7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
A single bubble path instability in dilute surfactant solution
Yoshiyuki Tagawa, Ami Funakubo, Shu Takagi and Yoichiro Matsumoto
The University of Tokyo, Faculty of Engineering, Department of Mechanical Engineering
Hongo 731, Bunkyoku Tokyo, 1138656, Japan
tagawa@fel.t.utokyo.ac.jp
Keywords: Single bubble, Surfactant effect, Zigzag/Spiral motion, Slip condition, 3D measurement
Abstract
Path instability of the single bubble is sensitive to the contamination of water. In this research, surfactant effects on bubble 3D
motion in quiescent water are experimentally investigated. Using two highspeed digital cameras, measurements of 3D trajectories,
velocities, and aspect ratios are carried out with a tank of 1.0m height filled with super purified water and a small amount of
surfactant. Experimental parameters are bubble size for 14mm and surfactant concentration 0150ppm which controls surface slip
conditions of the bubble from freeslip to noslip. Due to Marangoni effect, addition of surfactant in gaswater system yields
immobilization of the bubble surface.
First, results of bubbles in 1.92.1mm diameter without shape oscillation are analyzed. There is a onetoone correspondence
between bubble rise velocities and trajectories. It indicates that the slip condition decides not only the drag force but also the wake
structure. The maximum amplitude of the horizontal projection of the spiral motion is 4.7 times as large as the minimum one and
corresponds to not the maximum rise velocity but the maximum horizontal velocity. As a result of the comparison with simple
models, the mechanism of the trajectory change by surfactant is explained by effects of drag force and the direction of motion
affected by nonaxisymmetric wake structure.
Second, all measured trajectories are plotted on two dimensional field of bubble Reynolds number Re and instantaneous
boundary slip condition. In freeslip and noslip condition, bubble motions are dependent onRe. However, in halfslip condition,
bubble motions are spiral and almost independent from Re. Therefore, bubbles in certain condition move along trajectories changing
from spiral to zigzag. These interesting motions are caused by changing slip condition.
Introduction
Millimetersized bubbles in water follow not straight
trajectories but zigzag or spiral. These interesting motions
are shown in various industrial processes which require
better understanding (Fan & Tsuchiya 1990, Ohl et al. 2003).
However, due to complex coupled action of surfactant
effects, shape oscillation, and wake instability, the path
instability mechanism is not fully understood.
A small amount of surfactant is able to change bubble
behaviors drastically. For example, a bubble in aqueous
surfactant solution rises much slower than one in purified
water. This phenomenon is explained by the Marangoni
effect. Due to the gradient of surfactant surface
concentration, a variation of surface tension along the
bubble surface appears and causes a tangential shear stress.
This shear stress results in the decrease of the rising velocity
of the bubble. Here, the surface slip condition is changed
from freeslip condition (zeroshear stress along the surface)
to noslip condition ( no slip velocity ). (Magnaudet and
Eames, 2000).
Using super purified water whose specific residence is
18.2MQcm and organic particles are less than 10 p.p.b.,
Duineveld (1995) found that the Marangoni effect
influences not only the rising velocity but also path
instability of bubbles in quiescent water. Critical reynolds
number which is threshold beyond occurrence of path
instability is reduced by a small amount of surfactant
impurities. Mougin & Magnaudet (2002) considered the
bubble as a spheroidal body of fixed shape and solved
numerically the coupled fluidbody system. The results are
agreed with experimental observations of Ellingesn & Risso
(2001). They found that without shape oscillation of bubbles,
the path instability occurs. The main factor is wake
instability which leads a double threaded wake. Lunde &
Perkins (1998) and de Vries (2001) visualized the
counterrotating double threaded wake structure and pointed
out that 3D motion corresponds to the wake structure one by
one. The main difference between spiral motion and zigzag
motion is periodic vortex shedding. Yang & Prosperetti
(2007) carried out linear stability analysis of the flow past a
spheroidal bubble and Magnaudet & Mougin (2007) did
numerical simulation of fixed spheroidal bubble in uniform
flow. Both researches revealed that not surface slip
condition of the bluff body but the vorticity in a base flow
has an important role to decide the property of wake
structure.
However, 3D paths, velocities and shapes of freely
rising bubble in a surfactant solution are not investigated in
detail. The purpose of this study is experimental
investigation of the surfactant effect to path instability and
to get insight into instability mechanism of different
boundary conditions. Experimental Parameters are
surfactant concentration and bubble size.
