I n d . Eng. Chem. Res. 1996,34,3454-3465
3464
Nonlinear Oscillations and Breakup of Conducting, Inviscid Drops in an Externally Applied Electric Field Osman A. Basaran* Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6224
Tadeusz W. Patzek Department of Materials Science and Mineral Engineering, University of California at Berkeley, Berkeley, California 94720
Robert E.Benner, Jr.?and L. E. Scriven Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455
The equilibrium shapes of a conducting drop in a uniform electric field E, are virtually prolate spheroids and are stable with respect to all infinitesimal-amplitude disturbances. Inviscid oscillations about these states are analyzed by solving numerically Bernoulli’s equation for drop shape and Laplace’s equation for the velocity potential inside and the electrostatic potential outside the drop. The drops are impulsively set into motion by either a step change in electric field strength from E!’ for time t < 0 to E‘:’ for t z 0 or subjecting them at t = 0 to an impulse of magnitude 42 in velocity potential proportional to the second spherical harmonic. For initially spherical drops that are set into oscillation by impulsively changing the field strength, the oscillation frequency computed for small field strengths accords with domain perturbation results. When the field strength E‘:’ is sufficiently large, the drops do not oscillate but become unstable by issuing jets from their tips, which are computed here for the first time. Whereas drops set into motion by a step change in field strength do evolve in time through a succession of virtually spheroidal shapes as surmised by previous spheroidal approximations, ones set into motion by a n initial velocity potential disturbance exhibit radically different dynamics. Moreover, such drops go unstable by issuing jets from their conical tips when 42 is large but fission when 42 is small. 1. Introduction
The dynamic response of electrified drops has been studied for over a century since Rayleigh’s (1882) pioneering analysis of the infinitesimal-amplitude, inviscid oscillations and stability of an isolated, charged conducting drop. Interest in the subject has broadened over the years because of fundamental and technological applications in fields as diverse as nuclear fission (Bohr and Wheeler, 1939),cloud physics (Beard et al., 19891, separations and mass transfer operations (Ptasinskiand Kerkhof, 19921, and containerless processing in low gravity (Carruthers and Testardi, 1983). However, despite the importance of the subject and the large number of studies that have been devoted to it, a comprehesive theory of finite-amplitude oscillations and breakup of even an inviscid, perfectly conducting drop in an externally applied, uniform electric field has been lacking and is presented in this paper. A perfectly conducting drop of density Q floats freely in an insulating ambient fluid of permittivity E , as shown in Figure 1. The effect of gravity is henceforward considered to be negligible because the gravitational Bond number G gR2AQfu,whereg is the gravitational acceleration, R is the radius of a sphere having the same volume as the drop, AQ is the density difference between
* Present address:
School of Chemical Engineering, h d u e
University, West Lafayette, IN 47907. Phone: 317-494-4061. Fax: 317-494-0805. E-mail:
[email protected]. + Present address: Sandia National Laboratories, Albuquerque, NM 87185. 0888-5885l95l2634-3454$Q9.0OlO
I.:
FLUID
Figure 1. Definition sketch: a perfectly conducting drop immersed in an externally applied electric field.
the drop and the ambient fluid, and u is the surface (interfacial)tension of the drop-ambient fluid interface, is taken to be vanishlingly small, G -K 1. If the drop is immersed in a uniform electric field fi-, the drop, which is spherical in shape when the field strength 8, is zero, can take on a succession of equilibrium shapes provided that each time the field strength is changed it is varied by an infinitesimal amount and sufficient time is allowed between successive changes for viscosity to damp out any transients. By approximating the equi-
0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3455 3,
I
UNSTABLE
p, 2.5 m
TURNING POINT: STABILITY UMlT
YlKSlS (1981) BASARAN AND SCRIVEN Ll8W
/
tn
\
through a sequence of spheroidal shapes in time. By means of this spheroidal approximation, Brazier-Smith (1971) was able to surmise that the frquency of oscillation of the fundamental mode decreased compared to that calculated by Rayleigh (1879) in the absence of electric effects as the field strength increased. Here the frequency calculated by Rayleigh is the celebrated result given by
=
I
C ~ T I C A LFIELDJ STRENGTH = 0.325
,kn - 1xn + ~ w ~ R ~ ) =, 2, ...
