, given by Eqs. (29)-(34) and (38)-(43), has well-defined parity with respect to z, for the equations and boundary conditions are invariant under the change of sign of z. Therefore, the eigenmodes can be classified in either antisymmetric or symmetric, according to the parity of the shape of the interface with respect to z. Note that both the mean axial velocity and the electric potential have opposite parity to that of the shape of the interface, since applying a derivative with respect to z to any quantity changes its parity. Owing to the analogy in the treatment of both kinds of modes, the analysis presented below only deals with the antisymmetric ones. Afterwards, a recipe is provided to obtain the analogous results for the symmetric eigenmodes. Using Eq. (35) to evaluate the right-hand side of Eq. (39) yields - 2gf2A(e,~z:) = 2AzI2(Ae) ~ A,x2sin[x,(z + A)], n= 1 (44) which is the inhomogeneous term of the differential equation (39). Therefore, a general solution of this inhomogeneous problem is v~ = w h + v~ p, (45) where w h is the general solution of the homogeneous problem, i.e., Eq. (39) with Z = 0; and v} p is a particular solution of the inhomogeneous one. The general solution of the homogeneous problem is 2 ~h = ~ [s~Cj cosh(/cjz) + N'j sinh(rcjz)], (46) j=l where _+ ~cj (with j = 1, 2) are the four complex roots of the biquadratic equation C2 K:4 q'- Cl K2 qC O = O. (47) 274 F.~ Garcia et al./Journal of Electrostatics 42 (1997) 259-278 0.8 ~,l i i i ~ d ~ t i i t i i i i , J 0.6- ,~, F-t~ _ 0.2 0.4 -- "~"~..C = 0 Ac ~ - c=o.2"~-.~ , /~ o.1 0.2 -- "~-,.., .-.,'~fC= 0.2 _ co o.o ~ ~ o.o c~ -0.2 ~- C=O -0.4 -0.1 --~-C = 0.2 iO~ 6 7" iO~ 2 -0.8 r I I I I I I J i i i [ i i i i i 2 3 4 5 A Fig. 6. Oscillation frequency (dashed) and growth factor (solid lines) of the first mode versus the slenderness, for ,8 = 0.58 and X = 2.4, as given by the averaged model. The labels on the curves show the Ohnesorge number (C = 0 and C = 0.2). In order to show the effect of viscosity in the presence of an electric field, a plot of the first eigenvalue (m = 1) versus A is presented in Fig. 6, for inviscid (C = 0) and viscous (C = 0.2) liquids. These curves have been computed with the averaged model, for/~ = 0.58 and ;( = 2.4. Observe that the viscosity has no influence on the stability of the bridge, since the critical slenderness Ac is not affected by the value of C. However, the dynamics changes dramatically. When the liquid is viscous, the stable zone of the mode (A < Ac) separates into two zones of different behavior. For short enough bridges, there is a zone of damped oscillations (c~ < 0, co ~ 0). Greater values of A lead to lower-frequency oscillations, until a second zone of pure damping (c~ < 0, co = 0) is reached. The more viscous the liquid is, the shorter the liquid bridge must be to see oscillations. Finally, in the unstable zone (A > Ac), the changes introduced by the viscosity are only quantitative: the growth factor ~ decreases as C increases. In the above-cited experimental configuration [5] the viscosity of the liquid leads to a value of the Ohnesorge number C = 4.6. H~reafter, we will restrict our discussion to that value. When such a bridge is stable, it 'is observed in the experiments that all disturbances are aperiodically damped. Besides, the breaking is appreciably slowed down by the effect of viscosity. No oscillations have been observed in any case. In Fig. 7 the growth factor of the first antisymmetric mode is plotted versus A for C = 4.6,/~ = 0.58, g = 0 and 2.4, as given by each 1-D model. Note that, as expected, the stability limits are the same as in the inviscid case. For the studied range of slenderness, the eigenvalue is real, which means that this mode does not oscillate. When the bridge is stable, all disturbances are aperiodically damped. When it is unstable, the predicted growth factor is real, but significantly smaller than for inviscid liquids. This explains the slowing down that we have observed in the breaking of liquid bridges in our laboratory, using the experimental setup described in Ref. [5]. Notice that the results given by the different models are close, the differences giving an estimate of the errors. For our viscous liquid bridge, the results of the Lee model are F.J. Garcia et al./'Journal o/Electrostatics 42 (1997) 259 278 275 0.04 0.02 0.00 (3I, -0.02 -0.04 -0.06 ........ ~-~ I -0.08 , i _ _ 2 3 10 W¢" [ .......... ,00 0I 0.015 - - -- i 0.010 4.3 4.5 4.7 F ..... T-- 4 5 6 7 8 9 A Fig. 7. Growth factor of the first antisymmetric mode as a function of the slenderness, for C = 4.6, # = 0.58, Z = 0 and Z = 2.4. Given by the Lee and averaged (short dashed), Cosserat (point-dashed), and parabolic (solid lines) models. A detail is magnified. f 0.8 0.4 0.0 i i -I .0 -0.5 0.0 z/A Fig. 8. Shape of a half of the interface of the first antisymmetric mode (m = 11 versus z'A. for .4 = 10, # = 0.58, and Z = 0, t, ..., 6 as given by the parabolic model for C = 4.6. the same as the ones of the averaged model, within the plot accuracy. For large C without electric field [12], the 3-D solution is between the averaged and parabolic ones when A > Ac. The Cosserat model clearly underestimates the correct values of~, owing to the inconsistency in the derivation of its viscous terms [15]. Although the 3-D solution is not available for Z -¢ 0, a detail of the 1-D predictions is included in the figure, which shows similar relative positions of the curves. The differences give us an estimate of the error of the 1-D approximations. The effect of the electric field on the shape of the interface for viscous liquids is shown in Fig. 8. In the absence of electric field, the effect of viscosity is to increase the 276 F.J. Garcia et al./Journal of Electrostatics 42 (1997) 259 278 typical axial length 21-12]. For C = 4.6, it is evident that 2 -- A. In this case, increasing Z cannot affect 2. Instead, the main effect of the electric field on the bridge shape is that the positions of the greatest and smallest cross sections are closer to the disks. This is also observable for any value of C as long as 2 -~ A. 5. Conclusions A set of new 1-D models for viscous jets, also extended to liquid bridges, has been generalized to include the effect of dielectric forces. A linear stability analysis based on these models has been performed. In the inviscid case, a quantitative comparison of the 1-D results with the 3-D ones show the relationship between the error of such models and the typical axial length 2. For slender enough bridges, 2 increases with the electric field, and the error decreases accordingly. In the viscous case in the presence of electric fields, no 3-D solutions are available. Similarly to the case )~ = 0, the linear results obtained with the 1-D models predict that the viscosity effect does not change the stability criteria. On the contrary, the dynamics is greatly affected, as viscosity inhibits the oscillations of stable liquid bridges and slows down the rupture of unstable ones, in qualitative agreement with the experiments done in our laboratory. It can be concluded from the comparison with the known exact 3-D linear solutions, that the 1-D models are quite adequate to deal with the first stages of development of perturbations. In general, the main aspects concerning the relative error of these models without electric field remain valid in its presence. The accuracy of these models improves in the presence of an AC axial electric field, since the latter makes the typical axial length increase. Besides, they allow us to study the dynamics of viscous liquid bridges subjected to axial electric fields, which for the moment has defied a 3-D approach. Experiments are now under way to compare the growth rate of the instability in the linear stage with the predictions made by these 1-D models. Acknowledgements The authors are grateful to Angel Sanz and Francisco Medina for helpful discussions. This work has been supported by the Spanish Direccidn General Interministerial de Ciencia y Tecnologia under contract PB93-1182. Nomenclature ~',, d, sgj, ~, Co, Cl, c2, fin, h,, ~,, x,, ~c, Kj f, wo, auxiliary constants, coefficients, and functions linear counterparts of F, I/P, Wo, W2, spatial dependence off, ~, Wo, We, q5 F.J. Garcia et al./Journal of Electrostatics 42 (1997) 259-278 277 C F L m n V-n r R t V W Wo, W2 2 6Q A ~;in ~;0 17 2 ~Lmax A Ac P O" ~b qo 0 Z