Bifurcation diagrams of axisymmetric liquid bridges subjected to axial electricfieldsAntonio Ramos, Heliodoro González, and Antonio Castellanos Citation: Physics of Fluids (1994-present) 6, 3580 (1994); doi: 10.1063/1.868416 View online: http://dx.doi.org/10.1063/1.868416 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/6/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Linear oscillations and stability of a liquid bridge in an axial electric field Phys. Fluids 13, 3564 (2001); 10.1063/1.1416183 Experiments on dielectric liquid bridges subjected to axial electric fields Phys. Fluids 6, 3206 (1994); 10.1063/1.868103 Experiments on the stability of a liquid bridge in an axial electric field Phys. Fluids A 5, 1081 (1993); 10.1063/1.858625 The effect of an axial electric field on the stability of a rotating dielectric cylindrical liquid bridge Phys. Fluids A 2, 2069 (1990); 10.1063/1.857792 Sensitivity of liquid bridges subject to axial residual acceleration Phys. Fluids A 2, 1966 (1990); 10.1063/1.857672

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Bifurcation diagrams of axisymmetric liquid bridges subjected to axial electric fields

Antonio Ramos Departamento de Electronica y Electromagn.etismo, Universidad de Sevilla, Sevilla 41012, Spain

Heliodoro Gonzalez Departamento de Electronica y Electromagnetismo, Universidad de Sevilla, and Departamento de Fisica Aplicada Universidad de Sevilla, Sevilla 41012, Spain

Antonio Castellanos Departamento de Electronica y Electromagnetismo Universidad de Sevilla, Sevilla 41012, Spain

(Received 7 March 1994; accepted 2 June 1994)

The stability of dielectric liquid bridges between plane parallel electrodes when an electric potential difference is applied between them is studied for an axisymmetric configuration regarding arbitrary volume, axial gravity, and unequal coaxial anchoring disks attached to the electrodes. The stability is determined from the bifurcation diagrams related to the static problem. 1\\'0 mathematical approaches are presented which are different in scope. First, the Lyapunov-Schmidt projection technique is applied to give the liquid bridge bifurcation diagrams for the bridge considered as an imperfect cylindrical one. The imperfection parameters, i.e., the relative difference of radii to the mean diameter, the deviation from the cylindrical volume, and the gravitational Bond number, are assumed to be small. Second, a Galerkin/finite element technique is used to obtain numerically bifurcation diagrams for arbitrary values of all the parameters. Agreement between both methods is good for small enough values of the imperfection parameters. The effect of the polarization charges existing at the free surface is highlighted. As in the absence of applied electric field, the gravitational Bond number and the relative difference of radii separately decrease the stability of the liquid column, but both effects conveniently combined may cancel out.

I. INTRODUCTION

The stability of liquid bridges is well established in a general configuration regarding arbitrary volume, axial ac­celeration of the reference system and different anchoring disks (Coriell, Hardy, and Cordes,l Martinez,2 Meseguer,3

Meseguer et al.,4 Martinez,s Meseguer, Sanz, and Perales,6 Slobozhanin and Perales7

). On the other hand, the effect of electromagnetic forces on the stability and dynamics of liq­uid columns, conducting or nonconducting, has been inves­tigated by many authors, owing to a great variety of practical applications, like ink jet printing or crucible-free materials processing techniques based on the floating-zone method. In this field some attention has recently been given to poorly conducting liquid jets (Saville8,9) and liquid bridges (Sanka­ran and Saville10) or nonconducting liquid bridges (Gonziilez et alY), which are shown to be strongly stabilized by axial electric fields. Recently, the competition between dielectric forces versus gravity forces in the stabilization of the latter case has also been studied (Gonzalez and Castellanos12 and Ramos and Castellanos13

). The effect still not considered in this context is that of unequal anchoring disks. Its role is to some extent analogous to that of axial gravity, and experi­mentally it is easier to control the anchoring disks than the Bond number. Depending on the orientation of the gravity field with respect to the anchoring disks, both destabilizing influences must be added or, on the contrary, cancel out. For bridges of cylindrical volume, small differences in radii would generate a pressure field term proportional to the axial coordinate in a basic state defined by a truncated cone shape, which is also the dependence induced by axial acceleration.

In this work we discuss how an axial electric field affects this behavior. We restrict our study to axisymmetric modes of bifurcation, so we are considering the minimum volume sta­bility curve. Nonaxisymmetric modes of instability are of importance when we are studying the maximum volume sta­bility curve (Slobozhanin and Perales 7) or the C mode in rotating columns (Brown and Scriven14 and Vega and Perales15

). TYpically, the process of elongating a liquid col­umn with the aid of electric field lead to the minimum vol­ume stability curve.

Strong DC electric fields lead in general to injection from at least one of the electrodes and, consequently, to the existence of space charge density and current at the bulk of the liquids. An additional current also exists due to dissocia­tion of the impurities. Thus, in order to avoid phenomena due to residual conduction and space charge in the system, it is necessary to apply AC fields of period much shorter than the charge relaxation times of both liquids from a practical point of view. If in addition the typical mechanical time is much greater than the period of the applied field, the system only "feels" its RMS value and a model with DC fields is appli­cable. The experiments described in Gonzalez et al. 11 satisfy these conditions.

