7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Bubble dynamics in a quiescent liquid pool
Saeid Vafaei and Dongsheng Wen
School of Engineering and Materials Science, Queen Mary University of London, London, UK.
Keywords: Bubble Formation, Surface Wettability, Surface Tension, YoungLaplace Equation
Abstract: In this study, the formation of bubbles on top of a submerged micrometersized nozzle
is investigated both experimentally and theoretically. The experimental study is conducted on
submerged nozzles of radius of 55 jo under low gas flow rate conditions (0.015 0.83 ml/min).
The bubble dynamics is recorded by a high speed optical camera and detailed characteristics of
bubble formation such as the variations of instantaneous contact angles, bubble heights and the
radii of contact lines are obtained. Using experimentally captured values of the height of bubble
and the radius of contact line, the YoungLaplace equation is solved, which is found to be able to
predict the bubble evolution quite well until the last milliseconds before the detachment. A
differential equation is derived based on a force balance analysis between gas/liquid hydrostatic
pressure, pressure raise due to gas flow rate, surface tension, and buoyancy force on a slice of a
bubble. An analytical expression is obtained by taking integral of the differential equation over
the whole bubble, which relates different bubble parameters such as the contact angle, radius of
contact line, and bubble volume. The experimental bubble volume is compared with the
analytical expression and good agreement is observed.
Keywords: Bubble Formation, Surface Wettability, Surface Tension, YoungLaplace Equation
Introduction
The dynamics of bubble formation play a significant role in various applications involving
dispersion of gas bubbles in liquids. It is a complicated phenomenon that affected by a variety of
parameters such as gas flow rate, liquid and gas properties, wettability, contact angle hysteresis,
operating conditions, and orifice diameter and its depth in liquid [13].
Experimental Setup
The experimental setup used in this investigation is detailed in Figure 1, which includes a gas
supply system, a micrometersized nozzle and a measurement system. The nozzle is made of
stainless steel with an internal diameter of 110 jnm and outside diameter of 210 /m. It is
submerged into a square glass container in the size of 20 by 20 mm. The glass container is filled
with quiescent deionized water at a height of 20 mm and open to the atmosphere under ambient
temperatures. The air flow is supplied by a pressurized air cylinder through a pressure reduction
valve and flows vertically into the orifice. The flow rate is controlled by a mass flow controller
(model F200CV002 of Bronkhorst) in the range of 0.0150.83 ml/min. A high speed camera
(1200 frame/sec) and an optical microscope head are used to capture bubble dynamics; the
images are stored in the PC for later processing. Figures 2 and 3 show the variation of the radius
of contact line and height of bubbles with time and gas flow rate.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Bubble shape prediction
Mathematically, the YoungLaplace equation represents the balance between the surface
tension and external forces such as gravity. This equation can be interpreted as a mechanical
equilibrium condition between two fluids separated by an interface. The YoungLaplace equation
shows that the pressure difference across the interface is equal to the product of the curvature
multiplied by the gasliquid surface tension. The YoungLaplace equation has been solved to
predict the shape of axisymmetric liquid pendants and sessile drops on an ideal solid surface [4
7]. In addition, the prediction of axisymmetric bubble shapes by the YoungLaplace equation
under quasi steady state conditions has been examined by experiments [45, 8] and numerical
simulations based on the volume of fluid method [9], with good agreement.
In this paper the YoungLaplace equation and force balance method are applied to predict the
bubble formation on a submerged orifice. It is assumed that the solid surface is perfectly smooth
and homogeneous without any contact angle hysteresis, so the bubble shape is axisymmetric. As
in the experiments, the gas flow rate is assumed to be steady, and is slow enough to ignore the
gasliquid shear stress and the momentum change in the bubble. Therefore, the bubble formation
takes place in a quasisteady state equilibrium condition. In this case, the equilibrium of the
surface pressure and gravitational forces on the liquidgas interface of the bubble can be
described by the YoungLaplace equation. As confirmed by experiments presented in the next
section, the shape predicted by the YoungLaplace equation is satisfied up to the point of the
critical equilibrium, where the bubble has the maximum volume and eventually will be detached.
Assuming that the bubble is growing in a quasisteady state, the YoungLaplace equation on
the interfacial surface can be written as
1 1
Ap =( + )lg (1)
R, R2
where R1 and R2 are the radii of curvatures, i.e. R1 is the radius of curvature describing the
latitude as it rotates and R2 is the radius of curvature in a vertical section of the bubble
describing the longitude as it rotates. The center of R, and R2 are on the same line but different
location. Ap is the pressure difference between the gas, p, and liquid phase, p, (see Figure 4).