u9 O, Qm Ohnesorge number (p/(paR) ~/z) nondimensional shape of the interface length of the liquid bridge index of modes vector normal to the interface nondimensional capillary pressure jump nondimensional radial coordinate radius of the anchors of the liquid bridge nondimensional time nondimensional radial velocity nondimensional axial velocity nondimensional mean axial velocity nondimensional velocity variables in the parabolic model nondimensional axial coordinate nondimensional growth factor nondimensional permittivity of the outer zone (eo/~i,) relative error of the eigenvalue f2 applied to a quantity, jump across the interface nondimensional permittivity permittivity of the liquid permittivity of the outer zone small amplitude of perturbations typical nondimensional axial length nondimensional wavelength of maximum growth in a jet slenderness (L/2R) critical value of the slenderness dynamic viscosity of the liquid density of the liquid surface tension of the interface nondimensional electric potential potential difference between electrodes electric bond number (Sin~2R/aL 2) nondimensional oscillation frequency complex eigenvalue (~ + ira) References [1] J.M. Perales, J. Meseguer, Theoretical and experimental study of the vibration of axisymmetric viscous liquid bridges, Phys. Fluids A 4 (6) (1992) 1110 1130, [2] H. Gonz/tlez, F.M.J. McCluskey, A. Castellanos, A. Barrero, Stabilization of dielectric liquid bridges by electric fields in the absence of gravity, J. Fluid Mech. 206 (1989) 545-561. 1-33 A. Ramos, A. Castellanos, Shapes and stability of liquid bridges subjected to AC electric fields, J. Electrostatics 26 (1990) 143-156. 1-43 A. Ramos, A. Castellanos, Bifurcation diagrams of axisymmetric liquid bridges of arbitrary volume in electric and gravitational axial fields, J. Fluid Mech. 249 (1993) 207-225. 278 F..~ Garcla et aL/Journal of Electrostatics 42 (1997) 259 278 [5] A. Ramos, H. Gonzfilez, A. Castellanos, Experiments on dielectric liquid bridges subjected to axial electric fields, Phys. Fluids 6 (9) (1994) 3206-3208. [6] H. Gonz~lez, A. Castellanos, F.M. McCluskey, A. Gafi~in, Small oscillations of liquid bridges subjected to AC fields, in: Synergetics, Order and Chaos, World Scientific, Singapore, 1988. [7] A. Sanz, The influence of the outer bath in the dynamics of axisymmetric liquid bridges, J. Fluid Mech. 156(1985) 101 140. [8] A. Borkar, J. Tsamopoulos, Boundary-layer analysis of the dynamics of axisymmetric capillary bridges, Phys. Fluids A 3 (12) (1991) 2866--2874. [9] J. Tsamopoulos, T.Y. Chen, A. Borkar, Viscous oscillations of capillary bridges, J. Fluid Mech. 235 (1992) 579-609. [10] M. Higuera, J.A. Nicolfis, J.M. Vega, Linear oscillations of weakly dissipative axisymmetric liquid bridges, Phys. Fluids 6 (2) (1994) 438 450. [11] J.A. Nicolfis, Hydrodynamic stability of high-viscosity cylindrical liquid bridges, Phys. Fluids A 4 (8t (1992) 1620-1626. [12] F. Javier Garcia, A. Castellanos, One-dimensional models for slender axisymmetric viscous liquid bridges, Phys. Fluids 8 (11) (1996) 2837 2846. [13] Lord Rayleigh, On the instability of jets, Proc. Lond. Math. Soc. 10 (1878) 4-13. [ 14] J. Meseguer, The breaking of axisymmetric slender liquid bridges, J. Fluid Mech. 130 (1983) 123 151. [15] F. Javier Garcia, A. Castellanos, One-dimensional models for slender axisymmetric viscous liquid jets, Phys. Fluids 6 (8) (1994) 2676 2689. [16] H.C. Lee, Drop formation in a liquid jet, IBM J. Res. Develop. 18 (1974) 364--369. [17] J. Eggers, T.F. Dupont, Drop formation in a one-dimensional approximation of the Navier-Stokes equation, J. Fluid Mech. 262 (1994) 205-221. [18] A.E. Green, On the nonlinear behavior of fluid jets, Int. J. Eng. Sci. 14 (1976) 49-63. [19] H. Gonz~ilez, A. Ramos, A. Castellanos, Parametric instability of dielectric, slightly viscous liquid jets under AC electric fields, Phys. Fluids 9 (6) (1997) 1830-1837. [20] J.A.F. Plateau, Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity, Ann. Rep. Smithsonian Institution, 1863, pp. 207-285. [21] N.K. Nayyar, G.S. Murty, The stability of a dielectric liquid jet in the presence of a longitudinal electric field, Proc. Phys. Soc. Lond. 75 (1960) 369-373.