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Dynamics of Drop Formation in an Electric Field

Dynamics of Drop Formation in an Electric Field
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  Dynamics of Drop Formation in an Electric Field Patrick K. Notz and Osman A. Basaran 1 Purdue University, School of Chemical Engineering, West Lafayette, Indiana 47907-1283 Received October 19, 1998; accepted January 28, 1999 The effect of an electric field on the formation of a drop of aninviscid, perfectly conducting liquid from a capillary which pro-trudes from the top plate of a parallel-plate capacitor into asurrounding dynamically inactive, insulating gas is studied com-putationally. This free boundary problem which is comprised of the surface Bernoulli equation forthe transient drop shape and theLaplace equation forthe velocity potential inside the drop and theelectrostatic potential outside the drop is solved by a method of lines incorporating the finite element method for spatial discreti-zation. The finite element algorithm employed relies on judicioususe of remeshing and element addition to a two-region adaptivemesh to accommodate large domain deformations, and allows thecomputations to proceed until the thickness of the neck connectingan about to form drop to the rest of the liquid in the capillary isless than 0.1%of the capillary radius. The accuracy of the com-putations is demonstrated by showing that in the absence of anelectric field predictions made with the new algorithm are inexcellent agreement with boundary integral calculations (Schul-kes, R. M. S. M.  J. Fluid Mech.  278, 83 (1994)) and experimentalmeasurements on water drops (Zhang, X., and Basaran, O. A.  Phys. Fluids  7(6), 1184 (1995)). In the presence of an electric field,the algorithm predicts that as the strength of the applied fieldincreases, the mode of drop formation changes from simple drip-ping to jetting to so-called microdripping, in accordance withexperimental observations (Cloupeau, M., and Prunet-Foch, B.  J. Aerosol Sci.  25(6), 1021 (1994); Zhang, X., and Basaran, O. A.  J. Fluid Mech.  326, 239 (1996)). Computational predictions of theprimary drop volume and drop length at breakup are reportedovera wide range of values of the ratios of electrical, gravitational,and inertial forces to surface tension force. In contrast to previ-ously mentioned cases where both the flow rate in the tube and theelectric field strength are nonzero, situations are also considered inwhich the flow rate is zero and the dynamics are initiated byimpulsively changing the field strength from a certain value to alarger value. When the magnitude of the step change in fieldstrength is small, the results of the new transient calculationsaccord well with those of an earlier stability analysis (Basaran,O. A., and Scriven, L. E.  J. Colloid Interface Sci.  140(1), 10 (1990))and thereby provide yet another testament to the accuracy of thenew algorithm.  © 1999 Academic Press  Key Words:  electrohydrodynamics (EHD); drop formation; fi-nite element method (FEM).1. INTRODUCTION The application of electric fields during the formation of drops from capillary tubes, nozzles, or orifices is in use in anumber of technological situations including spray coating (6),inkjet printing (7), spraying of agricultural chemicals (8), sep-aration processes (9), and mass spectrometry (10), amongothers. Electric fields also play an important role in processesof natural occurrence because of their influence on liquid dropsin fields as diverse as meteorology (11) and nuclear physics(12). Although the first scientific observations of the dynamicsof formation of drops in an electric field date back almost to thebeginning of this century to the pioneering studies of Zeleny(13, 14), a comprehensive theoretical understanding of thephenomena have heretofore been lacking. Remedying this sit-uation when the drop liquid is highly conducting is the goal of this paper.The equilibrium shapes and stability of both free and sup-ported—pendant and sessile—drops in an electric field havebeen studied exhaustively. Early theoretical works focused onsituations in which departures from spherical, cylindrical, orplanar base states were either (a) arbitrary but infinitesimal inamplitude, as in Rayleigh’s (15) pioneering study of the sta-bility of an isolated, perfectly conducting, charged drop, or (b)finite in amplitude but entailed assumptions made  a priori about the symmetry of those deformations, as in the cleversurmize made by Taylor (16) in the now celebrated spheroidalapproximation he used to determine the deformation and sta-bility of a perfectly conducting, uncharged drop immersed in auniform external electric field. A century after Rayleigh’sstability analysis which inaugurated the birth of the science of electrohydrodynamics (EHD), Miksis (17), Joffre  et al.  (18),and Basaran and Scriven (19) developed numerical methods tosystematically calculate without any simplifying assumptionsthe equilibrium shapes and stability of electrified drops.Miksis (17) theoretically determined the equilibrium shapesof free drops of a dielectric liquid surrounded by a dielectricmedium that are subjected to an electric field. Joffre  et al.  (18)theoretically determined the equilibrium shapes of conductingdrops that are pendant from a capillary and surrounded by adielectric fluid in the presence of an electric field. These 1 To whom correspondence should be addressed.Journal of Colloid and Interface Science  213,  218–237 (1999)Article ID jcis.1999.6136, available online at http://www.idealibrary.com on218 0021-9797/99 $30.00Copyright © 1999 by Academic PressAll rights of reproduction in any form reserved.  authors also carried out experiments and showed that theirtheoretical predictions were in good agreement with their ex-periments. Basaran and Scriven (19) theoretically determinedthe equilibrium shapes of conducting drops and soap bubblesthat are either pendant from the top plate or sessile on thebottom plate of a parallel plate capacitor. These authors alsodetermined stability limits with respect to field strength beyondwhich stable equilibrium shapes would no longer exist and thedrops would presumably go unstable by issuing jets from theirpointed tips. Harris and Basaran (20, 21) extended these worksby carrying out a systematic study of the shapes and stability of both conducting and nonconducting electrified drops that arependant from either the top plate of a parallel plate capacitor ora metal capillary. In a series of papers, Wohlhuter and Basaran(22) and Basaran and Wohlhuter (23) theoretically studied theeffect of linear and nonlinear polarization on the shapes andstability of pendant and sessile dielectric drops that are sur-rounded by a dielectric fluid in an electric field.Sherwood (24) theoretically studied in the creeping flowlimit the transient deformation and breakup of insulating andsemi-insulating, or leaky dielectric, free drops in an electricfield and identified different modes of breakup that depend onthe physical properties of the drop and host fluids. Haywood  et al.  (25) studied without the creeping flow approximation thetransient deformation of electrified drops for both liquid–liquidand liquid–gas systems. Although these investigators were notable to follow the drops to breakup, they showed that goodagreement exists between their computations and previousexperimental and theoretical results. Feng and Scott (26) usedfinite element analysis to theoretically determine finite ampli-tude prolate and oblate deformations suffered by semi-insulat-ing drops surrounded by a semi-insulating ambient fluid in anelectric field. Basaran  et al.  (27) and Feng and Leal (28)studied the large-amplitude oscillations of inviscid, conductingdrops that are set into motion by a step change in velocitypotential and/or electric field strength. Basaran  et al.  (27)showed that when the field strength is sufficiently large, thedrops can go unstable by issuing jets from their conical tips, inaccord with experiments.In recent years, there have been a number of studies aimedat understanding the physics of drop formation from a capillaryin the absence of electric fields. Zhang and Basaran (2) exper-imentally studied the effects of the inner and outer capillaryradii, surface tension, liquid viscosity, flow rate and surfactantson dynamics of formation of drops of Newtonian liquids intoair. Among other things, these authors reported that for lowviscosity liquids, the thread which connects the about to detachprimary drop to the liquid remaining in the capillary alwaysbreaks at its bottom first. For high viscosity liquids, the liquidthread becomes quite long and thin before breaking. Shi  et al. (29) showed that the threads formed when a high-viscosityfluid drips from a capillary can spawn a cascade of so-calledmicrothreads with each subsequent microthread’s radius beingsmaller by an order of magnitude compared to that of theprevious one. Schulkes (1) used the boundary integral methodto solve for the potential flow inside a growing drop to theo-retically predict the dynamics of formation and breakup of inviscid liquid drops from a capillary. Brenner  et al.  (30) didboth experiments and solved a set of one-dimensional slender- jet equations (cf. 31 and 32) to study the dynamics of formationof drops of low viscosity liquids and focused primarily on thedynamics for times approaching and after breakup. A thoroughreview of drop formation can be found in the recent paper byEggers (33).Whereas the theoretical understanding of the dynamics of drop formation has advanced a great deal in the past decadethanks in part to studies cited in the previous paragraph, thishas not been the case for drop formation in the presence of anelectric field despite the large amount of attention devoted to itin the literature. A liquid that drips from a capillary in theabsence of electric field continues to do so when subjected toan electric field of low strength (cf. 4, 34, and 35). If the flowrate is kept low but the field strength is increased, the mode of drop emission from the capillary can change from dripping toEHD jetting from a pendant drop that remains attached to thecapillary (35). However, under conditions that are identical tothose of EHD jetting, a regime can also be attained wheredroplets that are much smaller in size than the capillary radiusinstead of jets are emitted from the tip of a