Experimental investigation on the Electrohydrodynamic motion and

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Proc. 2016 Electrostatics Joint Conference
1
Experimental investigation on the
Electrohydrodynamic motion and Shape
Deformation of a sedimenting Drop under
Uniform Alternating Electric Field
Rajorshi Paul*, Guttapalli Naveen Kumar*, Shubhadeep Mandal*, N.K. Kishore†,
SauravPramanik†, Suman Chakraborty*
*Department of Mechanical Engineering, Indian Institute of Technology Kharagpur,
Kharagpur-721302, West Bengal, India
†Department of Electrical Engineering, Indian Institute of Technology Kharagpur,
Kharagpur-721302, West Bengal, India
Abstract—A leaky dielectric Newtonian drop suspended in another leaky dielectric
Newtonian liquid medium deforms as it settles under the influence of a uniform alternating
electric field. As a result of this deformation, the terminal velocity is affected. In this study we
present a detailed experimental investigation of the effect of horizontally directed uniform
alternating electric field on the settling velocity and deformation characteristics of silicone oil
drop falling in castor oil medium. For the fluid pair used, the drop deforms to a prolate shape
on application of electric field which retards the drop motion in the vertical direction. There
is a marked decrease in drop velocity with increase in field strength of the alternating electric
field.
I. INTRODUCTION
Elecrohydrodynamics (EHD) is the study of motion of fluids subjected to electric
fields. There are several potential applications of EHD ranging from different industrial
processes to modern day microfluidic devices. These include the separation of emulsions
of oils into the individual oils, transport of biological cells by enclosing them in droplets,
enhanced heat and mass transfer due to mixing inside the drops and so on [1-4].
When a neutrally buoyant drop is subjected to an externally applied uniform electric
field, the droplet deforms to ellipsoidal shape [5-17]. Allan and Mason experimentally
observed that the droplet can deform to prolate (major axis of the ellipsoid is aligned in
the direction of external electric field) or oblate (major axis of the ellipsoid is
perpendicular to the direction of external electric field) shape in the presence of steady
uniform electric field [20]. Taylor in his classical study proposed the leaky dielectric
model and theoretically obtained that the prolate or oblate shape deformation is decided
by the electrical conductivity, permittivity and viscosity of the droplet and suspending
medium [17-19]. The leaky dielectric model assumes small but finite electrical
conductivities for both the liquids. The non-zero conductivities of the fluid pair give rise
to tangential electric stresses at the interface which drives fluid motion thereby generating
Proc. 2016 Electrostatics Joint Conference
2
hydrodynamic stresses. The resulting shape of the drop is established as a balance
between the electric and the hydrodynamic stresses. Droplet dynamics in alternating
electric field was first considered by Torza et al. [16]. They obtained that the deformation
can be decomposed into a steady component and a time-dependent component which
oscillates at a frequency twice that of the frequency of imposed electric field. This was
experimentally verified later by Vizika et al. [15]. All of these studies investigate the
effect of the electric field on neutrally buoyant droplets and the influence of gravity has
not been accounted for.
Droplet having density different than the suspending medium moves vertically in the
presence of gravity and the sedimentation/terminal velocity of the droplet is given by the
Hadamard-Rybczynski solution [21]. In this case, the droplet shape has been found out to
be spherical. Xu and Homsy experimentally investigated the settling velocity of a
Newtonian droplet under the combined influence of steady electric field and gravity [2223]. They have obtained that velocity of the droplet is significantly affected by the shape
deformation and interfacial charge convection effects. In a very recent study,
Bandopadhyay et al. studied the lateral migration of a sedimenting droplet in the
combined presence of tilted electric field and gravity [24]. Till now, most of the study on
droplet sedimentation in gravity has been focused on the dynamics of the droplets under
the influence of steady electric fields. There have been studies on the deformation of
droplets in alternating electric fields. However, the settling of droplets under the
influence of alternating electric fields has not been studied in detail. In this paper we have
presented an experimental study on the effect of alternating electric fields on the settling
velocity of droplets.
