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Experimental investigation of drag coefficient of free-falling
deformable liquid gallium droplet
Citation for published version:
Mohamad, M, Mackenzie Dover, C & Sefiane, K 2018, 'Experimental investigation of drag coefficient of free-
falling deformable liquid gallium droplet', European Physical Journal: Applied Physics, vol. 84, 10903.
https://doi.org/10.1051/epjap/2018180271
Digital Object Identifier (DOI):
10.1051/epjap/2018180271
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Peer reviewed version
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European Physical Journal: Applied Physics
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Download date: 09. Aug. 2022
EPJ manuscript No.
(will be inserted by the editor)
Experimental Investigation of Drag Coefficient of Free-Falling
Deformable Liquid Gallium Droplet
M. Sofwan Mohamad
1,2
, C.M. Mackenzie Dover
1
and K. Sefiane
1
1
Institute for Multiscale Thermofluids, School of Engineering, University of Edinburgh, Faraday Building, King’s Buildings,
Edinburgh United Kingdom. EH9 3DW
2
School of Mechatronic Engineering, Universiti Malaysia Perlis (UniMAP), Kampus Pauh Putra, 02600 Arau, Perlis Malaysia.
Received: date / Revised version: date
Abstract. In this paper, the effect of shape and deformation on the drag coefficient of a free-falling liquid
gallium droplet in water in a terminal state is investigated experimentally. The temperature of the dispersed
and continuous liquid was varied in order to examine the effect on the liquid-metal droplets. The falling
droplets were imaged using a high speed camera and a simple model was developed to predict drag
coefficient over a Reynolds number range of 10
3
< Re < 10
4
. The drag coefficients of the deformed liquid
gallium droplets were found to be larger than that associated with a solid sphere and the associated Weber
number is below 4.5. It was found that the shape of all droplets in our experiment were oblate-spheroid. A
correlation has been established to predict the aspect ratio of a liquid gallium droplet moving in quiescent
water. The deformation is highly dependent on interfacial surface tension and inertial force, while the
viscosity ratio and pressure distribution have negligible effect.
Key words. Droplet, Drag coefficient, Deformation, Aspect ratio, Multiphase flows
1 Introduction
The motion and deformation of liquid droplets is impor-
tant to numerous industrial applications and has attracted
considerable research interest as a result. In industrial en-
gineering, understanding the dynamics of liquid droplets
is relevant to many applications such as the liquid sprays
injected in to combustion engines[1], ink-jet printers[2],
micro-fluidics[3] and cooling systems[4]. Among various
droplet materials, gallium and its alloys have drawn huge
attention recently because of the associated physical and
chemical properties such as low-melting points (liquid state
at or near room temperature) and high thermal conduc-
tivities. Such characteristics offer tremendous opportuni-
ties for developing advanced technologies in newly emerg-
ing areas such as electronic cooling[5,4], 3D printing and
printed electronics[6], flexible devices[7], soft actuators for
robotics[8] and adhesion[9].
Hydrodynamic drag is of major importance to count-
less industrial practices as it is one of the most significant
parameters that govern the movement of a droplet through
a liquid and reducing it could lead to a substantial energy
saving. To date, a substantial amount of literature can be
found on this topic. However, reported data on the dy-
namics of deformable droplet are limited. The study of
R. Clift et al.[10], E. Loth[11] and Wegener et al.[12] pre-
sented an extensive review of theory, experimental data
and relevant approximations representing the characteris-
tics of single droplet in fluid systems. As yet, there has
been no systematic investigation on the effect of liquid
gallium droplet morphology on the velocity and drag co-
efficient to the knowledge of the authors. In the following,
the dynamics of a free falling droplet of gallium in water
are studied and the effect of droplet morphology on the
velocity and drag coefficient are quantified. Characteris-
ing the relationship between the dynamics of the gallium
droplets and the medium it is moving through facilitates a
superior control of forces that inhibit droplet motion and
the correlation can be applied to a number of systems.