(1)
where n is the mode number and n = 2 is the lowest or fundamental mode of oscillation. While Brazier-Smith 0 (1971) also showed that an initially prolate spheroidal I 0 0.06 0.1 0.16 03 0.26 0.3 0.36 0.4 0.45 0.5 drop subjected to an electric field having sufficient APPLIED FIELD STRENGTH, E, strength should go unstable, the validity of the spheFigure 2. Bifurcation diagram in parameter space: aspect ratio roidal approximation and the mechanism(s) of instabilas a function of applied field strength for the prolate family of ity of drops in an electric field have remained as open equilibrium shapes. problems in electrohydrodynamics. Experimental studies on drops immersed in a uniform librium drop shapes as spheroids, G. I. Taylor (1964) electric field have revealed that when the field strength surmised that this family of prolate shapes loses stabilbecomes sufficiently large, the tips of the drop first take ity at a turning point in field strength, as shown in on a conical appearance and the drop disintegrates by Figure 2. The so-called Taylor limit of stability has issuing jets from these regions of high curvature (Nolan, since been confirmed by various approximations (e.g., 1926; Macky, 1931; Taylor, 1964; Garton and Krasucki, Brazier-Smith, 1971) and rigorous numerical analyses 1964; Inculet and Kromann, 1989). To gain insights (Miksis, 1981; Basaran and Scriven, 1982; Adornato and into this instability, Brazier-Smith and Latham (1969) Brown, 1983). As discussed by Basaran and Scriven and Brazier-Smith et al. (1971b) employed a marker(1989),those members of the prolate shape family whose and-cell type approach, albeit in the inviscid limit, t o aspect ratios &/6 are smaller than 1.85 are stable with investigate the deformation of a single and a pair of respect to all infinitesimal-amplitude perturbations. conducting drops, respectively, in an external electric Here the aspect ratio is defined as the ratio of the field. Although the algorithms used by these investigadistance between the poles of the drop to the width of tors predicted the formation of conical tips, they were the drop in the equatorial plane or, for a truly spheroidal unable to simulate the issuance of jets seen in experidrop, the ratio of the semimajor axis length to the ments and carried out too few calculations to gain semiminor axis length. Of greater interest are ocilladetailed insights into the underlying dynamics. tions about these equilibrium states and stability of In the limit in which viscous forces dominate inertial these shapes with respect to finite-amplitude disturones, Re 0, Shenvood (1988) used a boundary integral bances, which are the subjects of this paper. approach to theoretically study the deformation of drops Motivated by practical applications (see, e.g., Basaran having finite as well as infinite conductivities in an et al., 19891, the dynamics of drops that are impulsively externally applied, uniform electric field in the creeping set into motion by one of two means are analyzed in flow limit. Although this approach does not allow the this paper. With the first means, the motion is induced study of drop oscillations, Shenvood (1988) showed that for times 2 0 by causing a step change in the electric two distinct mechanisms of drop instability exist defield strength from for z < 0 to E?)for z 0. The pending on the values of the conductivities and permitconsequence of imposing too large a value of J???) can be tivities of the drop and ambient fluids, as seen in drop breakup, which is also explored here. With the experiments (see, e.g., Torza et al., 1971). In some cases second means, the drop is subjected a t = 0 to an the drops became unstable to a varicose or sausage impulse in velocity potential proportional t o the second instability whereas in others they became unstable by spherical harmonic. Although the dynamics of drops forming conical ends. Once again, Shemood's algorithm subjected to velocity potential disturbances in the too was not successful a t predicting the issuance of jets absence of electric fields is a classical problem in fluid from the drop tips. mechanics (Lundgren and Mansour, 1988; Patzek et al., To go beyond Shenvood's (1988) work, Haywood et al. 1991; Pelekasis et al., 19921, the analogous problem in (1991) used a finite volume technique t o account for the the presence of electric fields has heretofore not been effect of finite inertia in studying the deformation of studied. drops having finite conductivity in an externally applied, There have been numerous studies to date which have uniform electric field. These