Experimental Thermal and Fluid Science 55 (2014) 150–157
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Experimental Thermal and Fluid Science
journal homepage: www.elsevier.com/locate/etfs
Experimental investigation of a drop impacting on wetted spheres
Gangtao Liang, Yali Guo, Xingsen Mu, Shengqiang Shen ⇑
Key Lab. of Ocean Energy Utilization and Conservation of Ministry of Education, School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history:
Received 26 December 2013
Received in revised form 15 March 2014
Accepted 16 March 2014
Available online 24 March 2014
Keywords:
Drop impact
Wetted sphere
Spreading
Splashing
a b s t r a c t
Numerous experiments were performed to investigate a heptane drop impact dynamics on wetted
spheres using a high speed camera. Outcomes after impact include spreading at a low impact Weber
number and splashing at a high value. Limits between the two outcomes can be greatly affected by
the sphere-drop curvature ratio ranging in 0.090–0.448. Additionally, the spreading process on wetted
spherical surfaces is discussed in detail. The spreading factor defined as the ratio between the spreading
area and the drop surface area can be increased by increasing the curvature ratio or by reducing liquid
viscosity, while the effect of the increment in the Weber number is minor. It is found that the spreading
factor follows a linear law with dimensionless time, which is confirmed by the butanol drop spreading as
well. Finally, concerning different curvature ratios and fluids, many coefficients with respect to the linear
law are obtained to predict the spreading scale by regressing the experimental data.
Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction
The drop impact phenomenon is common in industry, such as
drops impact on surfaces of heat transfer tubes in horizontal-tube
falling film evaporators [1], spraying cooling [2], ink jet printing
[3], plasma spraying technique [4] as well as various fire safety situations [5], etc. Rein [6] summarized the main studies focused on
the liquid drop impact phenomenon and made comprehensive reviews about this subject. Some research in the public literature
precisely inspires the present investigation.
For the drop impact on dry solid surfaces, Fukai et al. [7] presented a theoretical study on the spreading process accounting
for the surface tension, and obtained the recoiling occurrence
and mass accumulation around the spreading film periphery. Their
results also show that the dependence of the maximum spreading
radius on time is non-monotonic. Later, Fukai et al. [8] proposed
another theoretical model, considered the presence of inertia, viscosity, gravitation, surface tension and wetting effects. Their theoretical model predicts well the deformation of the impacting drop,
not only in the spreading phase, but also during recoiling and oscillation. Rioboo et al. [9,10] conducted many experiments and provided qualitative and quantitative analysis. The results show that
outcomes after impact include splashing, rebound, partial rebound
and deposition. The time evolution of the spreading factor is divided into four distinct phases: the kinematic phase, the spreading
phase, the relaxation phase and the wetting/equilibrium phase. Xu
⇑ Corresponding author. Tel.: +86 0411 84708464.
E-mail addresses: gtliang@126.com (G. Liang), zzbshen@dlut.edu.cn (S. Shen).
http://dx.doi.org/10.1016/j.expthermflusci.2014.03.008
0894-1777/Ó 2014 Elsevier Inc. All rights reserved.
et al. [11] investigated the influence of the surrounding gas pressure on splashing limits and found a striking phenomenon: splashing can be suppressed by decreasing the surrounding gas pressure.
Afterwards, Xu et al. [12,13] reported the interplay between substrate roughness and the surrounding gas pressure. They associated two distinct types of splashing with each parameter:
prompt splashing is due to surface roughness, while corona splashing is resulted from instabilities produced by the surrounding gas.
Bi et al. [14] experimentally found that liquid viscosity plays a
decisive role in the spreading process, and surface tension has a
leading influence on the recoiling process. Both the two properties
jointly determine the oscillation characteristics. For a single drop
impact on hot surfaces, Negeed et al. [15] discussed thermal properties of the hot surface and drop characteristics on the drop evaporation. Later, Negeed et al. [16,17] presented solid–liquid contact
time and the maximum drop spreading diameter, concerning effects of the surface roughness amplitude, the oxide layer thickness,
the We, and surface superheat.