The pressure on the gas side of the interface at a given z inside the bubble can be written as,
20
P, (z) = + Po + gz + gh (2)
PO
where the first term on the right hand side is the pressure difference at the bubble apex, Po is
ambient pressure, and the last two terms are the hydrostatic gas and liquid pressures respectively
with h the hydrostatic head.
On the liquid side of the interface the pressure can be written as:
p,(z) = Po + p,g(h+ z) (3)
The relationships below are used to substitute for the radii of curvature,
R, = ds dO and R2 = r /sin (4)
where 0 is the running angle and r is the radius of the bubble at location z. Substituting
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
equations (57) into equation (1), the YoungLaplace equation is obtained as,
dO 2 gz sin (5)
( p Pg ) (5)
ds Ro arg r
The YoungLaplace equation can be solved, with the following system of ordinary
differential equations for axisymmetric interfaces, to obtain the bubble shape.
dr
= cosO (6)
ds
dz
= sin O (7)
ds
dV
= r 2 sin 0 (8)
ds
where dV is the differential bubble volume. This system of ordinary differential equations
avoids the singularity problem at the bubble apex, since
sin 0 1
(9)
r ,=o R
Knowing two parameters of the bubble shape such as the radius of contact line, height of
bubble (see Figures 23), the system of ordinary differential equations (58) is solved to obtain
the axisymmetric bubble shape, using the following boundary conditions [45, 8, 1012].
r(0) = z(0) = 0(0) = V(0) = 0 (10)
The average gas flow rate is calculated by multiplying the bubble frequency and detached
bubble volume, Q, = JV.
Figure 5 compares the theoretical prediction from the YoungLaplace equation with the
experimental results for bubble evolutions under a gas flow rate of 0.83 ml/min. It is clearly
demonstrated that based on the two inputs from experiments, the YoungLaplace equation can
accurately predict the bubble evolution under current experimental conditions. Figure 6 shows
the evolution of bubble formation for 0.254 ml/mim gas flow rate. Figure 7 shows the variation
of radius of triple line in touch with substrate. The validity of the YoungLaplace equation can be
satisfied up to the detachment period, where the bubble has almost the maximum volume and
would depart by supplying further gas amount or introducing small perturbations around the
bubble. In the last milliseconds closing to the departure, the bubble start being stretched and
consequently the viscosity plays an important role. Such a good agreement before the bubble
departure is expected for bubble formation under low flow rate conditions as the gasliquid shear
stress becomes negligible. As gas flow rate increases, the increase of gasliquid shear stress
could invalidate the YoungLaplace equation. The detailed description of bubble dynamics is
presented below.
Having built the confidence of the YoungLaplace prediction, the detailed characteristics
of bubble formation such as bubble volume and the radius of curvature at apex that are difficult
to determine accurately from the experiments will be revealed. The trend of variation of bubble
contact angle with time can be seen in Figure 8. The experimental data show that the gas flow
rate affects the bubble volume detachment, the beginning of rapid expansion of bubble and
bubble expansion rate (see Figure 9).
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
.Since the YoungLaplace equation can not predict the bubble volume at the last
milliseconds, the bubble departure volume is calculated by summing the bubble volume at the
last point that can be predicted by the YoungLaplace equation and the integral of gas flow rate
at the last moments of the detachment, d Qdt, tdwhere is the detachment time obtained from
experiment, and t, is the last moment of the bubble formation that can be predicted by Young
Laplace equation. The amount of this integral is equivalent to the area of triangle as shown by
dash lines in Figure 9. Figure 10 illustrates current experimental findings on the bubble departure
volume for a 110 micrometersized nozzle. It clearly shows that the characteristics of weak
dependence of flow rate reported by Datta et al [13] is still kept under our low flow rate
conditions ( < 0.83 ml/min). The detached volume decreases under lower flow rates, reaching a
minimum value around a flow rate of 0.5 ml/min, and start increasing afterwards. However
within the whole range, the bubble departure volume only varies within a 10% of the average
value. Such a behavior is understandable due to the dual roles that the flow rate may play here.
At one side, increasing flow rate contributes to the increase of the flow amount into the bubble;
however at the other side, it would contribute to the necking process, which has a tendency of
reducing the bubble volume. The departure volume appears to be a combination of these two
effects.