2 Experimental Setup
An experimental setup was designed to measure the ter-
minal velocity of the falling droplets, a schematic of which
is shown in Figure 1. The setup comprises a square cross
section, straight-walled column with closed bottom filled
with water. The column is 1000 mm in height and 70.45
mm in inner side width. These dimensions were selected
such that a droplet having diameter of less than 7 mm
can fall through the continuous liquid with minimal wall
effects, terminal velocity could be reached and end effects
could be avoided[13,14]. The column is made of clear Per-
spex, allowing the motion of the falling droplets to be
recorded by a high speed camera. The bottom part of the
column is inclined towards a central discharge point where
a liquid gallium retrieval mechanism is located. The liq-
uid gallium retrieval mechanism consists of two ball valves.
2 M. Sofwan Mohamad et al.: Drag Coefficient of Free-Falling Deformable Liquid Gallium Droplet
This allowed for the liquid gallium to be removed with a
minimal amount of water being discharged from the col-
umn. An electric heater is fitted at the bottom part of
the column and the water temperature inside the tank
is controlled by a proportional-integral-derivative (PID)
controller connected to two thermocouple probes that are
situated at the top and bottom of the chamber. The water
in the tank can be heated up to about 80
◦
C. In order to
minimise the effect of thermal convection during experi-
ments, the heater was turned off after heating the water
to the required temperature. A few minutes after that,
the temperature of water was found to be homogeneous
within ±1
◦
C.
-
+
V
LED Backlight
10uL
ºC
ºC
70.0
30.0
PID controller
Power supply
PID
controller
Heater
Syringe pump
TC 1
TC 2
Syringe
Droplet
Perspex tank
Valves
PC control
High speed
camera
40
Coil heater
Nozzle
Fig. 1: Experimental set up.
Liquid gallium was heated to the desired temperature
and then fed from a syringe in to the column of water.
A precise volume of liquid gallium was released gradu-
ally, directly below the water surface to create a droplet
with the help of a programmable syringe pump (Cole-
Parmer Touch-Screen Syringe Pump 78-8110C). The ac-
curacy of the syringe pump is ±0.355% with reproducibil-
ity of ±0.05%. Droplets were increased in volume until
they detached from the needle due to their weight and
needles of different diameters could be attached to change
the size of the examined droplet. The droplet behaviour
during free fall was recorded by a high speed CMOS cam-
era (Nanosense MKIII) with frequency up to 1024 fps at
a resolution of 1280x1024 pixels. The camera was cou-
pled with an Edmund optics double gauss lens (54-691)
to produce low distortion images. A custom made 100 W
LED light (Maxilux ProStrip120-High Power) was placed
at about 40 mm behind the column to illuminate the test
region.
The image sequences were converted into 8-bit grey
scale, background-subtracted and thresholded such that
Table 1: Equivalent diameter of liquid gallium droplets
used in experiment.
Volume (uL) Equivalent diameter (mm)
17 3.19
60 4.86
90 5.56
spheres appeared as a black dot on a white background
to allow for tracking and quantification of droplets using
an in-house ImageJ macro. Information such as the rela-
tive position, velocity and diameter were obtained using
automated analysis of this type. Figure 2 shows a series
of images pre- and post-processing.
(a) Original image (b) Treated image
Fig. 2: Shape of liquid gallium droplets at terminal condi-
tion.
3 Results & Discussion
3.1 Diameter of the droplet
The diameter of the droplet can be calculated from a
known dispersed volume of gallium. Volume equivalent di-
ameter, d
eq
=
3
q
3V
4π
can be calculated with the assump-
tion that the droplet is a solid sphere symmetric about the
major and minor axis plane of observation. Table 1 sum-
marizes the calculated diameter of liquid gallium droplets
of various volumes. The calculated equivalent diameters
are a convenient parameter to represent the droplets as
well as to understand the effect of deformation on drag
coefficient.