The organization of the article is as follows_ In Sec. II the physical system under investigation is presented, as well as the governing equations and boundary conditions giving its mathematical description. Section III is devoted to the local bifurcation analysis and the determination of the ana­lytical expression for the critical electric field as a function of the three imperfection parameters. Section IV describes the

3580 Phys. Fluids 6 (11), November 1994 1070-6631/94/6(11 )/3580/11 /$6.00 © 1994 American Institute of Physics

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FIG. 1. Schematic of the liquid bridge.

numerical method used to study the bifurcation diagrams for arbitrary values of all the parameters. Results and compari­son between both approaches is made in Sec. V. Finally the main conclusions are drawn in Sec. VI.

II. FORMULATION OF THE PROBLEM

The most general configuration to be investigated con­sists in a liquid bridge placed between two plane parallel electrodes a distance L apart and subjected to a DC potential difference. The bridge is anchored to the electrodes at two circular coaxial regions of different radii Rl and R 2 • Anchor­age may be achieved with the help of rings of negligible height glued to the electrodes. The contact lines are thus fixed and the contact angles are free to some extent (see Fig. 1). The electrodes are wide enough to disregard any border effect. The bridge, of density pi, may be surrounded by an­other immiscible liquid of density pO. Both are assumed in­compressible and perfect insulators. The forces acting upon the bridge are due, in the bulk, to axial gravity and, at the interface, to capillarity and jump in the dielectric constant. Electrostrictive effects do not influence the statics nor the dynamics of the system owing to the incompressibility of the liquids and are not considered.

Static configurations must satisfy the pressure equilib­rium

p-pgz=I1""'const, (1)

where p stands for the pressure (modified by the electrostric­tive effect of the electric field), z is the axial coordinate, and II takes two different values for each medium.

For the electric problem we only need to consider the electric potential <I> in both media. Axisymmetric shapes are described by

F(r,z)==r- f(z)=O. (2)

The governing set of partial nonlinear differential equations is quite similar to that presented in Gonzalez and Castellanos12 and Ramos and Castellanos:13

(3)

x ~ X 2 ilII+Bz+ 2: il[e(V<I>Y]-IVFI2 il[e(VF·V<I» ]

+V·n=O at r=f, (4)

Phys. Fluids, Vol. 6, No. 11, November 1994

1 fA ., r= 'JA dz f-,

~ ~A

along with the boundary conditions

f(±A)=l±H,

<I>(r,A)=A, <I>(r,-A)=-A,

J<I> -=0 at r=O, ar lim <I>=z, r-+ OO

il <I> = 0 at r = f,

VF·il(eV<I»=O at r=f,

(5)

(6)

(7)

(8)

(9)

(10)

(11)

where n is the outward normal vector to the interface defined as n==VFIIVFI evaluated at F=O and the notation il means increment of any quantity across the interface, also in the outward direction. These equations have been made nondi­mensional by scaling lengths with the mean radius R==(RI +R2)/2, the electric field with its value at r-+oo,

E 00 = <I> oiL, which imposes <l>oRl L as the scale for the elec­tric potential, the pressure with the capillary jump across the interface, aiR, and permittivities with that of the inner medium, e i. Six non dimensional parameters appear in the above equations:

• the slenderness, A==LI2R; • the volume 1', relative to 71"R2 L (for cylinders, r= 1); • the gravitational Bond number B==(pi_ po)gR210",

where 0" is the surface tension between both liquids;

• the relative permittivity of the outer to the inner me­dium, {3==eO/ei . Notice that the nondimensional permittivity is in general written as e, but {3 stands for the outer medium

and 1 for the inner one, i.e., ile= {3-1;

• the electric Bond number, x==ei<l>6RIO"L 2; and • the relative difference of radii to the mean diameter

III. PERTURBATIVE ANALYSIS FOR ALMOST CYLINDRICAL BRIDGES

A. Description of the method

Bifurcation diagrams in the special case of a liquid bridge of nondimensional volume close to r= 1 may be ob­tained from a perturbative analysis starting from the cylin­drical basic solution, by considering gravity forces, deviation from cylindrical volume, r==r-l, and difference in radii as small imperfections. Values of order unity for B, T, or H come out of its range of validity. In spite of this restriction, the analysis is useful to determine the role of each parameter in the stability of the bridge. For bridges of equal radii and cylindrical volume the method has been already applied (Gonzalez and Castellanos12

). Here we will shortly desc"ribe the essential features and how the method is adapted to ac­count for unequal anchoring disks and noncylindrical vol­umes.

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Let us introduce for convenience in this section a new expression for the free surface, given by

Hz f(z) = f*(z)+ A'

that transforms (6) in

f*(±A)=l.

(12)

(13)

The transformation will be useful to stress the twofold effect of unequal anchoring radii: (i) the appearance of a z-dependent pressure of capillary origin, similar to the gravi­tational pressure term, Bz, at the Young-Laplace equation; and (ii) the existence of a uniform polarization charge den­sity at the interface from the boundary condition (11). Both circumstances have opposite influence on stability.

The system can be expressed formally as jT(X,17)=O, where jT is an operator acting on a functional space ..%'1 whose elements are

x= [<pi(r ,z), <P°Cr ,z) ,f* (z) ,Lln]. (14)

The components of this vector are subjected to the above­stated boundary conditions. The symbol 17 stands for any parametrical dependence of the operator that we might em­phasize.