Analytical expression of bubble volume
The total volume of the bubble can be determined by integrating over the shape of the bubble
predicted above. A useful analytical expression of the volume as a function of the radius of
contact line, bubble height, surface tension, radius of curvature at the apex, and gas flow rate can
be obtained as described below. In this method the force balance is applied on a slice of a bubble
along the vertical axis to find an expression between the volume and other parameters of a
bubble, as schematically shown in Figure 4, assuming an axisymmetric shape.
The force balance equation is written along the vertical direction as
SF = dFb(z) F(z + dz)+ F (z) F,(z)sin (11)
F,(z + dz) sin( + dO) = 0
whereFb is the buoyancy force, F, is the force due to the pressure and F is the force due to the
surface tension. Equation (11) can be simplified as
dFb (z) d[Fp (z) F,(z) sin 8] = 0 (12)
where the individual elements are as follows.
The buoyancy force:
dFb = ( pg )gZ dz (13)
The force due to the hydrostatic pressure
F,(z) =[p(Z) p,(z)]r2 (14)
The force due to the surface tension
F,(z) = cg 27 (15)
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
where the gas and liquid pressure on the gasliquid interfacial surface are given by equations (2
3). By substituting equations (1012) into equation (9), the resultant equation is
(p P,)g7 dz d[(p (z) p,(z))7r2 2agrsr sin 0]= 0 (16)
By integrating equation (16) along the z axis, an analytical expression is obtained as
27
(p, Pg )gV+ r (p, p,g)g(3 2 lgrzs sin0= 0 (17)
Ro
Equation (17) gives an analytical expression between parameters of a bubble under low gas
flow rate conditions. This equation also shows that there are various forces acting on the bubble
including the buoyancy force, (p, p,)gV, the force due to the YoungLaplace pressure,
20
21g ,, the hydrostatic force of the gas and liquid phase, (p, pg)g'0r2, and vertical
Ro
component of the surface tension force, 2lgrd;r sin 00.
Figure 11 illustrates the trend of dominant forces during a bubble formation. The force due to
the Laplace pressure is equivalent with vertical component of surface tension as long as the
gravity force is negligible. The force due to Laplace pressure and vertical component of surface
tension force respectively act upward and downward. After the rapid expansion of bubble, the
effect of buoyancy force due to the increase of bubble volume increases, which pushes the
bubble upward. Consequently, the contact angle start being increased and the vertical component
of surface tension force increases. The force due to Laplace pressure keeps decreasing as the
radius of curvature at the apex increasing. The effect of buoyancy force not only changes the
trend of the variation of contact angle but also varies the forces acting on bubble.
Figure 12 shows comparison between bubble volume obtained by analytical expression of
equation (17) and numerical solution of the YoungLaplace equation. An excellent agreement
between these two is observed.
Detailed description of bubble dynamics
The growth begins at t=0 by splitting the previous bubble from the middle of the necking part
[14]. The upper part of the bubble is separated and starts moving upward as a bubble; meanwhile
the lower part forms a small bubble at the top of nozzle. At t =0 the contact line is almost pinned
at the inside wall of the nozzle. Evolution of bubble formation for 0.25 ml/min gas flow rate (see
Figures 67) is explained as
a) t = 0 7.4 ms. The very first capture image shows already a curved interface, although
immediately after the bubble departure the interface suppose to be flat. The contact angle and
radius of curvature at apex decrease and the radius of contact line increases, however the
expansion rate is negligible because of the high capillary pressure.
b) t = 7.4 524.94 ms. In this phase, the bubble contact line remains pinned at a fixed
location on the nozzle wall and the contact angle and radius of curvature at apex keeps
decreasing. The bubble volume growth rate is still small. The bubble profile remains spherical
due to the negligible buoyancy. Up to this point, the vertical component of surface tension force
which acts downward almost is equal to upward force due to the Laplace pressure force (see
Figure 14). At the end of this period the bubble shape is hemispherical, the contact angle is 90
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
degree and the radius of curvature at the apex is the minimum.
c) t = 524.94 583.27 ms. In this period the contact line of the bubble is sliding from the
position on the nozzle wall to the outer radius of the nozzle. Such a period is typically very short.
