In the present study, the deformation of a droplet is numerically modeled by considering the dynamic model for electric charge migration at the drop interface under the effect of a uniform electric field. The drop and its ambient are both considered behaving as leaky dielectric fluids. Solving the charge conservation equation at the interface, which is the most important part of this study, the effect of conduction and convection of charges on different deformation modes will be explored. In this work, the interface is followed by the level set method and the ghost fluid method (GFM) is used to model the jumps at the interface. Physical properties are also chosen in a way that solving the charge conservation equation becomes prominent. The small drop deformation is investigated qualitatively by changing various effective parameters. In cases, different patterns of charges and flows are observed indicating the importance of electric charges at the interface. It is also shown that the transient behavior of deformation parameter can be either a monotonic or a nonmonotonic approach toward the steady-state. Moreover, large drop deformations are studied in different ranges of capillary numbers. It will be shown that for the selected range of physical parameters, considering the dynamic model of electric charges strongly affects the oblate deformation. Nevertheless, for the prolate deformation, the results are approximately similar to those obtained from the static model.

References

1.
Jeong
,
S. I.
, and
Seyed-Yagoobi
,
J.
,
2002
, “
Experimental Study of Electrohydrodynamic Pumping Through Conduction Phenomenon
,”
J. Electrostat.
,
56
(
2
), pp.
123
133
.
2.
Pearson
,
M. R.
, and
Seyed-Yagoobi
,
J.
,
2011
, “
Experimental Study of EHD Conduction Pumping at the Meso-and Micro-Scale
,”
J. Electrostat.
,
69
(
6
), pp.
479
485
.
3.
Park
,
J. U.
,
Hardy
,
M.
,
Kang
,
S. J.
,
Barton
,
K.
,
Adair
,
K.
,
Kishore Mukhopadhyay
,
D.
,
Lee
,
C. Y.
,
Strano
,
M. S.
,
Alleyne
,
A. G.
,
Georgiadis
,
J. G.
, and
Ferreira
,
P. M.
,
2007
, “
High-Resolution Electrohydrodynamic Jet Printing
,”
Nat. Mater.
,
6
(
10
), pp.
782
789
.
4.
Basaran
,
O. A.
,
Gao
,
H.
, and
Bhat
,
P. P.
,
2013
, “
Nonstandard Inkjets
,”
Annu. Rev. Fluid Mech.
,
45
(
1
), pp.
85
113
.
5.
Castellanos
,
A.
,
2014
,
Electrohydrodynamics
, Vol.
380
,
Springer
, Berlin.
6.
Laser
,
D. J.
, and
Santiago
,
J. G.
,
2004
, “
A Review of Micropumps
,”
J. Micromech. Microeng.
,
14
(
6
), p.
R35
.
7.
Saville
,
D.
,
1997
, “
Electrohydrodynamics: The Taylor-Melcher Leaky Dielectric Model
,”
Annu. Rev. Fluid Mech.
,
29
(
1
), pp.
27
64
.
8.
Rayleigh, L., 1882, “
On the Equilibrium of Liquid Conducting Masses Charged with Electricity
,”
Philosophical Magazine
,
14
(87), pp. 184–186.
9.
O'Konski
,
C. T.
, and
Thacher Jr
,
H. C.
,
1953
, “
The Distortion of Aerosol Droplets by an Electric Field
,”
J. Phys. Chem.
,
57
(
9
), pp.
955
958
.
10.
Allan
,
R.
, and
Mason
,
S.
,
1962
, “
Particle Behaviour in Shear and Electric Fields. I. Deformation and Burst of Fluid Drops
,”
Proc. R. Soc. London. Ser. A.
,
267
(
1328
), pp.
45
61
.
11.
Taylor
,
G.
,
1966
, “
Studies in Electrohydrodynamics. I. The Circulation Produced in a Drop by Electrical Field
,”
Proc. R. Soc. London. Ser. A.
,
291
(
1425
), pp.
159
166
.
12.
Melcher
,
J.
, and
Taylor
,
G.
,
1969
, “
Electrohydrodynamics: A Review of the Role of Interfacial Shear Stresses
,”
Annu. Rev. Fluid Mech.
,
1
(
1
), pp.
111
146
.
13.
Torza
,
S.
,
Cox
,
R.
, and
Mason
,
S.
,
1971
, “
Electrohydrodynamic Deformation and Burst of Liquid Drops
,”
Philos. Trans. R. Soc. London. Ser. A, Math. Phys. Sci.
,
269
(
1198
), pp.
295
319
.
14.
Ajayi
,
O.
,
1978
, “
A Note on Taylor's Electrohydrodynamic Theory
,”
Proc. R. Soc. London. A
,
364
(
1719
), pp.
499
507
.
15.
Feng
,
J. Q.
, and
Scott
,
T. C.
,
1996
, “
A Computational Analysis of Electrohydrodynamics of a Leaky Dielectric Drop in an Electric Field
,”
J. Fluid Mech.
,
311
(
1
), pp.
289
326
.
16.