authors did not study drop considered small- and finite-amplitude excursions about oscillations although their method should be capable of the stable members of the prolate shape family. These simulating such motions. Haywood et al. too identified studies can be best categorized with the aid of an two distinct mechanisms of breakup. effective Reynolds number defined as Re Feng and Beard (1990) used a multiple-parameter (l/v)-, where v is the kinematic viscosity Y ,id@, domain perturbation technique to study the inviscid p is the viscosity, and is the density. By assuming oscillations of a charged conducting drop that is levithat viscous forces are negligible compared to inertial tated against the force of gravity by an externally ones, Re 00, Brazier-Smith et al. (1971a) and Brazierapplied, uniform electric field. Because of limitations Smith (1971) theoretically studied the infinitesimal- and that are inherent to all perturbation analyses, this work, although rigorous, is restricted to small drop deformafinite-amplitude oscillations, respectively, of inviscid, conducting drops in a uniform electric field. In both of tions and small values of the applied field strength; viz., these papers, the drops were also restricted t o evolve when G = 0 and the drop carries no net charge the U
-
z
z
e:'
z
-
3456 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
e:'
initial drop shape is spherical, i.e. = 0, and E?) 4:'. When E':) is less than some critical value E:), however, the drop disintegrates by fissioning no matter how large the value of &. The emission of microjets from the tips of liquid drops is also of interest in the so-called electrostatic spraying or atomization of liquids from nozzles (see, e.g., Cloupeau and h n e t - F o c h , 1989, 1990). Electrostatic atomization is currently receiving a great deal of attention worldwide because it allows the production of nearly monodisperse droplets. However, as in the disintegration of free drops, when an axisymmetric jet issuing from the tip of a pendant drop of a liquid that is flowing out of a nozzle breaks up to form a cloud of small droplets, the physics of aerosol production is a fully three-dimensional problem. Moreover, both free drops (e.g., Garton and Krasucki, 1964) and their pendant counterparts (Cloupeau and h e t - F o c h , 1989)produce an axisymmetric andor a steady jet over very narrow ranges of the parameter space. Therefore, there is a need to model the dynamics of electrified drops and the
r,E,F,F,
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3463
0.l68
-26
is
2.1E EJ ]m]m]Fl Fl 2*6El ~ 1 FlE] ~ El 1 211;ln,
-26
-26
12€ j ,€
,,,,
2666 , , , , ,, 0 8; l~
~
,,,,
,,,,
3.677
3.907
4296
4.689
6.494
6.616
6.749
6.917
6.348
6.164
0
@
-26
-26
0
26-26
0
26-26
0
26-26
0
26-2.6
0
26-26
0
26
Figure 14. Time sequence of shapes of an electrified drop that is set in motion by an impulse in velocity potential: E,
= 0.2 and 42 =
0.85. Times are shown above each drop shape because the time step size is chosen adaptively. 1.5
I
BREAKUPBY FISSIONING
-
"\I
10 DROP BREAKS
I
c' DROP OSCILLATES
1
LUNDQRENAND MANSOUR: 1.218 f 0.01
THIS WORK: 1.205 f 0.006 0.25
re U
''STABLE" OSClUATlONS
o ~ ~ " " " ' " " ' " " " " ' " ' ' ' ' ~ ' ' ~ 0
0.06
0.1
0.15
0.2
ox
n, 0.3
a
APPLIED FIELD STRENGTH
Figure 16. Amplitude of velocity potential disturbance required -2.4
0
24-2.4
0
2.4
Figure 16. Time sequence of shapes of an electrified drop that is set in motion by an impulse in velocity potential; E, = 0.05 and = 1.2. Times are shown above each drop shape because the time step size is chosen adaptively.
to break up a drop. Initial condition: static drop a t equilibrium under the externally applied field of strength equal to the abscissa value.
Acknowledgment
$2
ejection of jets from their tips in three dimensions. The problem of microdroplet generation from the breakup of the microjet is still an open problem of fluid mechanics. On the basis of experimental observations, Hayati et al. (1987) have argued that electrical shear stresses are essential for maintaining stable microjets during electrostatic atomization. Therefore, it is somewhat puzzling that the simulations of Sherwood (19881,who took account of electrical shear stresses by allowing finite conductivity in the drop and continuous phases in contrast to the present paper, did not predict jetting once the drop developed conical tips. Thus, it also needs to be investigated whether this is a deficiency inherent in current boundary element simulations or whether finite inertia is required before a drop can eject a jet from its highly curved tips.