For B = H = r= 0 the cylindrical bridge with uniform electric field is a possible solution for arbitrary values of /3, A, and x:

Xc =[z,z,l, -1- hC/3-1)]. (15)

Other solutions close to this one are x=xc+eXo, where e is a small amplitude of deformation and

Xo = [<P~(r ,z), <Pg(r,z) ,foCz), Ll no]. (16)

The linear part of jT at xc, is Axo=O, with

_V2 0 0 0

0 _V2 0 0

a a2 A= - (2AXoLlE)- ° l+azz -1

. az

0 0 -(~) J~A dz· 0

(17)

where J'\ dz· means integration of a given function of z over the interval [-A,AJ. Now Xo must satisfy the linearized boundary conditions

<Po(r,±A)=O,

<po(O,z)=O(I), lim <Po(r,z)=O,

Ll<Po=O,

( a<po) afo Ll E - -LlE -=0

. ar az'

fo(±A)=O.

3582 Phys. Fluids, Vol. 6, No. 11, November 1994

(18)

(19)

(20)

(21)

(22)

The bifurcation points defined in the space of parameters {/3,A,X} by this problem may be found in Gonzalez and Castellanos,12 as well as the associated eigenfunction xo.

Let us fix f3 and A, so that the symbol 17 represents X=X-Xo, B, r, and H. We look for solution sets (x,x,B,H,r) of jT(x,x,B ,H,r)=O which are a local extension of the bifurcation set (xc ,0,0,0,0). With a natural definition of an inner product also given in the above-mentioned work, the operator A, of Fredholm type, is self-adjoint and has a one-dimensional kernel whose elements are proportional to xo. If we perform the decomposition x=xc +eXo+x1 and .'Y" (X,17) into its linear and nonlinear parts, the Lyapunov­Schmidt technique (see for example Myshkis et at. 16) allows us to rewrite the original problem as

.iJT(Xc + xl.. + eXo, 17) + !{Ie e, 17)Xo= 0,

!{ICe, 17) =0,

(23)

(24)

with !{I a scalar function. The last equation is usually called the bifurcation equation, giving the relation between the con­trol parameter X, the amplitude of deformation of the bifur­cated family e, and the imperfection parameters B, H, and r. By expanding all the functions with respect to these five parameters,

i,j,k,l,m

(25)

we arrive at an infinite set of linear nonhomogeneous recur­sive problems:

'Pijklm(r, ±A) = 0,

(26)

(27)

'Pijklm(O,Z)=O(l), lim 'Pijklm(r,z)=O, (28) r-+ OO

Ll 'Pijklm= hlNlm(Z) ,

Ll( a'Pijklm) -Ll aUijklm =h(2) ( , E ar E az ijklm z),

( d2 ) ( fJ'Pijklm)

.1 + d? Uijklm- 2AXOLl E ---a;- ,

- 'Tfijklm -!{Iijklmfo(z) = h}fklm(Z),

Uijklm(±A)=O,

(29)

(30)

(31)

(32)

(33)

where X.L ,ijklm= ('P~jklm' 'Pljklm ,Uijklm ,LlP ijklm) and the non­homogeneous terms h~jklm(Z), S = 1,2,3,4 come from the ex­pansion of the nonlinear part of the operator and boundary conditions. These latter may be found in Gonzalez and Castellanos12 in the case 1= m = 0, corresponding to a liquid bridge with equal radii and cylindrical volume.

It may be demonstrated that the coefficients !{Iijklm have the form

Ramos, Gonzalez, and Castellanos

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./. -JAd{ihl3l -2A [minh (2) (di o 'l'ijkllll- -A Z ° /jkllll Xo \1'0 /jkllll + dz

_~ (1) ~ffiin) ]} ar hijkllll . (34)

From some symmetry properties verified by the extended nonlinear equation (23) we can predict which of these coef­ficients are nonzero. The resulting bifurcation equation (24) to the lowest order is

e( I/IllOOOX+ 1/101001 r+ I/I03000e2) + I/I00100B + I/I00010H = O. (35)

At this order of approximation the linear system needs to be solved only for (ijklm) =(02000), which is done using the so-called method of lines.

B. Comparison between the roles of Hand B

Let uS consider first the case r= 1. Equation (35) is analogous to the bifurcation equation discussed in the afore­mentioned work. For B = H = 0 it represents a pitchfork dia­gram in the (x,e) plane and nonzero values break it into two isolated branches of solutions, one of which exhibits a turn­ing point. This latter branch describes stable equilibria up to the turning point, where a change in stability takes place according to the bifurcation theory of nonlinear differential equations (Iooss and Joseph17). The location of the turning point determines the new stability criterion. All these fea­tures are given qualitatively in any of the bifurcation dia­grams presented in Fig. 2. The pitchfork diagrams, corre­sponding to B = H = 0 are plotted with the help of dashed lines. For nonzero imperfection parameters the resulting iso­lated branches are represented by solid lines if stable and by dotted lines if unstable. Further comments about this figure will be made later.

The dependence in H enters in the same way as B, the other imperfection parameter. Let us rewrite the last two terms as

I/I00100B + I/IooOlOH = 1/100100 ( B - a ~), a= - A ifJo0010.