II.
EXPERIMENTAL SETUP
The experiments were conducted in a 14 cm X 14 cm X 20 cm Perspex cell fitted with
two square brass electrodes 5 mm thick having dimensions 12 cm X 12 cm. The distance
between the two electrodes was maintained constant at 12 cm. The edges of the brass
electrodes were filed in order to remove any sharp edges. The surfaces of the electrodes
were polished. All electrical connections were made using thick copper wires in order to
avoid corona formation due to application of high voltage. Drops were injected using a 5
ml hypodermic syringe. To minimize the effect of the side wall, droplets are injected
from the centre of the cross-section. The electric field is supplied through a High Voltage
Testing transformer rated 60 KV. The frequency of oscillation of the electric field is kept
at power frequency ~50 Hz constant throughout the experiments. An LED lamp placed
behind the cell was used for illumination. The experiments were recorded using two
different types of cameras. For observing the settling, a Nikon digital single lens reflex
(DSLR) camera was used. For capturing the droplet deformation, a high speed camera
capable of recording at 200 frames per second, integrated with a computer, was used. A
schematic of the set-up is shown in Fig. 1.
For the experiments, the pair castor oil (dispersing phase) and silicone oil (dispersed
phase) have been used. Both the fluids are known to be Newtonian and incompressible in
nature. The densities of the oils used are similar but the continuous medium (castor oil) is
much more viscous compared to silicone oil (See Table 1). As a result, the drop settles
slowly. The electrical conductivities of these oils are very small. Thus they act as good
Proc. 2016 Electrostatics Joint Conference
3
dielectric materials and can sustain very high electric fields.
Fig. 1. Schematic of the experimental set-up
III. EXPERIMENTAL METHOD
Three different sets of experiments were conducted. In the first set of experiments, the
drop falls in absence of any electric field. In the second set of experiments, the settling
velocity of the drop was observed with the help of a DSLR camera. The last set of
experiments involves observing the drop deformation with the help of the high speed
camera. Prior to starting the experiments, the camera is adjusted and focused for the best
possible recording. The camera and the LED light source are positioned at the safe
distance from the cell in order to avoid sparking from the connecting wires. This can
damage the instruments. The drop is injected using a syringe from the center of crosssection of the cell. The volume of injected oil cannot be controlled in a syringe. The size
of the injected drop was calculated from image processing later. After the drop falls
through a certain height, the voltage is applied through the transformer. The cell is tall
enough so that the drop attains a steady settling velocity before it reaches the bottom. The
drop size was varied for each voltage value from ~1 mm diameter to ~4 mm diameter.
The value of the voltage was increased in steps of 5 kV from 5 kV to 50 kV. As the
density difference between the experimental fluids is very small, the duration of each
experiment varies from ~2 minutes to ~10 minutes depending on the drop size. The
experimental videos were processed using the Image Processing Toolbox in MATLAB.
The results obtained from MATLAB were cross-checked by manual processing to ensure
the accuracy of the processing.
IV. RESULTS
A. Velocity of settling drop without any imposed electric field
Before investigating the pivotal effect of alternating electric field on the settling
velocity of the droplet, we perform experiments on the settling velocity of the droplet in
the absence of electric field. The terminal velocity of a settling drop is attained due to the
balance between the drag force and the gravitational force on the drop. As the size of the
drop increases, the weight of the drop increases more than the surface area dependent
Proc. 2016 Electrostatics Joint Conference
4
drop increases, the weight of the drop increases more than the surface area dependent
drag force. Hence, the settling velocity of the drop increases with increase in drop
diameter. The terminal velocity ∞ for a drop settling in another immiscible fluid
medium in absence of any other body force is given by the Hadamard-Rybczynski
solution [21]:
2  + 1 2 ( − 1)
∞ = �
�
9  + 2

(1)
3
Here, λ and κ are the ratio of the viscosities and densities respectively of the drop to the
dispersing medium, µ and ρ are the viscosity and density of the dispersing medium, R is
the drop radius and g is the acceleration due to gravity. In this case, the drop shape is
determined by the balance between the hydrodynamic stresses and the surface tension at
the interface and the analytical solution shows that the drop is perfectly spherical.