This research was supported by the NASA Fund for Independent Research, the University of Minnesota Computer Center, and the Division of Chemical Sciences, Office of Basic Energy Sciences (BES), U.S. Department of Energy (DOE), under Contract DEAC05-840R21400 with Martin Marietta Energy Systems, Inc.
Nomenclature
&, alb = dimensional and dimensionless drop aspect ratio c,(t) = amplitudes or
coefficients of linear modes, dimensionless E , = dimensional and dimensionless electric field strength e,, E, = dimensional and dimensionless electric field E'," = dimensional and dimensionless electric field strength for time < 0 E:), E':' = dimensional and dimensionless electric field strength for time h 0
e,,
zi),
3464 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
E(C)= electric field strength at which the mode of breakup ~
changes from fissioning to jetting from conical tips, dimensionless E, = normal component of electric field, dimensionless fl0;t) = drop shape function, dimensionless g = acceleration due to gravity G = gR2Ae/u= gravitational Bond number, dimensionless G(t)= integration constant in Bernoulli's equation, dimensionless H = local mean curvature of the interface, dimensionless K = reference pressure, dimensionless n = mode number of oscillation n = unit normal vector N';" = number of elements inside the drop in the rdirection, dimensionless pro'= number of elements outside the drop in the rdirection, dimensionless No = number of elements in the 0-direction, dimensionless P, = Legendre polynomial of order n r = radial spherical coordinate, dimensionless r, = radius of large sphere enclosing drop, dimensionless R = radius of sphere having the same volume as the drop; also, characteristic length Re 3 ( 1 I v ) a = Reynolds number, dimensionless S(t) = dimensional and dimensionless drop surface area z, t = dimensional and dimensionless time 0, U = dimensional and dimensionless electrostatic potential Us= electrostatic potential at drop surface, dimensionless 0 , v = dimensional and dimensionless fluid velocity vs = velocity of points on drop surface, dimensionless u, V = dimensional and dimensionless drop volume V, = initial drop volume, dimensionless z = axial coordinate, dimensionless
s('t),
Greek Symbols 6 = Dirac delta function E
= permittivity of ambient fluid
0 = meridional spherical coordinate, dimensionless p = viscosity v = kinematic viscosity
e = density Ae = density difference between drop and surrounding
fluid m
u = characteristic time
u = surface
or interfacial tension = characteristic field strength UE = charge density, dimensionless 4,4 = dimensional and dimensionless velocity potential 42 = amplitude of velocity potential disturbance, dimensionless 6 ,w = dimensional and dimensionless angular frequency of oscillation
Literature Cited Adornato, P. M.; Brown, R. A. Shape and stability of electrostatically levitated drops. Proc. R . SOC.London, A 1983,389, 101. Basaran, 0. A. Nonlinear oscillations of viscous liquid drops. J . Fluid Mech. 1992,241,169.
Basaran, 0. A.; Scriven, L. E. Profiles of electrified drops and bubbles. In Proceedings of the Second International Colloquium on Drops and Bubbles; LeCroissette, D. H., Ed.; Jet Propulsion Laboratory: Pasadena, CA, 1982. Basaran, 0. A.; Scriven, L. E. Axisymmetric shapes and stability of charged drops in an external electric field. Phys. Fluids A 1989,1 , 799. Basaran, 0. A.; Scriven, L. E. Axisymmetric shapes and stability of pendant and sessile drops in an electric field. J . Colloid Znterface Sci. 1990,140,10. Basaran, 0. A.; DePaoli, D. W. Nonlinear oscillations of pendant drops. Phys. Fluids 1994,6,2923. Basaran, 0.A.; Scott, T. C.; Byers, C. H. Drop oscillations in liquidliquid systems. AIChE J . 