In some industrial equipments, the impact target surfaces are
not always planar. For example, in horizontal-tube falling film
evaporators, liquid drops impact on heat transfer tubes. The target
surfaces are curved instead of planar. Concerning a drop impact on
curved dry surfaces, Pasandideh-Fard et al. [18] simulated a 2 mm
water drop impact on tubes with the diameter in 0.5–6.35 mm and
low velocity of 1 m/s. They found that drops landing on the largest
tube cling to the solid surface, but for smaller tubes, there are not
enough surface areas for the liquid to remain attached, and drops
fall off after impact, disintegrating into several smaller drops. Hung
and Yao [19] studied experimentally water drops with a diameter
G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157
151
Nomenclature
A
d
h
k
Oh
Ra
Re
t
v
We
area, mm2
diameter, mm
film thickness, mm
coefficient
Ohnesorge number, l/(qrddrop)1/2
surface roughness, lm
Reynolds number, qvddrop/l
time, ms
impact velocity, m/s
Weber number, qv2ddrop/r
Greek symbols
d
dimensionless film thickness, h/ddrop
l
liquid viscosity, Pa s
q
liquid density, kg/m3
of 110–680 lm impacting on isothermal cylindrical wires. Their results show that outcomes after impact include disintegration and
dripping. Smaller drops are disintegrated if the incoming drops
have high velocity or the wire diameter is small. Larger dripping
drops are formed when the velocity is low or the wire diameter
is large. Shen et al. [20] studied influences of several dimensionless
parameters on the drop deformation after impact on a two-dimensional round surface using lattice Boltzmann implementation of
the pseudo-potential model. Four typical deformation processes
can be found in their research: moving, spreading, nucleating
and falling. In Chow and Attinger [21], the drop diameter was
80 lm and the target sphere diameter was in the range of 0.06–
10 mm. Their visualization experiments show that the sphere curvature has no significant influences on the maximum spreading
factor for a substrate-drop curvature ratio below 0.3. Hardalupas
et al. [22] reported experiments on liquid drops with the diameter
160–230 lm impacting on small solid spheres with the diameter
0.8–1.3 mm at impact velocity 6–13 m/s. They observed a retraction of the liquid crown at low drop impact velocity and disintegration from cusps located on the crown rim at high impact velocity.
They also pointed out that the increase in the sphere curvature
promotes the splashing onset. Bakshi et al. [23] reported experimental investigations of drops with the diameter 2.4–2.6 mm
impacting onto a spherical target of 3.2 mm in diameter. Spatial
and temporal variations of the film thickness on the target surface
were measured. Three distinct temporal phases of the film dynamics are clearly visible from their experimental results: initial drop
deformation, inertia dominating and viscosity dominating.
However, the research above mentioned is limited to a drop impact on dry surfaces. For the impact on wetted surfaces, i.e., solid
surfaces covered by thin liquid films, a lot of work was also completed. Cossali et al. [24] and Motzkus et al. [25] defined the impact
target as a thin liquid film when the dimensionless film thickness d
varies in the range of 0–1. Rioboo et al. [26] found experimentally
three outcomes including deposition, crown formation without
splashing and splashing by varying impact velocity (0.44–3.14 m/
s), the drop diameter (1.42–3.81 mm) and the dimensionless film
thickness (0.004–0.189). Particularly for a dimensionless film
thickness less than 0.02, crowns without splashing could almost
no longer be observed. Okawa et al. [27] and Shi et al. [28] gained
the same results by experiments and three-dimensional simulations. Cossali et al. [24] and Vander Wal et al. [29] associated
splashing with the production of satellite drops separating from
the crown liquid sheet after the impact, which were named as secondary drops. Cossali et al. [24] firstly distinguished two kinds of
splashing: prompt splashing and delayed splashing. The prompt
r
s
U
-
surface tension, N/m
dimensionless time, vt/ddrop
spreading factor, As/Adrop
sphere-drop curvature ratio, ddrop/dsphere
Subscripts
c
critical
drop
liquid drop
h
horizontal
s
spreading
sphere solid sphere
v
vertical
splashing is associated with ejected drops from the crown edge
when it is still advancing, while the delayed splashing occurs near
or after the crown maximum expansion and is associated with the
crown wall breakup. Motzkus et al. [30] also demonstrated outcomes of coalescence, prompt splashing and delayed splashing in
their work.