Nomenclature
C Concentration of surfactant (ppm)
U Rise velocity of bubble (mm/s)
V Horizontal velocity of bubble (mm/s)
Uto, Bubble speed (mm/s)
Ro Volume equivalent radius (mm)
f Frequency of periodic bubble motion (1/s)
T Periodic time of bubble motion (s)
A Amplitude of periodic bubble motion (mm)
Re Reynolds number (Re=2plURo/i1)
Eo E6tv6s number (Eo=4g(pipg)(Ro)2/r)
Mo Morton number (Mo=g u4(plpg)/ p 2a)
St Strouhal number (St=2fRo /U)
We Weber number ('. =, rfRo /a)
CD Drag coefficient
ka Absorption rate constant (m3/(mol s))
Greek letters
p Density (kg/m3)
p Viscosity (Pa s)
,P Desorption rate constant (mol/m3)
F Surface concentration of surfactant (mol/ m2)
Subsripts
1 Liquid
g Gas
Experimental Facility
Experimental apparatus is shown in Fig.l. A single
nitrogen gas bubble is generated at the bottom of test section
of 100xl00xl000mm3. Its 3D trajectory is measured by two
cameras (MotionPro 10000, Redlake co.) following up the
bubbles with ZAxis stages (KT45AB20100, THK Co.).
Views of two cameras are at right angles to each other.
Superpurified water ( MilliRX 12a, MilliQ SP,
Millipore co.) and teflon tubes are used to avoid any
impurities as far as possible. Specific resistance of the water
is more than 18.2 MQcm and less than 20 ppb of total
organic carbon (TOC). Surfactants we used are Triton
X100 and 1Pentanol. Properties are shown in table 1. The
concentration of solution is set from 0 ppm 150 ppm.
Bubble generator is constructed with an audio speaker
(F200 Technics co), a pressure controller and an electric
valve. This equipment can change bubble sizes for 15mm
by adjusting frequency of the speaker. Tomiyama et al.
(2002) pointed out that the bubble path depends on the
initial shape. In this experiment, we observed "large initial
shape deformation" type bubbles.
Telecentric lenses (59LGG950 59LGH416) are used to
prevent the difference of bubble images due to the different
position in depthwise direction. A small space between a
camera and lenses causes high frequency image vibration.
To remove the noise, we put a certain marking behind test
section and calculated vibration of lenses. The bubble
position and diameter are represented by its centroid and
volume equivalent diameter, respectively. We set camera
conditions as shutter speed: 1/2000sec, frame rate: 250fps,
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
and spatial resolution: 22pm/pixel. About 2mm diameter
bubbles in 1Pentanol solution, Mo number is 2.6x1011and
Eo number is 0.54.
Measurement error of trajectories is 0.1mm, that of
velocity is 2.2mm/s. Aspect ratio y of bubble is the ratio of
major axis and minor axis.
Before 3D motion observation, we measured rise
velocity of straight rising bubbles to check surfactant effects
in purified water. Drag coefficient is calculated by
following equation.
SRsg
CD 83
3U2
Ro is equivalent bubble radius, g is gravity acceleration,
U is rise velocity. We compared our results with Moor
(1965), Mei et al.(1994), Duineveld(1995), Sanada et al
(2007) and Takagi et al. (2003) shown in Fig.2. In Re<250,
our results agree with results of Mei et al.(1994). However
in Re>250, our results become more close to the
Moore(1965)'s result. It is because Mei et al.(1994)
considered a bubble as a spherical shape while Moore
considered it as an ellipsoidal one. Around Re=500 (bubble
diameter is around 1.5mm), our results agree with other
experimental datas within 5%. Generally, the more bubble
diameter increases, the less Marangoni effect affects bubble
motions. Therefore, our results show that bubbles of more
than 2mm diameter in super purified water have a freeslip
boundary condition.
Diffuser
Back
Light
ZAxis stage
ZAxis
x
Fig. Experimental apparatus
1Pentanol TritonX100
Molecular
Molecular 88.15 '647
mass
Molecular CH(CH 4H C8H17C6H4(OC2
formula CH3(CH2)40H
formula H5)xOH(x 10)
Density 0.814 1.07
Diffusivity 1.1 1010m2/s 3.5 1010m2/s
Solubility Dissolve well Dissolve well
k, [m'/(mol s)] 5.08 50
fo [mol / m3] 21.7 6.6x104
F,,a[mol/m2] 5.9x106 2.9x106
Tablel Property of 1Pentanol and TritonX100
0.0 L
100
0 . . ,
200 300
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
(Fig.7). It means that surfactant absorption on the freeslip surface
causes larger horizontal movement while rising speed decreases.
The maximum value of the amplitude is larger as 4.7 times as the
minimum value.
The frequency of the trajectory is shown in Fig. 8. As a
contrary to the amplitude, a minimum value is taken around U=
280 mm/s. The maximum value is larger as 2.1 times as the
minimum.