pendant drop. Thismode of drop formation has been referred to as microdrippingby Cloupeau and Prunet-Foch (3). If the field strength is keptlow but the flow rate is increased, drop production occurs viathe breakup of an electrified jet.The various modes of EHD jetting and in particular thatfrom the tip of a so-called Taylor cone, or electrospraying, hasreceived the most attention to date (3, 35–38). Although therequirements for achieving an electrospray are stringent, thismode of EHD jetting is capable of generating an aerosol of nearly monodisperse drops in the size range from 5 nm to 100  m (38). As summarized in Ref. (3), during EHD jetting theflow can be (i) steady and axisymmetric, (ii) unsteady andaxisymmetric, and (iii) unsteady and nonaxisymmetric.The controlled production of monodispersed droplets froman electrified jet is also possible and was realized by Sato (39).He was able to achieve drop sizes as small as 28   m at a rateof 50 drops per second. This was done by operating in a regimewhere the breakup of the thread connecting the main drop tothe liquid in the capillary does not result in the formation of secondary or satellite droplets.More recently, Zhang and Basaran (4) have carried out aquantitative experimental study of the effects of axially appliedelectric fields on drop formation in the dripping regime. Theseauthors were able to record details of the entire drop formationprocess for a  single  drop by using a high-speed imager that iscapable of recording a frame every 1/12th of a millisecond andcapturing and storing in digital format more than 1000 frames.219 ELECTRIFIED DROP FORMATION  Zhang and Basaran (4) uncovered that as the field strengthincreases, the location at which the liquid thread connecting theprimary drop to the liquid in the capillary breaks first switchesfrom its bottom to its top. These authors also showed that byusing an electric field of moderate strength, a satellite dropletthat is created from the breakup of the liquid thread can becaptured by the rest of the liquid hanging from the capillarydue to the attraction between the positive (negative) inducedcharge on the satellite droplet and the negative (positive) netcharge on the liquid hanging from the capillary. Therefore, amonodispersed drop size distribution could be obtained by judicious use of an applied electric field.Takamatsu and co-workers (40, 41), Vu and Carlson (42),and Byers and Perona (43) have attempted to analyze thedynamics of drop formation in the dripping regime by means of macroscopic balances based on approximate expressions forelectrostatic, gravitational, and surface tension forces on thedrops and in which inertial forces are neglected. Not toosurprisingly, such approximations provide little insight into thedetails of the dynamics of drop deformation and breakup in thepresence of an electric field.More recently, Zemskov  et al.  (44) and Grigor’ev  et al.  (45)attempted to predict the size, dripping frequency and the netcharge of drops emanating from charged capillaries through theuse of theories that were less approximate than those cited inthe previous paragraph but which were nevertheless still highlyidealized. The analyses in both of these works were based onenergy minimization. Zemskov  et al.  (44) assumed the dropformation process to be quasi-static and then minimized thetotal energy at each instant in time assuming that the liquid isfed into the capillary slowly enough so that the volume of thedrop hanging from the capillary can be taken as constant overtime intervals of the order of the period of oscillation of thedrop in its fundamental mode. They thereby attempted tofollow the drop until it became unstable. Recent studies of dropformation in the absence of electric fields (cf. 2) and of theclosely related problem of deformation and breakup of stretch-ing liquid bridges (see, e.g., 46) make plain the deficiency of such an approach in the presence of flow. However, even in thelimit in which the flow rate is vanishingly small, such ananalysis cannot provide any information on the sequence of transient shapes that the drop would go through beyond thewell known stability limits of static pendant drops calculatedalmost a decade earlier by others (cf. 18 and 19). Grigor’ev  et al.  (45) made a number of assumptions in their analysis re-garding the shapes of drops as they detach from a capillary andthe distribution of the electric field outside the drops so as tomake the problem tractable by analytical techniques. In partic-ular, the geometrical assumptions they adopted regarding theshape of the drops that detach from and the liquid that remainsattached to the capillary are not valid over much of the param-eter space of interest. Clearly, the transient evolution of inter-face shapes and electric field distributions should be computedas part of the solution procedure.Wright  et al.  (47) simulated the breakup of a viscous dropthat is subjected to pressure impulses and a time dependentelectric field using an approximate one-dimensional model of the flow field and a two-dimensional description of the electricfield. Unfortunately, these authors made a number of incorrectassumptions in formulating their model. For instance, theydescribed the interface location in cylindrical coordinates (  x ,  z ) as  x    R (  z ), where  x  and  z  stand for the radial and axialcoordinate and  R  is the interface shape function, and thensimplified their equations by assuming that the slope of theinterface  R    dR  /  dz  is small. Clearly, this assumption is notvalid over most of the drop surface. Not too surprisingly,Wright  et al.  (47) found that the predictions of their model didnot agree well with experimental measurements.Plainly, a theoretical analysis in which the equations gov-erning the drop shape, the flow field inside and the electric fieldoutside a drop forming out of a capillary are solved needs to becarried out to determine the effect of an electric field on thedynamics in the dripping mode and how other modes such as jetting and microdripping might arise as field strength in-creases. This analysis is taken up in this paper for the situationin which a perfectly conducting drop of an inviscid liquid isformed into an ambient fluid that is perfectly insulating, e.g.air. Section 2 presents the governing equations and boundaryand initial conditions. The details of the numerical methodused are discussed in section 3. Section 4 presents computa-tional results for cases in which liquid is supplied at a constantflow rate through the capillary. Section 5 presents the results of an investigation of the dynamic response of an initially staticpendant drop to a step change in electric field strength. Con-cluding remarks form the subject of section 6. 2. MATHEMATICAL FORMULATION The system consists of a perfectly conducting drop of aninviscid, incompressible liquid of density    and surface tension    that is being formed into a dynamically inactive, insulatingambient gas of permittivity    by flowing it through a metalcapillary of radius  R  and vanishingly small wall thickness, asshown in Fig. 1. The flow rate far upstream of the capillary exitis constant in time and is given by     R 2 v ˜  , where  v ˜   is theaverage velocity. The capillary protrudes a distance  H ˜  1  fromthe center of the top plate of a circular parallel-plate capacitor.The two plates of the capacitor are horizontal with respect tothe direction of gravity and the capillary, the capacitor, and thedrop share a common axis of symmetry that lies along thedirection of gravity. The bottom plate of the capacitor isgrounded and the top plate and capillary are held at a potential U  o  above ground. In what follows, it is convenient to presentthe equations and results in dimensionless form. Here thecapillary radius  R , capillary time scale     R 3  /    , and top-plate220  NOTZ AND BASARAN  potential  U  o  are used as the characteristic length, time, andelectric potential scales. When a variable appears with a tildeover it and the same variable also appears without it, it is to beunderstood that the variable without a tilde is the dimensionlesscounterpart of that with a tilde.The liquid is taken to undergo irrotational motion inside theregion  V  ( t  ), where  t   denotes time, which consists of theinterior of the drop and capillary. In this case, the velocity isgiven by the gradient of the scalar potential    ,  v    , and itfollows from the continuity equation that     obeys Laplace’sequation:  2    0 in  V   t   . [1]The drop shape  S  f  ( t  ) is unknown  a priori  and is determinedby solving the augmented surface Bernoulli equation:     t    12      2  2   Gz   N  e  E  n2  0 on  S  f   t   , [2]where  E  n   n    E  is the normal component of the dimension-less electric field  E  on the drop surface with  n  taken as theoutward pointing unit normal,  2  is twice the dimensionlesslocal mean curvature,  G    gR 2    /     is the gravitational Bondnumber, and  N  e    U  o2  /2  R    is the electric Bond number. InEq. [2] and in what follows, it is convenient to use a cylindricalcoordinate system (  x ,  z ) based at the center of the contactcircle.The electric field of course vanishes inside the conductingliquid and is given by the gradient of a scalar potential  U   in theambient gas  V   ( t  ), viz.  E  U  . The electric potential, too,is governed by Laplace’s equation:  2 U   0 in  V   t   . [3]Table 1 summarizes the dimensionless groups that governthe dynamics of drop formation. An appreciation of the orderof magnitude of these dimensionless groups can be gained byconsidering a typical situation from the closely related butexperimental work of Zhang and Basaran (4). If the drop liquidis water and the drops are formed from a capillary with a radiusof 0.16 cm into air at room temperature, the gravitational Bondnumber  G    0.35. For this capillary, if the flow rate is 1 mLmin  1 , then the dimensionless inlet velocity  v    0.01. Whenthe outlet of the capillary is flush with the top plate and theplate separation is 4.8 cm, the geometrical parameters  H  1    0and  H  2  30. If the applied field strength is 5 kV cm  1 (or theapplied potential is 24 kV) and the ambient fluid has thepermittivity of free space,       o    8.8542    10  12 C 2 N  1 m  2 , the electric Bond number  N  e   22.14.Equation [1] is solved subject to the boundary conditionsthat (i) the fluid does not penetrate the walls of the capillary, FIG. 1.  