1/100100

(36)

where a depends in general on A and /3. Comparison of nonhomogeneous terms for the orders (00100) and (00010) gives

h (l) ~~O h(2l -0 h(3) -z h(4) -0 00100-' 00100-' 00100- , 00100-,

( 2) _ Lll: h(l) -0 h

00010-' 00010- A'

(37)

h (4) -0 00010- .

(38)

We thus conclude that the effect of the polarization charge at the interface, represented by the function h~~OlO' does not allow us to consider the combination H / A as a parameter equivalent to B. If this function were zero, we would obtain a = 1 and the combination H / A could be called "effective Bond number." The function a (A) is presented in Fig. 3 for

Phys. Fluids, Vol. 6, No. 11, November 1994

€

(a)

€

(b)

/.' ~,"

/: / :

{ "--- -I - - -=--=-~=-==--=~-=-

. ...... \ Xo X

'. \ ".\

'.:-,.

./ /

...­...-" ,/

Xo (-. - ~.~ ~~~~~~-

'\ X "

"-. ........

........ .....

_v .. ;.. -",...:.-

€ •. ~./ A3 ,"/

(c)

.......... / - - - -X-;;', -r-=-=X ,'. ". " . .:....::.­

~

FIG. 2. Qualitative behavior of the local bifurcation diagrams in the (s,x) plane for fixed /3, B, H, and A when this latter parameter increases (A[<Az<A3). The solid (dotted) lines are stable (unstable) equilibria. Dashed lines correspond in each case to the bifurcation diagram without imperfection (pitchfork type) and XO is the critical electric Bond number for a perfectly cylindrical bridge. Before a certain "compensation slenderness," Acomp, (diagram (a)], the critical field, whose location is determined by the turning point, is greater than Xo. For A2"" Acomp (diagram (b)] the critical field approaches to Xo and coincides exactly with it at Acomp. Beyond this value [diagram (c)], the critical field increases again but the amphoric shape of the interface is inverted.

FIG. 3, The coefficient a as a function of the slenderness for several values of the permittivity ratio.

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0.1

8=0.1 H=0.314

0.05 8=0.02

B-aH/A H=0.1

0

-0.05

-0.1 4 6

A

FIG. 4. TWo examples of dependence with the slenderness of the combina­tion B -a(f3,A)HIA, which determines the global importance of both im­perfection parameters. The combination is zero for a certain slenderness if B and H have equal sign.

some representative values of (3. The dielectric forces have thus influence upon: (i) the value of the critical field XO for which a perfectly cylindrical liquid bridge has marginal sta­bility, and (ii) the relative importance between the imperfec­tion parameters Band H / A, measured by a. Increasing A causes the coefficient a to diminish and the effect of H/A, always destabilizing, is less important. For A>7I' the coeffi­cient a becomes lower than 1, i.e., below the value given by a purely hydrostatic phenomenon. From this figure we can determine the ratio B/H as a function of A that produces full compensation (within our first-order approximation to the bifurcation equation). Conversely, the individual effects are added if Band H have opposite sign. In Fig. 4 we present the combination B-a((3,A)H/A vs A for two given values of Band H. Deviation from zero determines the importance of imperfections in the bifurcation equation, as a function of the slenderness. The global imperfection parameter is zero for some A, tend to B as A increases, and becomes very impor­tant as the slenderness decreases. Coming back to Fig. 2 we can observe the qualitative evolution of the bifurcation dia­grams when all parameters but the slenderness are fixed. In­creasing A we typically find several effects, namely, a shift to the right of the bifurcation point Xo and an increase in the curvature of the parabola, as stated in Gonzalez and Castellanos.12 The new feature is the existence of the above­mentioned "compensation slenderness" (close to A2 in Fig. 2) for which the shape of the bridge adopts exactly the form of a truncated cone. Beyond this slenderness the deformation changes sign.

The expression giving the critical electric Bond number as a function of the remaining parameters is

xc= Xo+ a(A,(3)[B-a(A,(3)H/A]2/3,

(39) 3 ( ", ,,;2 ) 1/3 a(A,f3) = - -::)2T'J '1'03000'1'00100 ,

2 1/111000 .

which is obtained from the bifurcation equation by consider­ing the location of the turning point in the plane (e,x), as discussed above. Representation of some stability curves in the plane (X-A) will be done in the next section for the sake

3584 Phys. Fluids, Vol. 6, No. 11, November 1994

TABLE I. Comparison between three different predictions for the critical slenderness of bridges of cylindrical volume in the absence of gravity forces and for different_values of H. Ac,l' numerical (Martinez2

); Ac,2, given by (40) (Mesegueil); Ac,3' from the condition Xc=O (see text).