Each value has the extreme value at rise velocity U=280 mm/s,
changes most greatly in 230260 mm/s, and has the flexion point
in this section if it is thought one curve. The width of the change
becomes small in 190230 mm/s. Therefore, it is expected that
there is a similar change in the flow structure around bubbles.
400 500
Fig.2 Plots of ReDrag coefficient CD (V: Our experimental
results, : Moore(1965) :Mei et al. (1994) 0:
Duineveld(1995) D: Sanada et a/.(1995), o: Takagi et al.
(2003))
Results and Discussion
Trajectories, velocities, and shapes of 1.92.1 mm bubbles in
1Pentanol solution are shown in Figure 35. In this diameter
range, shape oscillation does not affect bubble motions (Ellingsen
& Risso, 2001)
In super purified water, or 0 ppm solution, bubbles of freeslip
condition move along spiral path (Fig.3(a)) or transition path
from zigzag to spiral while there are no reverse transitions. The
rise velocity is constant at 356mm/s which agrees with the results
of Duineveld(1995). In 1Pentanol 150ppm solution, bubbles
always move along zigzag path (Fig.3(d)). The rise velocity
decreases immediately. After the velocity drop, it becomes
periodic around 167mm/s. The frequency of velocity fluctuation
is twice as much as that of path fluctuation. The shape of bubble
takes almost spheres (Fig.5(d)). When we put more surfactant
into the solution (200ppm), velocity and shape of bubble is same
as the result in 150ppm. Therefore, the slip condition is supposed
to be noslip.
In 1Pentanol 25100 ppm solution, Fig.4 shows that the rise
velocities are depend on surfactant concentration. The thicker
concentration becomes, the smaller rise velocity becomes and the
more shape becomes spherical. The slip condition becomes close
to noslip when surfactant concentration is high. In these slip
conditions, bubble motions are spiral.
Tsuge & Hibino (1977) found the following important relation
available in both purified water and surfactant solution
1000
5 5
E 0 E
0  ~0m5 10 1 5 0 5 10
x[mm] x[mm]
(a) Oppm (b) 25ppm
1000o
800
600
n400
200,
10
0 10
10 10 o
x[mm] y[mm]
St = 0.100CO 734
According to their results, the experimental data in purified
water and surfactant solution must be plotted within 30% of
equation(2). Fig.6 shows the line of equation(2) and our results.
Note that these data plots are around z=150mm as same as Tsuge
& Hibino(1977). All of our results satisfy this relation.
To reveal the relation between a slip condition and bubble
behavior, the amplitude and the frequency of bubble path in
rising distance more than 400 mm were plotted in Fig.7 and Fig.8,
respectively.
The maximum value of the amplitude is taken not at the
highest rise velocity U=360 mm/s but around U=280 mm/s
10 10
5 5
5 5
o0 5 0 5 1o 19 5 0 5 10
x[mm] x[mm]
(c) 75ppm (d) 150ppm
Fig. 3 Experimental results of trajectories; upper: whole view,
lower: horizontal projection ((a) Oppm (b) 25ppm (c)75ppm (d)
150ppm)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
400
300
200
100
0 200 400 600 800
Z [mm]
Fig. 4 Experimental results of rise velocities
150 200 250 300
U [mm/s]
Fig. 7 Rise speed U (mm/s) v.s. trajectory amplitude A (mm)
C
0
0
(a) Oppm (b) 25ppm (c) 75ppm (d) 150ppm
2.4 
2.0 igzag  Spiral
O Oppm
2 0 25ppm
1.6 5ppm
V 50ppm
A 75ppm
1.2 4i t l Oppm
S150ppm
0.8 I
150 200 250
U [mm/s]
8
, 6
150 200 250 300
U [mm/s]
350 400
Fig. 8 Rise speed U(mm/s) v.s. trajectory frequencyf(1/s)
300 350 400
Fig. 5 Pictures of bubbles and aspect ratios
To analyze these changes of trajectories, the relation between
amplitude and frequency is modeled first. Spiral motion bubble
moves on a circular trajectory in horizontal plane while zigzag
bubble goes on a line trajectory. Therefore,
V
Spiral : A= ,
27 f
Zigzag : A =
4f
V(mm/s) is horizontal velocity of bubble. When V is regarded as
constant, amplitude A is inversely proportional to frequency
Fig.9 shows the plot of experimental data and 3 curve lines of
constant V=80,120,160mm/s of equation(3).
When concentration changes from Oppm to 25ppm,
amplitude A is 1.6 times as large as amplitude calculated by
inverse curve of equation(3) of constant V Therefore, the
horizontal velocity Vincreases. However, the case in 25150ppm,
amplitudes are smaller than that calculated from equation(3). In
these concentrations, Vdecreases as concentration increases.