Fig. 2. Plot showing the variation of the settling velocity with drop radius for a drop falling under the influence
of gravity alone. The experimental data (in red) has been compared to theoretical results (in blue)
In Fig. 2 the variation of the settling velocity with the drop radius is shown. The points
plotted in red correspond to the experimentally obtained velocities and the dotted line
represents the theoretically predicted values calculated from eqn. 1. A close match is
obtained between the experimentally obtained values and those predicted by theory. The
settling of drops under the influence of gravity is a well studied phenomenon. Hence the
close agreement of experimental results with theory shows that the experimental
procedure and the post-processing followed in this study is fairly accurate.
B. Settling velocity under uniform horizontally directed alternating electric field
As the silicone oil drop settles in the immiscible castor oil medium, the velocity of the
drop increases to a steady value when the net force on the drop becomes zero due to
balance between the hydrodynamic drag and the gravitational force. In presence of an
electric field, the settling velocity becomes a function of both the drop size and the
strength of the electric field for fixed electrohydrodynamic properties. As the drop size
increases, the settling velocity increases as explained in section A.
Proc. 2016 Electrostatics Joint Conference
(a)
(b)
(c)
5
(a)
(b)
(c)
Fig. 3.Variation of settling velocity with drop speed. The image on the left shows the position of three drops of
different sizes at t=0. The image on the right shows the positions of the same drops at t=68 sec. The applied
electric field is 3.33 kV/cm.
Fig. 3 demonstrates the effect of drop diameter on the settling velocity of the drop in
presence of a horizontally directed uniform alternating electric field. Three drops of
different sizes subjected to the same electric field (3.33 kV/cm) are settling in the castor
oil medium separately. At t = 0, the drops are at the same location, but at t = 68 sec, the
drop having the maximum diameter falls through the largest distance.
In fig. 4 the droplet velocity variation with drop diameter for three different values of
electric field strength is shown. For a fixed value of drop diameter, fig. 4 depicts that the
settling velocity of the droplet decreases with increase in the field strength. Two
governing factors which affect the drop settling velocity in electric field are interfacial
charge convection and shape deformation. But Xu has previously shown that the effect of
charge convection at the droplet interface is less important in alternating electric field due
to the fact that there is no steady component of surface charge [23]. So, in the present
situation the shape deformation effect will be the deciding factor. We observe prolate
deformation of the silicone oil drop in castor oil medium. This prolate shape leads to
increase in cross-sectional area of the drop in the direction of settling which further
increases the hydrodynamic drag. This increase in hydrodynamic drag force on the
droplet finally reduces the droplet velocity for higher field strength.
Fig. 4. Variation of drop settling velocity with drop diameter for different electric fields
Proc. 2016 Electrostatics Joint Conference
6
Deformation of a settling drop under uniform horizontally directed alternating
electric field
When a silicone drop is injected into the castor oil medium using a syringe, the drop
falls under the effect of gravity. As the electric field is switched on, the drop deforms
which in turn affects the settling velocity of the drop. The deformation of the drop has
been observed to be dependent on the size of the injected drop and also on the strength of
the electric field. The deformation of the drop is expressed in terms of the deformation
ratio D given by:
C.
 =
 − 
 + 
(2)
Here, a is the length of the drop axis parallel to the direction of the electric field and b
is the length of the axis perpendicular to it as shown in Fig. 5.
Fig. 5. Drop deformation is quantified in terms of the deformation ratio which depends on the lengths of the
polar and equatorial axes of the drop.