From the above reviews on a single drop impact phenomenon,
we note that most impact targets are dry planar or curved surfaces,
and studies focused on the impact on wetted surfaces are limited
to planar wetted surfaces. Inspired by the above research, it is
found that there are few studies especially focused on the impact
dynamics for a single drop impinging on wetted curved surfaces.
In the previous study [31], the outcomes after a single drop impact
on wetted cylinders were presented. Later, the rebound and
spreading processes were discussed in detail in [32]. However,
the drop impact on cylinders is a three-dimensional problem. For
a drop impact on wetted spheres with a two-dimensional geometry, the impact behavior is still unclear. Thus, in the present research, outcomes after a single heptane drop impact on wetted
spheres are presented through experimental observations using a
high speed camera. In addition, the spreading factor is analyzed,
with respect to influences of the We and the sphere-drop curvature
ratio -.
2. Experimental apparatus and procedures
The experimental apparatus is similar with that in [31], and
shown in Fig. 1. The main components include a syringe, a hypodermic needle connected with the syringe by a latex tube to generate drops, a high speed camera, a wetted sphere, a xenon lamp
used to provide illumination for photography, a light diffuser,
and a data acquisition computer.
A single drop can be formed by forcing the liquid in the syringe
at a certain pressure through the stainless steel hypodermic needle. The needle is flat tipped, with an inner diameter of 0.50 mm.
The drop is formed at the tip of the needle and detaches when
the gravity exceeds the surface tension force. The impact behavior
is recorded by a Phantom V12.1 high speed camera with capacity
of 106 frames per second, equipped with a 100 mm, f-2.8 Tokina
macro lens. The camera is aligned horizontally. In order to obtain
photographs with sufficient image resolution, the shooting speed
is set as 10,000 frames per second, with 1024 512 pixels in each
image. The back light method is employed in the experiments to
expose the impact images and the cold light source is provided
by a xenon lamp XD-300 with a power of 350 W. A light diffuser
is used between the sphere and the xenon lamp to make the light
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G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157
Fig. 1. The schematic diagram of the experimental apparatus.
70
butanol
heptane
60
50
h (μm)
be distributed uniformly on the wetted spherical surface. Because
the whole impact process occurs in a very short time, the trigger
mode is selected as post trigger. Namely, after the drop impact
on the wetted sphere, the trigger is launched, then the signal is
delivered to the data acquisition computer and the impact process
is recorded.
Before each experiment, several drops are cleared away to ensure that the liquid remains free of any air bubbles. Seven spheres
are adopted with the diameter ranging from 4 mm to 20 mm. The
spheres are polished by the Cw 1500 silicon carbide electro coated
abrasive paper to assure that the average roughness of the sphere
surfaces Ra is less than 0.05 lm. Heptane is selected as the experimental fluid. The drop diameter can be acquired by pixel analyzing and calibration is performed by using a reference substance.
The software MATLAB 7.1 is used to fulfill the pixel analyzing process. The drop is similar to an ellipse and the diameter is measured
in both the horizontal and vertical directions. In Rioboo et al. [10],
Stow and Hadfield [33], the equivalent diameter of the ellipse is
defined as (d2hdv)1/3. The equivalent diameter of the heptane drop
is 1.79 mm. The uncertainty is 1 pixel, and the error is 0.025 mm,
which corresponds to a 1.40% relative error of the real diameter.
Hence, the sphere-drop curvature ratio is ranging from 0.090 to
0.448. The distance between the needle tip and the sphere is adjusted to vary drop impact velocity. The impact velocity v is derived by tracking the location of the drop centroid in two images
with 0.5 ms time spacing before impact, which ranges from
0.19 m/s to 2.14 m/s with an accuracy of ±0.05 m/s.