A liquid drop growing out of a capillary in the presence of anelectric field: definition sketch. The variables shown here have already beenmade dimensionless. TABLE 1Governing Dimensionless Groups Dimensionless group This work  G  Gravitational bond number  gravitational force/capillary force  gR 2    /     N  e  Electric bond number  electrical force/capillary force   U  o2  /(2  R   ) v  Inlet velocity  (inertial force/capillary force) 1/2  v ˜   /        R H  1  Capillary length  H ˜  1  /   R H  2  Distance between capillary outlet and bottom plate  H ˜  2  /   R H  3  Distance between inflow and outflow planes in capillary  H ˜  3  /   R R   Radius of cylinder along surface of which the electric field approaches a uniform, vertical electric field  R˜    /   R 221 ELECTRIFIED DROP FORMATION  S  c, i , (ii) far upstream of the capillary outlet, fluid velocityapproaches a uniform and constant profile in the  z  direction,(iii) fluid velocity obeys the kinematic boundary condition atthe free surface,  S  f  ( t  ), and (iv) the flow field is symmetricabout the common axis of the capacitor, capillary, and dropsystem,  S  sym , viz., n       0 on  S  c, i , [4] n        v  on  S  e , [5] n       n    v s  on  S  f   t   , [6] n       0 on  S  sym , [7]where  S  e  is a horizontal plane, henceforward referred to as theinflow plane, located sufficiently far upstream of the capillaryoutlet, viz.    z    H  3  1 (cf. Fig. 1 and Table 1) and  v s  standsfor the velocity of points on the free surface. Due to the choiceof length and time scales adopted here, the dimensionless inletvelocity  v  is equivalently the square root of a Weber number.The boundary conditions on [2] are that the three-phasecontact line is fixed and the drop shape is axisymmetric, viz., x s  e  x  at  z  0 on  S  f   t   , [8] n  x s  0  at  z   L  on  S  f   t   . [9]Here,  x s  is the position vector of points on the free surface S  f  ( t  ),  e  x  is the unit vector in the  x  direction, and  L  is instan-taneous length of the drop measured along the  z  axis.The boundary conditions on [3] are (i) the top plate  S  t  andcapillary  S  c,o  are maintained at unit potential  and   because theliquid is a perfect conductor and is in contact with the capillary,the free surface too is at unit potential, (ii) the bottom plate  S  b is grounded and thus held at zero potential, (iii) the electricfield far from the drop approaches a uniform, vertical field, and(iv) the electric potential is symmetric about  S  sym , viz., U   1 on  S  t  S  c,o  S  f   t   , [10] U   0 on  S  b , [11] n    U   0 on  S   , [12] n    U   0 on  S  sym , [13]where  S    is a cylindrical surface of large radius  R   1 that iscoaxial with the capacitor, capillary, and drop system (cf. Fig.1 and Table 1).The initial state is one in which a static drop that encloses avolume  V  o  is pendant from a tube. Moreover, the initial dropshape is taken to be either a section of a sphere having avolume no larger than that of a hemisphere or the equilibriumshape of a pendant drop for prescribed values of   N  e  and  G . Theequilibrium drop shape is determined by solving the aug-mented Young–Laplace equation: K   Gz   N  e  E  n2  2   0. [14]In Eq. [14]  K   is the unknown reference pressure, which issimply the pressure jump across the interface at  z    0. Thereference pressure is determined by the constraint that the dropvolume  V   remain fixed at the specified value  V  o : V   V  o  0. [15]Equations [14] and [15] are solved simultaneously here withEq. [3] to compute the equilibrium shapes of drops.Two different physical situations are of interest in this paper.In the first case, the liquid velocity everywhere is taken toequal zero for  t     0 and liquid flow in the tube is impulsivelystarted at  t     0, viz.,      x ,  z ,  t   0   0, n        x ,  z    H  3 ,  t   0    v . [16]When the initial drop shape is a section of sphere, both  G  and  N  e  are impulsively changed from zero to their desired values at t   0 and kept there for all  t   0. When the initial drop shapeis the equilibrium shape,  G  and  N  e  are kept fixed for times  t   0 at the values they had for  t   0. The dynamic responses thatresult from these two sets of slightly different initial conditionsare reported and contrasted in section 4.In the second case, no liquid is injected into the capillarythroughout the dynamics and the volume of the pendant dropremains equal to  V  o  until breakup. In this case, starting fromthe equilibrium shape corresponding to given values of (  N  e   N  e( b ) ,  G ), the dynamic response is computed of a drop to a stepchange in electric field strength, or a step change in electricBond number of magnitude    N  e , at  t     0, viz.,  N  e   N  e  b  for  t   0, and  N  e   N  e  b     N  e   N  e  a  for  t   0. [17]Drop responses that result from the initial condition given inEq. [17] are reported in section 5. 3. NUMERICAL ANALYSIS Equations [1–3] are solved in this paper by a method of linesusing the Galerkin/finite element method (G/FEM) (48) fordiscretization in space and an adaptive finite difference method222  NOTZ AND BASARAN
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