The analytical solution derived by Torza et al. shows that the drop shape oscillates at a
frequency twice that of the imposed electric field [16]. This has been validated by
experiments. Torza et al. predicted that the deformation of the drop is a superposition of a
steady component and a time varying component. In our experiments, the DSLR camera
has been set to record at 50 frames per second. As the frequency of the applied electric
field is 50 Hz, the camera captures a particular point in the cycle. Hence the deformation
recorded by the DSLR camera remains constant. As the frequency of the electric field is
high, the oscillatory part of the deformation is very small compared to the steady
component [16]. Hence, for all practical purposes, we have treated the deformation
observed from the DSLR camera as the steady component of the deformation.
As the electric field strength is increased the drop deformation increases. Fig. 6 shows
the variation of the drop deformation with the electric field strength. Here the drop sizes
are similar.
(a)
(b)
(c)
Fig. 6. Drop shape observed by increasing the electric field from 0 to 4.17 kV/cm (peak value). The drop
diameter is similar in all the three cases.
Proc. 2016 Electrostatics Joint Conference
7
For neutrally buoyant drops under the influence of alternating electric fields, it has been
shown that the steady part of the deformation  varies linearly with 0 2  where R is the
radius of the undeformed drop, as [16]:
 =
9
Φ (E0 2 R)
16
(3)
Here, 0 is the peak electric field,  is the permittivity of the dispersing medium,  is
the surface tension and Φ is a function of the ratios of conductivity, permittivity and
viscosity. Our study reveals a similar trend for the steady state deformation and the
factor 0 2  as shown in Fig. 7.
Fig. 7. Plot showing variation of drop deformation with 0 2 
However, a very important thing to note from the images presented in figure 6(b-c)
that there is an asymmetry in the droplet shape with respect to a horizontal plane. This
asymmetry in droplet shape is more for higher field strength as observed in figure 6(c).
Similar asymmetry in droplet shape is also reported by Xu and Homsy for settling of
droplet in steady electric field which is the coupled effect of gravity and electric field
[22].
TABLE 1: PROPERTIES OF EXPERIMENTAL FLUIDS
Experimental fluids
Castor oil
Silicone oil
Density (kg/m³)
958
1023
Viscosity (Pa s)
0.693
Electrical conductivity (S/m)
Dielectric constant
0.0232
−10
2.78 × 10
7.70
9.26 × 10−11
3.88
Proc. 2016 Electrostatics Joint Conference
8
V. CONCLUSION
Electrohydrodynamic motion and shape deformation of a sedimenting viscous drop in
the presence of externally applied alternating electric field are investigated
experimentally. Here we study the nonlinear coupling of gravity (acting vertically) and
alternating electric field (acting perpendicular to the gravity) on the motion and
deformation characteristics of a non-neutrally buoyant droplet. Towards elucidating the
effects of shape deformation and interfacial charge dynamics on the droplet motion, we
perform controlled experiments using silicone oil droplets suspended in castor oil
medium. We study the droplet motion for different strength of applied electric field but
with fixed frequency of 50 Hz. Present study shows that the horizontally applied electric
filed retards the drop motion in the vertical direction by deforming the drop in prolate
shape. There is a significant decrease in drop velocity with increase in strength of applied
electric filed. Though the drop deforms to a prolate shape, there is a noticeable
asymmetry in the deformed shape. These studies on the motion of drops under the
combined influence of gravity and electric fields would help in designing more efficient
strategies for the separation of oils.
VI.
ACKNOWLEDGEMENTS
We are grateful to Dr. Rochish M. Thaokar and Dr. Manu Vashishtha at the Chemical
Engineering department in the Indian Institute of Technology Bombay for helping us
carry out conductivity and permittivity measurements using a broadband Dielectric
Spectrometer. We would also like to thank the technical staff at the Electrical and
Mechanical Engineering departments in the Indian Institute of Technology Kharagpur for
assisting us while conducting the experiments.
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