Before the experiment, a thin liquid film is spread on the sphere
surface uniformly by using a high-quality painting brush. Both the
drop and the film are heptane in the experiments. When the liquid
film becomes stable and has a relatively uniform thickness, the
experiment can be started. The thickness of the liquid film is measured directly by comparing an image of the film surface to a reference image of the dry sphere. Before each impact test, the preexisting film thickness is measured by a SLR digital camera (Canon
EOS 5D Mark II) with a larger resolution than the high speed one.
The film thickness is in the range 12–33 lm with an accuracy
±4 lm, depending on the sphere diameter. Fig. 2 shows the preexisting film thickness at the impact point with different curvature
ratios. It can be seen that the larger curvature ratio results in the
smaller film thickness, caused by the larger curvature of the
sphere. Cossali et al. [24], Stow and Stainer [34] pointed out that,
when the surface roughness is comparable with the pre-existing
film thickness, the roughness influence becomes remarkable. However, the roughness 0.05 lm is much lower than the smallest film
thickness 12 lm. Besides, all the spheres are polished with the
same type of the abrasive paper, so the roughness influence is
not considered in this research. In order to further verify the proposed linear rule of the spreading factor, the other fluid of butanol
is selected, the pre-existing film thickness of which is also
presented in Fig. 2. Some experiments are done with small
impact velocity. Table 1 summarizes the liquid properties and
40
30
20
10
0
0.0
0.1
0.2
0.3
0.4
0.5
ϖ
Fig. 2. The pre-existing film thickness.
experimental conditions adopted in the present study, whilst Table 2 presents ranges of dimensionless parameters.
3. Results and discussions
3.1. Outcomes after impact
The number embedded in the following images is the evolution
time with the unit ms. The time sequence 0.0 ms is set as the drop
exactly contacts with the wetted surfaces.
For the small impact We corresponding to - = 0.149, the drop
spreading process can be observed in Fig. 3. When the drop touches
the wetted spherical surface, liquid in the drop that firstly contacting with the wetted surface begins to flow around on the surface
(0.4 ms). As the spreading continues, the periphery of the spreading film extends outward continuously (1.6 ms). Namely, the surface area of the wetted sphere covered by the spreading film
becomes larger. Then the periphery becomes more and more
inconspicuous and a stable film is formed on the wetted sphere
at last (11.0 ms). It is noted that the rebound phenomenon at the
low impact We still does not occur for the heptane drop through
many repeatable experiments.
For the same curvature ratio, by increasing the impact We to a
certain high value, the splashing phenomenon can be seen in Fig. 4.
An extremely thin jet is generated immediately after impact in the
region where the drop connects with the wetted spherical surface
Table 1
Liquid properties and experimental conditions.
Liquid
r (N/m) l (Pa s)
q (kg/m3) ddrop (mm) h (lm) v (m/s)
Heptane
Butanol
0.0201
0.0201
684
810
0.000409
0.00295
1.79
1.82
12–33
36–61
0.19–2.14
0.34–1.46
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G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157
Table 2
Ranges of dimensionless parameters.
Liquid
We
Re
Oh
-
Heptane
Butanol
2–279
8–156
569–6406
170–730
0.0026
0.0171
0.090–0.448
0.091–0.455
Fig. 5. Spreading of the heptane drop with - = 0.448 and We = 67.
Fig. 6. Splashing of the heptane drop with - = 0.448 and We = 278.
450
Oh = 0.0026 Present experiments
Oh = 0.0175 Wang and Chen(2000)
Oh = 0.0026 Asadi and Passandideh-Fard (2009)
Oh = 0.0018 Asadi and Passandideh-Fard (2009)
Oh = 0.0028 Motzkus et al. (2011)
360
270
We c
(0.1 ms). Then the jet develops into an unobvious liquid sheet at
0.3 ms. However, this liquid sheet only sustains an extremely short
duration and quickly cracks into many secondary drops (0.5 ms).
By 0.9 ms, the sheet has collapsed completely. Thus, the splashing
in Fig. 4 pertains to the typical prompt splashing defined in Cossali
et al. [24], caused by low viscosity of the heptane drop.