Feng
,
J. Q.
,
1999
, “
Electrohydrodynamic Behaviour of a Drop Subjected to a Steady Uniform Electric Field at Finite Electric Reynolds Number
,”
Proc. R. Soc. London. Ser. A
,
455
(
1986
), pp.
2245
2269
.
17.
Ha
,
J. W.
, and
Yang
,
S. M.
,
2000
, “
Deformation and Breakup of Newtonian and Non-Newtonian Conducting Drops in an Electric Field
,”
J. Fluid Mech.
,
405
, pp.
131
156
.
18.
Ha
,
J. W.
, and
Yang
,
S. M.
,
2000
, “
Electrohydrodynamics and Electrorotation of a Drop With Fluid Less Conductive Than That of the Ambient Fluid
,”
Phys. Fluids
,
12
(
4
), pp.
764
772
.
19.
Collins
,
R. T.
,
Jones
,
J. J.
,
Harris
,
M. T.
, and
Basaran
,
O. A.
,
2007
, “
Electrohydrodynamic Tip Streaming and Emission of Charged Drops From Liquid Cones
,”
Nat. Phys.
,
4
(
2
), pp.
149
154
.https://www.nature.com/articles/nphys807
20.
Lac
,
E.
, and
Homsy
,
G.
,
2007
, “
Axisymmetric Deformation and Stability of a Viscous Drop in a Steady Electric Field
,”
J. Fluid Mech.
,
590
, pp.
239
264
.
21.
Hua
,
J.
,
Lim
,
L. K.
, and
Wang
,
C. H.
,
2008
, “
Numerical Simulation of Deformation/Motion of a Drop Suspended in Viscous Liquids Under Influence of Steady Electric Fields
,”
Phys. Fluids
,
20
(
11
), p.
113302
.
22.
Van Poppel
,
B.
,
Desjardins
,
O.
, and
Daily
,
J.
,
2010
, “
A Ghost Fluid, Level Set Methodology for Simulating Multiphase Electrohydrodynamic Flows With Application to Liquid Fuel Injection
,”
J. Comput. Phys.
,
229
(
20
), pp.
7977
7996
.
23.
Teigen
,
K. E.
, and
Munkejord
,
S. T.
,
2010
, “
Influence of Surfactant on Drop Deformation in an Electric Field
,”
Phys. Fluids
,
22
(
11
), p.
112104
.
24.
López-Herrera
,
J.
,
Popinet
,
S.
, and
Herrada
,
M.
,
2011
, “
A Charge-Conservative Approach for Simulating Electrohydrodynamic Two-Phase Flows Using Volume-of-Fluid
,”
J. Comput. Phys.
,
230
(
5
), pp.
1939
1955
.
25.
Paknemat
,
H.
,
Pishevar
,
A. R.
, and
Pournaderi
,
P.
,
2012
, “
Numerical Simulation of Drop Deformations and Breakup Modes Caused by Direct Current Electric Fields
,”
Phys. Fluids
,
24
(
10
), p.
102101
.
26.
Pooyan
,
S.
, and
Passandideh-Fard
,
M.
,
2012
, “
On a Numerical Model for Free Surface Flows of a Conductive Liquid Under an Electrostatic Field
,”
ASME J. Fluids Eng.
,
134
(
9
), p.
091205
.
27.
Yang
,
Q.
,
Li
,
B. Q.
, and
Ding
,
Y.
,
2013
, “
3D Phase Field Modeling of Electrohydrodynamic Multiphase Flows
,”
Int. J. Multiphase Flow
,
57
, pp.
1
9
.
28.
Lanauze
,
J. A.
,
Walker
,
L. M.
, and
Khair
,
A. S.
,
2015
, “
Nonlinear Electrohydrodynamics of Slightly Deformed Oblate Drops
,”
J. Fluid Mech.
,
774
, pp.
245
266
.
29.
Mandal
,
S.
,
Bandopadhyay
,
A.
, and
Chakraborty
,
S.
,
2016
, “
The Effect of Uniform Electric Field on the Cross-Stream Migration of a Drop in Plane Poiseuille Flow
,”
J. Fluid Mech.
,
809
, pp.
726
774
.
30.
Mandal
,
S.
,
Bandopadhyay
,
A.
, and
Chakraborty
,
S.
,
2016
, “
Effect of Surface Charge Convection and Shape Deformation on the Dielectrophoretic Motion of a Liquid Drop
,”
Phys. Rev. E
,
93
(
4
), p.
043127
.
31.
Mandal
,
S.
,
Chakrabarti
,
S.
, and
Chakraborty
,
S.
,
2017
, “
Effect of Nonuniform Electric Field on the Electrohydrodynamic Motion of a Drop in Poiseuille Flow
,”
Phys. Fluids
,
29
(
5
), p.
052006
.
32.
Sengupta
,
R.
,
Walker
,
L. M.
, and
Khair
,
A. S.
,
2017
, “
The Role of Surface Charge Convection in the Electrohydrodynamics and Breakup of Prolate Drops
,”
J. Fluid Mech.
,
833
, pp.