RzlR I 0.97 0.9 0.8

H 0.Q15 0.053 0.11

Ac.1 2.98 2.79 2.56

Ac.2 2.99 2.80 2.58

Ac,3 2.99 2.82 2.64

of comparison with numerical data. Another accuracy test comes from the case of zero applied field and B =0 for which there are available data. In particular, Martinez2 gives stabil­ity curves of bridges with arbitrary volume and anchoring radii, and Meseguer3 gives analytical expressions for the critical slenderness of bridges of cylindrical volume:

A . =7I'-(3/2)4/3(H/7I')2/3 c,2 , (40)

where we have introduced our definition for H in his original expression. In Table I we present comparison of our critical slenderness, labeled A~,3' for a bridge having r=1, B =0,

and X=O with that calculated by both authors, called ACo1

and Ac,2, respectively. It is apparent that data calculated from Bq. (40) is closer to the results given by Martinez than ours, but the latter are acceptable up to H =0.1 (the error is 3%). It was shown by Gonzalez and Castellanos12 that in the case H =0, B *0 the bifurcation equation (35) in the space (e-x) may be used to give the bifurcation equation in the space (e-A). The diagrams are similar in both parameter spaces and we also obtain a critical slenderness as a function of the electrical parameter X from the location of a turning point. The reciprocal case B =0, H *0 is obviously obtained substi­tuting B by H/A, giving exactly (40). Unfortunately, we find the greatest accuracy from the diagram (e-x) for H=O and, as stated above, from the diagram (e-A) for B =0. Thus, for the general case with Hand B nonzero, neither of the pos­sibilities is preferable to the other.

C. Deviation from the cylindrical volume

Once we have studied the role of Band H in the stability of bridges of cylindrical volume we will consider the case B = H = 0 and 740. The nonhomogeneous problem of order (00001), i.e., that arising from the collection of all terms proportional to T in the expansion of the nonlinear extended problem, gives

h (l) -h(2l -h(3) -0 h(4) -1 00001 - 00001 - 00001 -, 00001 - . (41)

Substitution in (34) gives 1/100001=0. The nonhomogeneous problem has the following solution:

1 ( "" U00001(Z)=:Z 1-CL

n=2

. even

(42)

Ramos, Gonzalez, and Castellanos

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~l

0.9

~

0.55 10

;-.-- 2

,6'-0.1

3 4 A 5 6 7

FIG. 5. The coefficient K as a function of the slenderness for several values of the permittivity ratio.

even

(43)

7TOOOOl = Kl +c!2), (44)

where xn"""n7T!2A, In' and Kn are the modified Bessel func­tions of order n and first and second kind, respectively, Ho ({3,xll ) """-{3Kl (xn) /KO(xn)-!1(xn)/Io(xn), qn"""l -x~+)(o(D.€)2Xll/Ho({3,Xll)' and c"""1!2:.;=lq;,l. In (43) the upper (lower) radial dependence corresponds to the inner (outer) region. These expressions when multiplied by T de­scribe the shape, electric potential, and pressure deviations from the basic cylindrical solution due to a small excess or defect in relation to the cylindrical volume. Notice that the shape is symmetric with respect to the equatorial plane and the potential is antisymmetric.

The next coefficient in the expansion of 1/1 is 1/101001' as stated in (35), which is nonzero. The corresponding functions involved in this order are omitted, as well as the final analytical expression for the integral (34), and may be pro­vided by the authors to interested readers upon request.

The influence of T upon the stability of the bridge is to shift the critical electric field according to the following cri­terion:

K({3,A)= 1/101001. 1/111000

(45)

The factor K is represented in Fig. 5 as a function of A for several values of {3. The graphic has vertical logarithmic scale because the function has a strong dependence on {3, increasing considerably as {3-1 is small. For all tested val­ues of the permittivity ratio we found K>O. This implies that for a bridge with volume greater than that of a cylinder the critical electric field decreases and, conversely, removal of liquid from a marginally stable cylindrical zone leads to de­stabilization.

Phys. Fluids, Vol. 6, No. 11, November 1994

Again, (35) may be used to obtain the bifurcation dia­grams in the plane 8-A, instead of 8-)(. Putting )(=0 the result given by Meseguer3 is recovered:

(46)

IV. NUMERICAL ANALYSIS

The finite element method used here follows closely that derived in Ramos and Castellanos.13 In order to reduce the number of unknowns and to have numerical solutions that more exactly account for the field far from the axis we have included an analytical solution that matches the finite ele­ment approximation at a distance r=Roo from the axis, fol­lowing Orr et al. 18 The matching condition at r = Roo is that the potential and its normal derivative are continuous.

The liquid bridge profile is tesselated into curve seg­ments and the inner and outer media into quadrilaterals be­tween spines z=const (Saito and Scrivenl9

). The grid pitch is finer near the contact lines and increases in geometric pro­gression with increasing distance from them in the radial and axial directions. The quadrilaterals close to the contact lines have been divided into triangles. The introduction of these triangular elements eliminates some numerical oscillations of the electrical pressure close to the anchoring lines that ap­peared in a pure quadrilateral mesh (Ramos and Cas­tellanos13

). The quadrilateral and triangular elements are iso­parametric biquadratic of nine nodes or isoparametric qua­dratic of six nodes, respectively (Strang and Fix2o). The one­dimensional elements are quadratic and constitute the edge of the previous bidimensional elements.

With these assumptions the interface and the potential are represented, respectively, by

N

f(z)=~ fiVi[Z«;)], (47) ;=1

M

<P(r,z)=~ <P i wi[r(g,1]),z(g,1])] for r~Roo, (48) i=1

where g and 1] stand for the local coordinates of the isopara­metric transformation, v i and Wi represent the one­dimensional and two-dimensional basis functions, respec­tively, and fi and <Pi are the values of these basis functions at the nodes.