For the higher curvature ratio - = 0.448, the drop spreading
process is presented in Fig. 5. Compared with Fig. 3, experimental
observations in Fig. 5 suggest that the spreading qualitative behavior varies less with different curvature ratios. However, for the
drop splashing in Fig. 6 with - = 0.448, it appears a few differences. In Fig. 6, the secondary drops are much smaller than that
in Fig. 4. Moreover, the tiny secondary drops move horizontally
after impact in Fig. 6. While in Fig. 5, the secondary drops scatter
more irregularly.
When splashing occurs, the critical impact velocity can be acquired by fine adjustment of the impact height. Fig. 7 shows curves
of the critical Weber number Wec for heptane at different curvature ratios. It shows that the Wec decreases with the decrement
in the curvature ratio when the curvature ratio is larger than
0.224, while for - < 0.224 the Wec varies little and almost keeps
a constant 124. For the large curvature ratio, the sphere diameter
is small, so there is drop downward slippage after impact along
the wetted surface due to gravity. One part of the impact energy
has not been applied to overcome the adverse effect of the flow
resistance and surface tension force. Hence, more impact energy
is required to attain the splashing occurrence, i.e., the Wec is high
for the large curvature ratio. With the decrement in the curvature
ratio, such downward slippage is alleviated, as a consequence the
Wec is reduced. However, when the curvature ratio is less than
0.224, the wetted surface approximates to a planar surface and
the curve effect fades away gradually. Hence, the Wec does not
change with the curvature ratio. To make this interpretation be
more conclusive, several splashing limits for a single drop impact
on a horizontal liquid film (- = 0) in the literatures are selected
to make a brief comparison with the present results in Fig. 7. Refs.
[35–37] suggest that the splashing limits grow with the increment
in the Oh or the viscosity. In Fig. 7 it is noted that for Oh = 0.0175 in
Wang and Chen [35] and Oh = 0.0028 in Motzkus et al. [30], their
thresholds are larger than Oh = 0.0026 in the present experiments
with - < 0.224, while the critical value for Oh = 0.0018 in Asadi and
Passandideh-Fard [37] is smaller than the present value. However,
for the same Oh = 0.0026 in [37], their result agrees extremely well
with the present one. Thus, when the curvature ratio is larger than
180
90
0
0.0
0.1
0.2
0.3
0.4
0.5
ϖ
Fig. 7. Splashing limits of the heptane drop on wetted spheres.
0.224, the curve effect should be taken into account with respect to
the splashing process.
3.2. Spreading factor
The foremost parameter in drop spreading is the spreading
magnitude. A broad spreading can be desired to enhance the transport phenomena between the drop and the substrate in some
application aspects such as ink jet printing [38]. When the drop impinges on the horizontal surface, the spreading factor, which is defined as the ratio between the spreading film diameter and the
drop diameter, is used to measure the spreading magnitude [14],
whereas for the inclined surface, back and front spreading factors
are introduced [39]. However, when the drop spreads on the
curved surface, it is inappropriate to use these parameters to measure the spreading because the contact surface is not planar anymore: it is a spherical cap. Thus, in this research, the spreading
Fig. 3. Spreading of the heptane drop with - = 0.149 and We = 47.
Fig. 4. Splashing of the heptane drop with - = 0.149 and We = 191.
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G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157
1.0
ϖ = 0.448
ϖ = 0.224
ϖ = 0.149
ϖ = 0.090
hs/ t (m/s)
0.8
Fig. 8. The spreading model after impact.
area instead of the spreading diameter is used here. Fig. 8 is the
spreading model after impact, where the point O is the impact
point, the points A and B determine the raised spreading rim and
the point C is the foot point from O to the segment AB. The spreading height hs is the length of AB, so the spreading area As can be expressed as
As ¼ phs dsphere ;
0.6
0.4
0.2
0.0
0
2
4
6
8
10
t (ms)
Fig. 10. The specific value between hs and t with We = 42–67.