The subsequent pinning at the outer edge of the nozzle implies that the tube dimension will have
an important role in the dynamics of contact lines. Due to the drastic increase in the contact
radius, the radius of curvature at the apex increases and as a result the capillary pressure
undergoes a sharp decrease. This allows the bubble expansion rate to increase drastically and
accordingly the height of bubble and radius of curvature at the apex keep increasing. As bubble
volume increases, the effect of buoyancy force increases, and the difference between force due to
the Laplace pressure and the vertical component of surface tension increases (Figure 14). Still,
the buoyancy force is not big enough to push the whole bubble upward significantly, therefore
the contact angle keep decreasing. During this period, the bubble volume expansion rate, and
radius of contact line reaches approximately to the maximum (see Figures 13, 2). In addition, the
contact angle achieves their minimum values (see Figure 12).
d) t = 583.27 874.91 ms. During this time period, the contact line of the bubble remains
pinned at the outside radius of the nozzle. The bubble volume expansion rate, and radius of
contact line are relatively constant; however still the bubble height, radius of curvature at apex,
and bubble volume keep increasing due to the addition of gas molecules in the bubble. As bubble
volume increases, the effect of buoyancy force increases and the bubble starts being pushed
upward, becoming elongated; therefore the contact angle increases.
e) t = 874.91 962.4 ms. During this time period, the contact line of the bubble slides back
from the pinned position at the outside radius of the nozzle (see Figure 7), towards the center of
the nozzle. As time passes the bubble volume and consequently the buoyancy force keep
increasing and push the bubble upward. The contact angle keeps increasing and radius of contact
line decreasing and eventually necking starts. Up to this point the YoungLaplace equation is
able to predict the evolution of formation of bubble shape. The vertical component of surface
tension force is the maximum when contact angle is 90 .
f) t = 962.4 978.23 ms. During this time bubble volume and buoyancy force keep increasing
but since contact angle is being more than 90 degree, the vertical component of surface tension
decreases. So, the bubble begins to stretch even more upward and the effect of viscosity starts
playing a significant role, and consequently the YoungLaplace equation is not able to predict the
bubble shape anymore. The YoungLaplace equation is not valid in the detachment period due to
the influence of viscosity. Eventually upward forces will be dominant and bubble will detach.
Conclusions
This work investigates bubble dynamics on top of a submerged micrometersized nozzle under
low gas flow rate conditions. The experiment reveals detailed bubble dynamics by a high speed
optical camera, where the variations of instantaneous contact angles, bubble heights and the radii
of contact lines are obtained, as well as the identification of different stages for bubble growth.
Analytical work shows that the YoungLaplace equation can predict the bubble evolution quite
well using experimentally captured values of the height of bubble and the radius of contact line,
and a YoungLaplace based equation for bubble volume can be used to predict experimental
results.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Acknowledgement: The authors would like to extend their thanks to EPSRC for financial
support under Grant No: EP/E065449/1.
References
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Engineering, Vol. 8, pp.255368, 1970.
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conditions on bubble formation at orifice in an inviscid liquid. Transformation of bubble shape
and size, Colloids and Surfaces A: Physicochemical and Engineering Aspects, Vol. 218, n 13,
May 29, pp. 7387, 2003.
[3] Fan W., Jiang S., Zhu C., Ma Y., Li H., Study on bubble formation in nonnewtonian
fluids by laser image technique, Optics and Laser Techbology, Vol. 40, pp. 389393, 2008.
[4] Neumann A.W., Spelt J. K., Applied Surface Thermodynamics, M. Dekker, 1996.
[5] del Rio O.I.; Neumann A.W., Axisymmetric drop shape analysis: Computational methods
for the measurement of interfacial properties from the shape and dimensions of pendant and
sessile drops, Journal of Colloid and Interface Science, Vol. 196, pp 136, 1997.
[6] Vafaei S. Podowski M. Z.,The modeling of liquid droplet shape on horizontal and
inclined surfaces, 5th ICMF, Yokohama, Japan, May 30June 4, 2004.
[7] Vafaei S., Podowski M. Z., Analysis of the Relationship between Liquid Droplet Size and
Contact Angle, Advances in Colloid and Interface Science, Vol. 113, pp. 133,2005.
[8] Gerlach D., Biswas G., Durst F., Kolobaric V., Quasistatic bubble formation on
submerged orifices, International Journal of Heat and Mass Transfer, Vol. 48, pp 425438,2005.
[9] Gerlach D., Tomar G., Biswas G., Durst F., Comparson of volume of fluid methods for
surface tensiondominant two phase flows, International Journal of Heat and Mass Transfer, Vol.
49, pp. 740754, 2006.
[10] LonguetHiggins M. S., Kerman B. R., Lunde K., Release of air bubbles from an
underwater nozzle, Journal of Fluid Mechanics, Vol. 230, pp 365390,1991.