3.2 Droplet terminal velocity
Figure 3 shows co-plots of the velocity of free falling liquid
gallium droplets of various diameters in water as a func-
tion of their distance from the needle. It can be seen that
M. Sofwan Mohamad et al.: Drag Coefficient of Free-Falling Deformable Liquid Gallium Droplet 3
the velocity becomes terminal after 200 mm for all sizes of
droplets examined. After analysing the droplet’s velocity
under different thermal conditions, it was observed that
each droplet in the investigation could attain a terminal
condition after falling a distance of 200 mm. Hence, the
droplet image sequence ranging from 200 to 250 mm below
the needle was selected to calculate the average droplet
terminal velocity (u
T
) and drag coefficient (C
D
). In our
experiments we did not observe any significant difference
in u
T
by varying the temperature of liquid gallium and
water as depicted in Figure 4 (further discussion in Sec-
tion 3.4). However, it is obvious that the u
T
is dependent
on the droplet size.
50 100 150 200 250
Depth [mm]
0.45
0.50
0.55
0.60
0.65
0.70
Velocity [m/s]
3.19 mm
4.86 mm
5.56 mm
Fig. 3: Liquid gallium droplets velocity as a function of
their distance from the needle in water.
30
o
C 40
o
C 50
o
C 60
o
C 70
o
C
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Terminal velocity [m/s]
3.19 mm
4.86 mm
5.56 mm
Fig. 4: Liquid gallium droplets terminal velocity.
3.3 Drag coefficient
Calculating the C
D
was necessary to characterise the mo-
tion and the kinematics of the droplets. Firstly, a theo-
retical correlation to obtain C
D
as a function of Reynolds
number (Re) and fluid properties was developed. Then,
experimental values obtained by image processing were
used to calculate the C
D
and the effect of deformation is
subsequently investigated. A simple correlation based on
net gravitational force (gravitational-buoyancy) and drag
force balance for a sphere at terminal condition are given
by Eq. (1).
π(ρ
d
− ρ
c
)g
d
3
eq
6
=
1
8
C
D
πd
2
eq
ρ
c
u
2
T
(1)
Note that d
eq
is used in this equation to remove the effect
of deformation. Eq. (1) can be rearranged to calculate C
D
based on C
D
= f(Re) as follows:
C
D
=
4
3
(ρ
d
− ρ
c
)ρ
c
gd
3
eq
µ
2
c
/Re
2
(2)
where Re =
ρ
c
u
T
d
µ
c
, is the particle Re, µ
c
and ρ
c
are the
continuous phase fluid viscosity and density respectively
and d is the diameter of the droplet.
Next, C
D
was derived from the experiment. In order
to obtain the exact value of C
D
, the droplet deformation
was considered. Instead of d
eq
, the droplet’s actual frontal
diameter obtained directly from the images through im-
age processing was used. Figure 5 compares the C
D
calcu-
lated from Eq. (2) with those from experiment along with
the classical curve of C
D
of a solid sphere. An acceptable
agreement between theory and experiment was validated
by the minimal discrepancy between theoretical and ex-
perimental values. The differences in those values, may be
due to the effect of deformation that been ignored in the
derivation of Eq. (2). Moreover, it is noted that as the Re
increases, the experimental C
D
values deviated from those
calculated for an equivalent solid sphere and the difference
are more pronounce for larger droplets.
3.4 Shape & deformation of droplet
The C
D
of a falling droplet is known to be dependent
upon the droplet shape. The shape of droplets experi-
encing free fall in an infinite medium under the influence
of gravity are generally classified in three categories as
shown in Figure 6. These categories are defined by the
ratio of its largest horizontal dimension (equator), d
h
, to
its largest vertical dimension (polar), d
v
, known as aspect
ratio, E =
d
h
d
v
. Droplets are classified as spherical if the
aspect ratio lies within 10% of unity. For an aspect ratio
other than that, droplet is described deformed and can
be classified as an ellipsoid, which has two sub-categories
namely oblate-spheroid and prolate-spheroid.