ð1Þ
and the spreading factor U is defined as
U¼
As
:
Adrop
impact drop just spreads in a small region with the area about
12.5 mm2 and this curved region can be deemed as a flat surface,
so the spreading factors are almost the same with different curvature ratios. Nevertheless, after the initial stage, the effect of the
curvature ratio becomes prominent. As described in introduction,
the wetted surface is the solid surface covered by a very thin film.
If it is assumed that the impact point corresponds to the 0° included angle, when this pre-existing film becomes relatively stable,
the film thickness increases with angles for the left or right hemisphere due to gravity. Taking - = 0.224 for example, at the angles
45°, 90° and 135°, the film thickness is about 26 lm, 30 lm and
41 lm, respectively. The smaller curvature ratio means that there
is enough space for the drop to spread on the upper position of
the wetted sphere. While for the large curvature ratio, the
ð2Þ
The spreading factor in the following figures is only plotted if it
can be clearly measured.
Fig. 9 shows the spreading factor at different conditions. From
Fig. 9(a)–(c) it is found that the We has minor effects on the spreading factor in the present curvature ratio range, though some discrepancies emerge in Fig. 9(a) with - = 0.448, caused by the
relatively shorter lifetime of the spreading film. However,
Fig. 9(d) shows that the curvature ratio effect is remarkable. When
s is less than 0.7, the curvature ratio does not influence U, while for
s > 0.7, with the decreasing of the curvature ratio, the spreading
factor is increased. It is considered that at the incipient stage, the
10
3.0
ϖ = 0.224
ϖ = 0.448
2.5
8
6
Φ
Φ
2.0
1.5
4
We = 67
We = 116
We = 209
1.0
0.5
0.0
0.0
0.3
0.6
0.9
1.2
We = 8
We = 42
We = 87
2
0
1.5
0
1
2
3
τ
τ
(a)
(b)
10
4
5
12
ϖ = 0.112
heptane
8
9
Φ
Φ
6
4
We = 7
We = 38
We = 77
2
0
0
1
2
3
τ
(c)
4
6
ϖ = 0.448
ϖ = 0.224
ϖ = 0.149
ϖ = 0.090
3
5
0
0
1
2
3
4
5
τ
(d)
Fig. 9. The heptane spreading factor, (a)–(c) the We effect with different curvature ratios and (d) the curvature ratio effect with We = 42–67.
155
G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157
15
8
heptane
butanol
12
6
9
Φ
Φ
4
6
2
3
0
0
0
2
4
6
8
0
1
2
4
5
Fig. 14. Experimental data of the spreading factor for butanol.
Fig. 11. Experimental data of the spreading factor for heptane.
3.0
2.4
2.4
1.8
1.8
k
k
3.0
1.2
1.2
0.6
0.6
0.0
0.0
0.1
0.2
0.3
0.4
0.0
0.0
0.5
0.1
0.2
0.3
0.4
Fig. 15. The k value for butanol.
Fig. 12. The k value for heptane.
spreading rim arrives at the lower position quickly, where the film
is thicker and the spreading film will encounter more resistance, as
a consequence the spreading expands slowly. Therefore, the
spreading factor becomes larger with the decrement in the curvature ratio.
It is also noted that the spreading factor almost increases linearly with dimensionless time in Fig. 9. To verify this linear law,
the definition of the spreading factor in Eq. (2) is derived:
phs dsphere hs dsphere
As
¼
¼
:
2
Adrop
pd2drop
ddrop
ð3Þ
Substituting the impact velocity and time yields
U¼
hs dsphere
2
ddrop
¼
hs dsphere
vt
:
v tddrop ddrop
ð4Þ
Then combining the dimensionless time definition yields
U¼
dsphere
v ddrop
hs
s:
t
ð5Þ
For the given curvature ratio and the impact We, the first term
from the equation right side in Eq. (5) is a constant. If the second
term (hs/t) can keep a constant, the linear law of the spreading fac-
3.0
6.0
ϖ = 0.455
butanol
2.4
4.5
Φ
Φ
1.8
3.0
1.2
We = 37
We = 80
We = 130
0.6
0.0
0.0
0.4
0.5
ϖ
ϖ
U¼
3
τ
τ
0.8
1.2
1.6
ϖ = 0.455
ϖ = 0.228
ϖ = 0.152
ϖ = 0.091
1.5
2.0
0.0
0.0
0.5
1.0
1.5
τ
τ
(a)
(b)
2.0
2.5
3.0
Fig. 13. The butanol spreading factor, (a) the We effect with - = 0.455 and (b) the curvature ratio effect with We = 37–57.