[11] Padday J.F., The profiles of axially symmetric minisci, Phil Trans Roy Soc London Ser
A. Math Phys Sci, Vol. 269, pp 265293,1971.
[12] Marmur A., Rubin E., Chemical Engineering Science,Vol. 28, pp.1455, 1973.
[13] R. L. Datta, D.H. Napier, D.M. Newitt, Transactions of the Institution of Chemical
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collapse in a quiescent liquid pool, Physics of Fluids, Vol. 20, pp. 11210412, 2008.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
ALL FIGURES
Square Glass Container
Cylinder Gas Flow Controller
Figure 1. Schematics of the experimental setup.
o Qav=0.83 ml/min Qav=0.25 ml/min
1.2E04
1.OE04
8.0E05
6.0E05
O 0 0 6 6
O 0
000 *
0 0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8 1
Time (sec)
Figure 2. Variation of radius of contact line with time and gas flow rate.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
o Qav=0.83 ml/min Qav=0.25 ml/min
2.5E03
2.0E03 o
S1.5E03 
S1.0E03 o
0
5.0E04 "
*
0.0E+00 .*
0 0.2 0.4 0.6 0.8 1
Time (sec)
Figure 3. Variation of bubble height with time and gas flow rates.
Po
h
dz dFb
F (z+dz) dFb+dO
S\ Fo(z+dz)
Figure 4. Coordinate and applied forces of an axisymmetric bubble.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
x t0.0408 sec o t=0.282 sec
2.5E03
2.0E03
1.5E03
1.OE03
5.0E04
O.OE+00
O.E+00
Figure 5. Comparison between experimental
equation for a gas flow rate of 0.83 ml/min.
5.E04 1.E03
r(m)
bubble shapes and predictions of YoungLaplace
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
t=0 .......t:
t=524.94ms t:
 t=874.91 ms t:
2.5E03 I
2.0E03
1.5E03
N
1.OE03
5.0E04
O.OE+O0 F6
O.E+00
5.E04
r(m)
=7.4ms
=583.27 ms
=962.4 ms
1.E03
Figure 6. Evolution of bubble formation, predicted by YoungLaplace equation for gas flow rate
of Q, = 0.254ml/min.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
t0 7.4 ms
 t=583.27 ms  t874.91 ms
6.E05
4.E05
N
2.E05
0.E+00
t=524.94 ms
t=962.4 ms
4.0E05 6.0E05 8.0E05 1.OE04 1.2E04
r(m)
Figure 7 Variation of shape
ofQ, = 0.254ml/min.
150
S120
S90
S60
U 30
of bubble at the contact with solid surface for gas flow rate
o Qav=0.83 ml/min Qav=0.25 ml/min
o *
o
o
o
D *
0 0.2 0.4 0.6 0.8 1
Time (sec)
Figure 8. Variation of contact angle with time and gas flow rates.
I _
2.0E08
1.5E08
1.OE08
5.0E09
0.0E+00
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
SQav=0.83 ml/min Qav=0.25 ml/min
0
0 0 00
1
*
I
1
I 6** *
0 0.2 0.4 0.6 0.8 1
Time (sec)
Figure 9. Variation of bubble volume expansion rate with time and gas flow rates.
4.35E09
4.3E09
4.25E09
4.2E09
4.15E09
4.1E09
4.05E09 ,
0 0.25 0.5 0.75 1
Gas Flow Rate (ml/min)
Figure 10. Variation of Bubble volume detachment with average gas flow rate
0 0
0
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
0
vA
 
4.E05 
3.E05
2.E05
1.E05
0.E+00
Vertical Component of Surface Tension Force
Force Due to Laplace Pressure
Buoyancy and Gravity Forces
Contact Angle
150
100 O
50
50 0
0
0 0.2 0.4 0.6 0.8 1
Time (sec)
Figure 11. Variation of force due to Laplace pressure, vertical component of surface tension,
buoyancy, gravity, and contact angle with time, for Q, = 0.25ml/ min.
* Analytical Volume, Qav=0.25 ml/min
o Numerical Volume, Qav=0.25 ml/min
* Analytical Volume, Qav=0.83 ml/min
o Numerical Volume, Q=0.83 ml/min
4.E09
S2.E09
o
;>
O.E+00
0 0.2 0.4 0.6
Time (sec)
Figure 12. Comparison of bubble volume predicted by YoungLaplace equation and analytical
expression, given by equation (17).
/
A
A
0 / i
i 9
^ ^''
~o *~o
tr6 Oo
A
ii 0
0
Ii 0