The far field solution for r";3R", is a series given by the classical separation of variables technique applied to the Laplace equation:

<Pco=Z+ ~ Cft sin[xn(z+A)]Ko(xnr), (49) ,,=1

where xI! = n7T/2A, Ko is the modified Bessel function of second kind and order zero, and en is given by using the matching condition of continuous potential

(50)

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where <I> is the finite element expansion evaluated at the matching boundary.

The Galerkin weighted residuals of the augmented Young-Laplace equation and the Laplace equation are formed by weighting (4) by each fCZ)ViCZ) and (3) by each wi(r,z). Using the divergence theorem and boundary condi­tions to eliminate second-order derivatives we arrive at the following set of algebraic equations for fi and <I>i:

JA (r.;-;-;.z ffzvzi . yl + fzVi+ ~+(All + XAllE

-A yl+ fz

+BZ)fVi)dZ=O, i=I, ... ,N,

f V<I>,VWi d7+f3J V<I>,VWi d7 Vi Va

f a <I> 00

-f3 Wi -a- dS=O, i=I, ... ,M, r=R., n

(51)

(52)

where SUbscript z denotes differentiation with respect to z, llE stands for the electrical pressure, E(E;- E~)/2, and Vi' Vo are the inner and outer volumes. In (52) we have made use of the matching condition of continuous normal deriva­tive of the potential at r =Roo. More specifically, the last term in (52) divided by f327TRoo is

JA a<I>", Wi -a- dz

-A r * ~ xnK~(xnRoo) =£.; (<I>j-Zj)£'; A Ko(xnRoo) Ai,,rtj,n, (53)

j=l n=l

where primes denote derivatives with respect to the argu~ ment, j= 1, ... ,Mo mark the nodes at r=Roo, and

Ai,n= J~AWiSin[Xn(Z+A)]dZ. (54)

The series converges since <I>-z is continuous in (-A,A) and zero at z=±A. The volume constraint forms the neces-' sary equation to determine the unknown All and is given by (5).

This set of algebraic equations of (N + !VI + 1) unknowns can be regarded as a system of (N + 1) unknowns by consid­ering the electrical pressure as a function of the interfacial points, llE=IlE(ft, ... ,jN) (Ramos and Castellanos13). An auxiliary equation based on the arclength parameter is in­cluded to avoid a singular Jacobian matrix when limit points appear (Keller21)

(x-x*). dxl -As=O, ds .

x*

(55)

and the unknowns are supplemented with >.. the parameter that is being varied (either X, 7, or B). In (55), x=(j1> ... ,jN,All,>..) is the new set of unknowns, s is the arclength of the curve described by x, dX/ds is the unit tan­gent vector to the solution curve, and increments are taken with respect to a previous solution x*.

The (N +2) algebraic equations are solved by the New­ton method which is implemented as described in Ramos and

3586 Phys. Fluids, Vol. 6, No. 11, November 1994

~'-."" . ...... -- / f(A/2)

1.2 --

1.1 10

\ I \

~, ..... ~

11 X

~ >--~

12

FIG. 6. Sensitivity of f(N2) as a function of X to mesh refinement. ---, Nr/=3, N ro =5, N z =8; ---, N r;=4, N ro =7, N,=12; -'-, N rl=6, N ro =10, N z =16.

Castellanos.13 The numerical procedure for tracing the bifur­cation diagrams closely follows the works of Ungar and Brown22 and Basaran and Scriven.23 The stability of a par­ticular family is directly related to its location in the bifur­cation diagram (Iooss and Joseph17).

The location of the outer boundary was set at Roo=3. We found that the results for the dielectric pressure at the inter­face for this value of Roo and with the first five basis func­tions in the asymptotic expansion were in complete accor­dance with those obtained by Ramos and Castellanos13 for Roo=8 and without an asymptotic expansion. A reduction fac­tor of ~. in CPU time was found as a result of the decreased value of Roo by using the asymptotic expansion. The number of elements in radial direction inside and outside the liquid bridge was, respectively, Nri=4 and,Nro =7, and the number of elements in axial direction was N z=12. In the absence of electric field, the values of the critical volume as a function of Hand B obtained with the present discretization compare fairly well with those given by previous authors (Martinez;5 Meseguer et al. 6). In order to estimate the accuracy of the present discretization, the domain was tesselated into N ri =3, 4, 6 and N ro =5, 7, 10 rionuniform elements in the radial direction and N z =8, 12, 16 nonuniform elements in the axial direction. Calculations were made for a case of moderate deviation from the cylinder: r=1, B=O, A=4, H=0.2. In Fig. 6 we show the deformation of the interface at midheight, z=N2, between the upper electrode and the plane z=o as a function of X for the latter case with f3=0.55. The difference between the critical electrical Bond numbers is 1 % for the first two meshes and decreases to 0.3% for the two last meshes.

v. RESULTS

For the following results we will set the value of f3 at 0.55, which is the permittivity that appears mostly in previ­ous works.

A. Shapes with 7= 1

The effect of the electric field upon the eqUilibrium shapes is presented m Fig. 7. It shows two stable equilibria of a liquid bridge and the equipotentials of '¥=<I>-z (the

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2 __ -"

-2 '--_ ......

2 o 2 (a) r (b) r

FIG. 7. Equipotentials of'l'=<P-z for a liquid bridge with r=1, A=2.~, H=O.l, B=O subjected to an electric field: (a) X=O.Ol and (b) x=6: E~Ul­potentials values increase from the inner dotted line to the outer sohd lme.

perturbation to the linear potential) when it is subjected to two electric field strengths in the case A=2.S, H=0.1, B=O. We can see that there is an alignment of the shape with the axial electric field. This is the typical behavior and it ex­plains the stabilization effect of the axial electric field.