Figure 7 shows representative photographs of different
sizes of liquid gallium droplets used in our experiment as
4 M. Sofwan Mohamad et al.: Drag Coefficient of Free-Falling Deformable Liquid Gallium Droplet
10
3
10
4
Reynolds Number
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Drag Coefficient
3.19 mm
4.86 mm
5.56 mm
Solid sphere
30
40
50
60
70
Fig. 5: The relationship between drag coefficient and
Reynolds number (open symbols are experimental values
and filled symbols are theoretical values).
(a) Spherical (b) Prolate-spheroid
(c) Oblate-spheroid
Fig. 6: Schematic diagrams of typical droplet shape.
they fall through water and Figure 8 illustrates the dimen-
sions of those droplets obtained from image processing of
the images for 5 sequences for each diameter. The gallium
droplets are deformed even at the smallest diameter and
the value of aspect ratio is always above 1.1 (see Figure 9),
thus we can classify all of the droplets as oblate-spheroid.
The deformation of the droplets from spherical shape can
also be confirmed visually by looking the images in Fig-
ure 7. Due to aerodynamic and hydrostatic pressure, the
droplet aspect ratio continues to increase with droplet di-
ameter[15]. With the Re in the current investigation rang-
ing from 10
3
to 10
4
, this observation might extend the con-
clusion made by Taylor and Acrivos[16] to a wider range
of Re, stated that a droplet is likely to be deformed into
an oblate instead of a prolate-spheroid for all cases at low
Re.It is also noted that the temperature does not have
a significant effect on the steady state shape of a liquid
gallium droplet.
Deformation occurs because of the interplay between
pressure distribution and surface tension. The pressure
distribution produces local fluid-dynamic stresses, which
Table 2: Physical parameters of the continuous and dis-
persed fluid.
T (
◦
C) ρ
c
µ
c
ρ
d
µ
d
σ
c−d
30 995.7 0.798 6084 139.0 698.4
40 992.2 0.653 6077 137.7 698.1
50 988.0 0.547 6071 136.5 697.9
60 983.2 0.466 6064 135.2 697.5
70 977.8 0.404 6058 134.0 697.2
are controlled by the viscosity ratio, λ (which controls
the circulation inside droplet) and Re (which controls the
importance of viscosity). Surface tension is affected by
the Weber number (W e), the ratio of continuous fluid
stresses, which causes deformation to the surface tension
stresses, which oppose deformation. These independent
non-dimensional parameters can be calculated using the
following equations:
λ =
µ
d
µ
c
(3)
W e =
ρ
c
u
2
d
σ
c−d
(4)
where µ
d
is the dispersed fluid viscosity. σ
c−d
is the inter-
facial surface tension between the dispersed and continu-
ous fluid. This parameter could possibly be approximated
by utilizing Eq. (5)[17]. Table 2 summarises the physical
properties of liquid gallium and water at different temper-
ature that are relevant to the experiment.
σ
c−d
= σ
d
+ σ
c
− 2
√
σ
c
σ
d
(5)
Figure 10 shows the relationship between the aspect ra-
(a) 3.19 mm (b) 4.86 mm (c) 5.56 mm
Fig. 7: Shape of liquid gallium droplets at terminal condi-
tion.
tio and W e for different λ. In our experiments, viscosity
variation is caused by varying the temperature of the liq-
uid gallium and water as shown in Table 2, producing a
range of λ from 166 to 329. Internal circulation inside the
droplet caused by the λ, which produces high pressure at
the leading and trailing edge of the droplet, tends to cause
prolate shapes. On the other hand, fluid surrounding the
droplet also produces a high-pressure zone at the trailing
and leading edges of the droplet and a low-pressure zone
near the equator[18]. This opposes the effect of internal
circulation and tends to cause oblate droplet shapes[19]. In
our experiments, the viscosity of the dispersed fluid (liquid