29
53
.
33.
Das
,
D.
, and
Saintillan
,
D.
,
2017
, “
A Nonlinear Small-Deformation Theory for Transient Droplet Electrohydrodynamics
,”
J. Fluid Mech.
,
810
, pp.
225
253
.
34.
Hu
,
W. F.
,
Lai
,
M. C.
, and
Young
,
Y. N.
,
2015
, “
A Hybrid Immersed Boundary and Immersed Interface Method for Electrohydrodynamic Simulations
,”
J. Comput. Phys.
,
282
, pp.
47
61
.
35.
Safi
,
A.
, and
Turek
,
S.
,
2014
, “
GPGPU-Based Rising Bubble Simulations Using a MRT Lattice Boltzmann Method Coupled With Level Set Interface Capturing
,”
Computers & Fluids
,
124
, pp. 170–184.
36.
Liu
,
X. D.
,
Fedkiw
,
R. P.
, and
Kang
,
M.
,
2000
, “
A Boundary Condition Capturing Method for Poisson's Equation on Irregular Domains
,”
J. Comput. Phys.
,
160
(
1
), pp.
151
178
.
37.
Celik
,
I. B.
,
Ghia
,
U.
, and
Roache
,
P. J.
,
2008
, “
Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications
,”
ASME J. Fluids Eng.
,
130
(
7
), p.
078001
.
38.
Peskin
,
C. S.
,
1972
, “
Flow Patterns around Heart Valves: A Numerical Method
,”
J. Comput. Phys.
,
10
(
2
), pp.
252
271
.
39.
Yang
,
X.
,
Zhang
,
X.
,
Li
,
Z.
, and
He
,
G. W.
,
2009
, “
A Smoothing Technique for Discrete Delta Functions With Application to Immersed Boundary Method in Moving Boundary Simulations
,”
J. Comput. Phys.
,
228
(
20
), pp.
7821
7836
.
40.
Tryggvason
,
G.
,
Bunner
,
B.
,
Ebrat
,
O.
, and
Tauber
,
W.
,
1998
, “
Computations of Multiphase Flows by a Finite Difference/Front Tracking Method—I: Multi-Fluid Flows
,”
Lecture Series, von Karman Institute For Fluid Dynamics
, Rhode Saint Genese, Belgium, p.
7
.
41.
Benselama
,
A. M.
,
Achard
,
J. L.
, and
Pham
,
P.
,
2006
, “
Numerical Simulation of an Uncharged Droplet in a Uniform Electric Field
,”
J. Electrostat.
,
64
(
7–9
), pp.
562
568
.
42.
Zakinyan
,
A.
,
Tkacheva
,
E.
, and
Dikansky
,
Y.
,
2012
, “
Dynamics of a Dielectric Droplet Suspended in a Magnetic Fluid in Electric and Magnetic Fields
,”
J. Electrostat.
,
70
(
2
), pp.
225
232
.
43.
Soni
,
P.
,
Juvekar
,
V. A.
, and
Naik
,
V. M.
,
2013
, “
Investigation on Dynamics of Double Emulsion Droplet in a Uniform Electric Field
,”
J. Electrostat.
,
71
(
3
), pp.
471
477
.
44.
Taylor
,
G.
,
1964
, “
Disintegration of Water Drops in an Electric Field
,”
Proc. R. Soc. London. Ser. A. Math. Phys. Sci.
,
280
(
1382
), pp.
383
397
.
45.
Baygents
,
J.
,
Rivette
,
N.
, and
Stone
,
H.
,
1998
, “
Electrohydrodynamic Deformation and Interaction of Drop Pairs
,”
J. Fluid Mech.
,
368
, pp.
359
375
.
46.
Zhang
,
J.
, and
Kwok
,
D. Y.
,
2005
, “
A 2D Lattice Boltzmann Study on Electrohydrodynamic Drop Deformation With the Leaky Dielectric Theory
,”
J. Comput. Phys.
,
206
(
1
), pp.
150
161
.
47.
Salipante
,
P. F.
, and
Vlahovska
,
P. M.
,
2010
, “
Electrohydrodynamics of Drops in Strong Uniform DC Electric Fields
,”
Phys. Fluids
,
22
(
11
), p.
112110
.
48.
Xu
,
X.
, and
Homsy
,
G. M.
,
2006
, “
The Settling Velocity and Shape Distortion of Drops in a Uniform Electric Field
,”
J. Fluid Mech.
,
564
, pp.
395
414
.
49.
Feng
,
J. Q.
,
2001
, “
Application of Galerkin Finite-Element Computations in Studying Electrohydrodynamic Problems
,”
J. Electrostat.
,
51
, pp.
590
596
.
50.
Vizika
,
O.
, and
Saville
,
D. A.
,
1992
, “
The Electrohydrodynamic Deformation of Drops Suspended in Liquids in Steady and Oscillatory Electric Fields
,”
J. Fluid Mech.
,
239
(
1
), pp.
1
21
.
You do not currently have access to this content.