1989,35,1263. Beard, K. V.; Ochs, H. T., 111; Kubesh, R. J. Natural oscillations of small raindrops. Nature 1989,342,408. Bohr, N.;Wheeler, J. A. The mechanism of fission. Phys. Rev. 1939,56, 426. Brazier-Smith, P. R. The stability of a water drop oscillating with finite amplitude in an electric field. J . Fluid Mech. 1971,50, 417. Brazier-Smith, P. R.; Latham, J. Numerical computations of the dynamics of the disintegration of a drop situated in an electric field. Proc. R. SOC.London, A 1969,312,277. Brazier-Smith, P. R.; Brook, M.; Latham, J.; Saunders, C. P. R.; Smith, M. H. The vibration of electrified water drops. Proc. R . SOC.London, A 1971a,322, 523. Brazier-Smith, P. R.; Jennings, S. G.; Latham, J. An investigation of the behavior of drops and drop-pairs subjected to strong electrical forces. Proc. R . SOC.London, A 1971b,325,363. Carruthers, J. R.; Testardi, L. R. Materials processing in the reduced-gravity of space. Annu. Rev. Mater. Sci. 1983,13,247. Cloupeau, M.; h e t - F o c h , B. Electrostatic spraying of liquids in cone-jet mode. J . Electrost. 1989,22,135. Cloupeau, M.; Prunet-Foch, B. Electrostatic spraying of liquids: main functioning modes. J . Electrost. 1990,52,165. Feng, J. Q.; Beard, K. V. Small-amplitude oscillations of electrostatically levitated drops. Proc. R . SOC.London, A 1990,430, 133. Garton, C. G.; Krasucki, Z. Bubbles in insulating liquids: stability in an electric field. Proc. R . SOC.London, A 1964,280,429. Gresho, P. M.; Lee, R. R.; Sani, R. C. On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions. In Recent Advances in Numerical Methods in Fluids; Taylor, C., Morgan, K., Eds.; Pineridge Press: Swansea, UK, 1979; Vol. 1. Hayati, I.; Bailey, A. I.; Tadros, Th. F. Investigations into the mechanisms of electrohydrodynamic spraying of liquids. J . Colloid Interface Sci. 1987,11 7,205. Haywood, R. J.; Renksizbulut, M.; Raithby, G. D. Transient deformation of freely-suspended liquid droplets in electrostatic fields. AIChE J . 1991,37,1305. Inculet, I. I.; Kromann, R. Breakup of large water droplets by electric fields. IEEE Trans. Znd. Appl. 1989,25,945. Kistler, S. F.; Scriven, L. E. Coating flows. In Computational Analysis of Polymer Processing; Pearson, J. R. A., Richardson, S. M., Eds.;Applied Science Publishers: London, 1983. Lundgren, T. S.; Mansour, N. N. Oscillations of drops in zero gravity with weak viscous effects. J . Fluid Mech. 1988,194, 479. Luskin, M.; Rannacher, R. On the smoothing property of the Crank-Nicholson scheme. AppFicabZe Anal. 1982,14, 117. Macky, W. A. Some investigations on the deformation and breaking of water drops in strong electric fields. P m . R . SOC.London, A 1931,133,565. Miksis, M. J. Shape of a drop in an electric field. Phys. Fluids 1981,24,1967. Nolan, J. J. The breaking of water-drops by electric fields. Proc. R. Irish Acad. 1926,37,28. Patzek, T. W.; Benner, R. E.; Basaran, 0. A.; Scriven, L. E. Nonlinear oscillations of inviscid free drops. J . Comput. Phys. 1991,97,489.
Ind. Eng. Chem.Res., Vol. 34, No. 10, 1995 5465 Pelekasis, N. A; Tsamopoulos, J.A.; Manolis, G. D. A hybrid finiteboundary element method for inviscid flows with free surface. J . Comput. Phys. 1992,101,231. Ptasinski, K. J.; Kerkhof, P. J. A. M. Electric field driven separations: phenomena and applications. Sep. Sci. Technol. lW2,27,995. Rayleigh, Lord. On the capillary phenomena ofjets. Proc. R. SOC. London 1879,29,71. Rayleigh, Lord. On the equilibrium of liquid conducting masses charged with electricity. Philos. Mag. 1882,14, 184. Sherwood, J. D. Breakup of fluid droplets in electric and magnetic fields. J . Fluid Mech. 1988,188,133. Strang, G.;Fix,G. J. An Analysis of the Finite Element Method; Prentice-Hall: Englewood Cliffs, NJ, 1973.
Taylor, G.I. Disintegration of water drops in an electric field. Proc. R . SOC.London, A 1964,280,383. Torza, S.; Cox, R. G.;Mason, S.G. Electrohydrodynamic deformation and burst of liquid drops. Philos. Trans. R. Soc. London, A 1971,269,295.
Received for review January 9,1995 Accepted May 30, 1995@ IE9500376
@
Abstract published in Advance ACS Abstracts, September
1, 1995.