The local method has shown that the effect of a small difference in contact radius is very similar to that of an axial residual gravity. Figure 8 shows neutral stability curves in the (A,X) plane for different values of Hand B =0. For com­parison, the neutral stability curves (dashed lines for H=O, 0.01, O.OS, 0.1) obtained by the Lyapunov-Schmidt method are presented. A liquid bridge of given H is stable if the set (X A) belongs to the upper region delimited by its corre­sp~nding H curve. The curve H =0 gives the maximum limit of stable slenderness and the effect of H is to reduce signifi­cantlv the region of stability. From the comparison with the local- method we can see a good agreement for values of H.,;;;O.OS. For the curve H=O.l the relative difference in the critical A for given X is less than 4%.

3

FIG. 8. Neutral stability curves in the (X,A) plane for different values of H in the case r=1, B=O: (a) H=O; (b) H=O.01; (c) H=O.05: (d) H=O.l: (e) H=O.2. ---, the Lyapunov-Schrnidt method.

Phys. Fluids, Vol. 6, No. 11, November 1994

FIG. 9. Neutral stability curves in the (X,A) plane for different values of B and H when B/H=O.2: (a) H=O: (b) H=O.l: (c) H=O.25. Cylindrical vol­ume case. ---, the Lyapunov-Schmidt method.

The inclusion of a nonzero value of B changes these stability curves in two different ways depending on the sign of B, as indicated previously by the local results. Figure 9 shows neutral stability curves when Hand B compensate each other, for B / H =0.2. Also the case when both are zero is presented. For this chosen ratio of B / H the lowest perturba­tive approach predicts full compensation at A=4. As H in­creases, keeping B / H constant, the compensated point moves away from the curve of H = B = 0, so new terms in the local expansion are becoming important. The comparison of nu­merical and local results shows that both methods compare quite well for H=O.l in the computed range of A. For H=0.2S, the deviation is large when A<4 and is small when A>4. A possible explanation for this deviation may be that the perturbation parameter IB -aH/ AI is more important for A>4 than for A <4, as shown in Fig. 4. If we move along the neutral stability curves in the direction of increasing A, the liquid bridge neck moves upwards, from z<O to z>O. This is depicted in Fig. 10, where we see three liquid bridge shapes for three different values of A in the neutral stability curve of H =0.25, B =0.05. For A <4, the equilibrium shape is domi­nated by the effect of H; for A>4 the shape is dominated by the effect of B; and for A =4 the equilibrium shape tends to be a truncated cone since both effects balance.

Let us see some bifurcation diagrams computed numeri­cally for the case A =4. Figure 11 shows the quantity fCN2), which typically measures the bridge deformation, as a func­tion of X for four different cases: (a) H=B=O; (b) H=0.1, B=O; (c) H=0.1, B=0.01; Cd) H=O.l, B=0.02. The defor­mation parameter has been chosen because it is very sensi­tive to the effects of Hand B. The pitchfork bifurcation for H=B=O is broken when either H or B are different from zero, since they break the original symmetry with respect to the midplane z =0. However, there is a combination of Band H [case (d)] that is very close to the pitchfork bifurcation, and in fact, the bifurcation is now broken due to higher or­ders in the local approximation.

In Fig. 12 several neutral stability curves are presented.

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0.5 \ .- .... , ..... -.- I \

, , , I \ \

, I

\ ., , I

\ " i I \ i J 1 \ i i 1 \ i .. 1 \ ./ \. 1 \ 1

z/2A \ ! '. 1 \

/ \ 1 , \ 1 .' \' 0

/1 1\ , 1 1 \ ; \ J .. i \ J ..

I \ 1 .. i \ 1 .. i \ 1 " j \ I i i \ I i i \ I i \ \ I i

\ \ I , , \ I '. "'-. \ r _._ .....

-0.5 -1.5 -1 -0.5 0 0.5 1.5

fez}

FIG. 10. Critical equilibrium shapes for r-=l, H=O.25, B=0.05 and differ­ent values of A:-, A=2.5; ---, A=4; -'-, A=6.

As the ratio B / H decreases, the value of A needed for com­pensation increases. This behavior is in accordance with the local result that the relevant imperfection parameter is H/A. The figure also shows the cases when Hand B act in the same direction, always moving away from the curve of B = H =0. We have never observed that when Band H act in a compensated way the stability criterion is below that corre­sponding to B=H=O.

B. Shapes with #1

First, we show the comparison between the local and finite element results for zero values of H and B. Figure 13 depicts neutral stability curves in the (X,A) plane for differ­ent values of T. The comparison shows a deviation for r= 1.1 and 0.9 that is less than 4%. The effect of volume in the neighborhood of the cylinder is simply to shift the bifurca­tion diagram, with no change in the qualitative behavior.

However, new effects appear when '1'1 is not small. For zero values of Hand B, static considerations have shown that when the minimum volume stability limit is reached, two destabilizing ways are possible depending on the slen-

f{1I/2}

2 4 X

6 8

FIG. 11. The effect of Hand B: f(A/2) as a function of X for r-=l, A=4. ---, H=B=O; -'-, H=O.l, B=O; ''', H=O.l, B=O.Ol; -"'-, H=O.l, B =0.02. Solid line represents stable shapes.