156
G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157
tor with time can be affirmed. Fig. 10 presents the specific value
between hs and t with different curvature ratios. It suggests that
the second term (hs/t) varies less with time and the value grows
with the increasing of the curvature ratio. Hence, a linear spreading
model for a drop impact on wetted spheres can be proposed in Eq.
(6):
U ¼ ks;
ð6Þ
where k can be acquired by regressing the experimental data.
Fig. 11 provides the spreading factor data in the present experimental range shown in Tables 1 and 2. Due to influences of the
curvature ratio, the data are scattered. Hence, it is not proper to
fit all the data by using one correlation. Based on this situation,
more experiments are supplemented concerning different curvature ratios, in order to regress the k value for each curvature ratio
by utilizing the linear model in Eq. (6). Fig. 12 shows the k value for
heptane at different curvature ratios. It is found that the k value
fluctuates a little, but the overall trend is still increasing with the
decrement in the curvature ratio. This result coincides well with
that in Fig. 9(d).
Many experiments are also performed by adopting butanol as
the experimental fluid due to the same surface tension with heptane, so that the linear rule of the spreading factor can be further
confirmed. The experimental conditions are presented in Tables 1
and 2. These experiments are focused on the spreading process,
so the impact velocity is low and the splashing process is not considered in this research. Fig. 13 shows the spreading factor for
butanol at different conditions. The effects of the We and the curvature ratio are the same with that in Fig. 9: the spreading factor
can be increased by decreasing the curvature ratio, while the We
effect is minor. It is still found that the butanol spreading factor follows the linear rule with dimensionless time as well. Figs. 14 and
15 present the experimental data of the spreading factor and the
k value for butanol. It can be seen that the k value in Fig. 15 is more
inerratic than that in Fig. 12, while the k value still increases with
the decrement in the curvature ratio. Thus, Figs. 14 and 15 well
confirm the linear model proposed in Eq. (6) and this linear model
is recommendable for the spreading process after a single drop impact on wetted spheres.
Comparing Figs. 12 and 15, the k value for heptane is larger than
that for butanol. Table 1 indicates that butanol viscosity is much
higher than heptane viscosity. Higher viscosity means that more
impact energy will be decreased by viscous dissipation when the
drop spreads on the wetted surface, and the spreading extent is reduced a lot. Hence, the heptane spreading factor is larger than the
butanol value.
4. Conclusions
In this work, a series of experiments are performed focused on
the less concerned investigation of a single liquid drop impact on
the wetted spheres using a high speed camera. After the heptane
drop impacting on the wetted sphere, spreading and splashing
phenomena can be observed. The sphere-drop curvature ratio
can greatly influence the splashing thresholds: with the increment
in the curvature ratio larger than 0.224, the critical We can be increased due to the drop downward slippage, while the critical
We almost keeps a constant as the curvature ratio smaller than
0.224. The spreading process is quantitatively discussed in detail
and the spreading factor defined as the ratio between the spreading area and the drop surface area is introduced to measure the
spreading amplitude. Experimental results indicate that the
spreading factor can be increased by decreasing the curvature ratio, while the We influences are minor. The interesting linear model
of the spreading factor with dimensionless time is proposed, which
is also verified by the derivation in the definition of the spreading
factor. The k value corresponding to different curvature ratios is
obtained by regressing the experimental data. To further confirm
the linear rule, more experiments are conducted by adopting butanol as the experimental liquid. The trends are similar with that
when the liquid is heptane, while the k value is smaller caused
by its higher viscosity. Hence, the linear model proposed in this research is recommendable.
Acknowledgement
Support of the Key Project of the National Natural Science Foundation of China (No. 51336001) is gratefully acknowledged.
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