3588 Phys. Fluids, Vol. 6, No. 11, November 1994

20

X

10

4 5

FIG. 12. Several neutral stability curves in the (X,A) plane: (a) H=B=O; (b) H=O.25, B=0.036; (b /) H=O.025, B=-O.036; (c) H=0.25, B=O.05; (c /) H=O.25, B=-O.05; (d) H=O.i?r, B=O.l; (d /)H=O.17T,B=-O.1. Cy­lindrical volume case.

derness: a bifurcation point with a nonsymmetric bifurcating family for large A, or a limit point for small A (Sanz and Marti'nez24). This transition occurs for larger values of A as the electrical Bond number increases (Ramos and Castellanos13

). The effect of a residual value of H upon the bifurcation diagrams is either to break the pitchfork bifurca­tion giving rise to a limit point or to perturb the original limit point of the case H=O. In Fig. 14, we represent three bifur­cation diagrams for three different field strengths (x=0,3,7) in the case A=2.6. For each value of X, the deformation parameter f(N2) is depicted as a function of dor H=O and H=0.03. The sequence of diagrams with H=O shows how increasing X the bifurcation point reaches the limit point (Ra­mos and Castellanos13

). For X=7 the stability criterion is marked by the limit point while for X=O is marked by the bifurcation point. The nonzero value of H breaks the pitch­fork bifurcation of the case H =0 giving rise to two separated branches (the stable branch is the only one shown). As the electrical Bond number increases the effect of H upon the minimum stable volume decreases since the critical T is closer to the case H =0.

10t---t---

X 5r-r-r--r~d"-:r-:~h..f-

3 4 5 6

FIG. 13. Comparison of neutral stability curves in the (x,A) plane for dif­ferent values of 'T. ---, the Lyapunov-Schmidt results: (al r-=0.7; (b) r-O.8; (c) r-=0.9; (d) r-=1.0; (e) r-=l.1; (f) r-1.2; (g) r-=1.3.

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1.21----+-----/--/-::-""'-----1

f(A/2) ... c{ -

.::b--===--0.81----1---.....:::;..--+----..,

(a) 0.4 L--_...I-___ --L ____ -'

1.21--+----~+----..,

f(A/2)

(b) 0.4

1.2

f(A/2)

0.8

0.4 (c)

0.6 T

0.8

FIG. 14. The effect of X on the stability: fCN2) as a function of T for A=2.6. (a) X=O; (b) X=3; (c) X=7 . ... , diagram for H=O, ---, diagram for H =0.03. Solid lines represent stable shapes. D, bifurcation point; 0, turning point.

Figure 15 shows the minimum critical volume as a func­tion of A for three different values of the electrical Bond number (x=0, 2, 4) when the parameters Hand B compen­sate each other (H=O.I, B=0.05) and when they act in the same direction (H=0.1, B=-0.05). The cases when H=B=O are also presented. A liquid bridge of given Hand B is stable if its volume is above the corresponding neutral stability curve. The stabilizing effect of the axial electric field makes the stability region increase. We can see that as X increases the critical point at which Band H compensate each other moves in the direction of increasing A. Again, when Hand B act in a compensated way, the minimum stable volume is always greater than that of the case B = H =0.

VI. CONCLUSIONS

The Lyapunov-Schmidt projection technique has been applied to give the bifurcation diagrams of axisymmetric liq­uid bridges between unequal disks, in the presence of re­sidual gravity, when subjected to an axial electric field, in the vicinity of the cylindrical solution. The analytical expression for the critical electrical parameter may be approximated in the general case by

Phys. Fluids, Vol. 6, No. 11, November 1994

T

2.5 A 3

,. , ,. Q' ,. ,

,i ,. /

I , ,. /

b/ /

3.5 4

FIG. 15. Neutral stability curves in the (T,A) plane for different values of H, Band X. --,X=O; ---,X=2; -'-, X=4. (a) H=B=O; (b) H=0.1, 8=0.05; (c) H=O.I, 8=-0.05.

Xc= XoCB,A) - K(,B,A)[ or-I] + a(,B,A)( B

-a(,B,A) ~), (56)

where the functions XO and a are given by Gonzalez and Castellanos12 and a and K have been presented in this work. The Galerkinlfinite element method is used to solve the prob­lem for arbitrary values of the parameters and to extend the study to noncylindrical volumes. Results show a good agree­ment between both methods for small deviations of the cy­lindrical shape. The effect of H upon the stability of liquid columns subjected to electric fields is very similar to that of B, a behavior already known in the absence of electric fields. For quasicylinders it has been shown that the combination B-a(A)H/A (where a includes a genuine effect due to po­larization charges and is of order unity) is the new imperfec­tion parameter having the same influence in the stability as B in the case of equal disks. In the general case, the combined effect of gravity and radius difference may give rise to com­pensation but the new stability criterion is never below the stability criterion in their absence. Sensitivity of the mini­mum stable volume to these imperfection parameters de­creases as the applied field increases.

ACKNOWLEDGMENT

This work was supported by the Spanish Government agency Direccion General de Ciencia y Tecnoiogia (DGCYT) under Contract No. PB90-0905.

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