All of our results and previous results show that rise velocity
decreases when surfactant concentration increases. We focused
on surfactant effect to horizontal velocity.
c
350 400
0.1 C 1
Fig.6 Drag coefficient vs. Stroual number St
Fig.6 Drag coefficient CD v.s. Strouhal number St
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Fig.10 shows the relation between rise velocity U(mm/s) and
horizontal velocity V(mm/s). When rise velocity increases,
horizontal velocity increases in U>280mm/s while it decreases in
U<280mm/s. The maximum value of V(mm/s) corresponds to
the maximum value of amplitude A(mm) and minimum value of
frequency( 1/s).
To discuss about wake structure and drag force, two models
about the spiral motion in U>190mm/s are shown below. In the
first model, horizontal velocity is described as
V = ,iU2
(U,, =to U2 +V2 ) (4)
U,,a is the bubble speed. If U,, is constant, V increases as U
decreases. This model assumes that only the direction of bubble
motion changes while the bubble speed is constant. The direction
is affected by nonaxisymmetric property of doublethreaded
wake but not by drag force. Therefore, this model assume that the
drag force is unchanged but only the nonaxisymmetric property
of wake changes.
The second model consider the direction of motion as a
constant value;
V = UtanO
0 is an angle between the direction of bubble motion and
vertical direction. This model assumes that the speed of bubble
changes but the direction is constant.
To compare these models with experimental data, several
curves of equation(4) and (5) are plotted on fig.10.
In U>280mm/s, experimental data Vis 0.9 times as big as
V calculated by equation(4). However, compared with
equation(5), V is twice as large as V calculated from constant 0.
Therefore, in this region, growth of Vis mainly caused by change
of 0. Therefore maximum value of V and A, and minimum value
off are occurred by the change of nonaxisymmetry of the wake
structure.
In U<280mm/s, compared with equation(5), V is smaller
than Vfrom the equation(5). This is because decrease of 0. Both
U,ta and 0 decrease. Therefore, amplitude A decreases because
drag increases and the direction of motion becomes more
vertical.
4 6 8 10 12
f[1/s]
Fig. 9 trajectory amplitude A(mm) v.s. trajectory frequencyf(1/s)
150 200 250 300 350
U [mm/s]
Fig. 10 Rise speed U (mm/s) v.s. Horizontal speed V(mm/s)
Let us discuss the relation between the slip conditions,
Reynolds number Re and bubble motion modes (straight,
spiral, or zigzag).
To evaluate surface slip condition quantitatively,
normalized drag coefficient CD is defined as follows.
C C
S D,xp Dean
C. C.
CD, clean and CD, rigd mean drag coefficient of clean bubble
and rigid sphere with the same volume, respectively. CD,,x
is instantaneous drag coefficient calculated by experimental
data from equation(l). CD = 0 and 1 means freeslip
condition and noslip condition respectively.
Fig.11 shows the map of bubble motion modes on
ReCD field. Bubble motions of clean bubble (CD =0) in
600
consistent with previous investigations (Clift et al. 1978,
Magnaudet and Eames, 2000). Those of noslip bubbles
(CD =1), zigzag motions are shown 280
consistent with results of rigid spheres (Jenny et al., 2004).
In freeslip and noslip condition, bubble motions are
dependent on Re. However, bubble motions in halfslip
condition 0.4
spiral motions are stable and independent from Re. Results
in 1pentanol and tritonX100 solutions are consistent.
Therefore, slip condition changes Redependency of bubble
motion modes.
Fig.12 shows typical single bubble trajectories. Fig.
12(a) shows a trajectory changing from zigzag to spiral in
super purified water. This motion has been reported by
previous researches (Ellingsen and Risso, 2001). We also
observed a motion changing from spiral to zigzag in Triton
X100 0.27 ppm solution (Fig.12(b)). As previously
described, this transition has never been observed in
purified water. To analyze the mechanism of the change, we
focused on boundary condition on the bubble surface. The
condition is related to bubble rise velocity. Fig. 13 shows
rise velocities of same size of bubbles in 3 kinds of
solutions. In super purified water, rise velocity becomes
constant and bubble motion is spiral. In Triton X100 2.7
ppm solution, rise velocity decreases and becomes constant
but bubble motion is zigzag. In Triton X100 0.27 ppm
solution, around 130 mm distance, rise velocity is almost
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
same as that in super purified water and
similar spiral. Around 550 mm distance
decreases and is same as that in 2.7 ppm
solution. Its motion is similar zigzag. This g
of velocity along distance indicates that
gradually changes from freeslip to nos
phenomenon had been reported by Zhang an
about smaller bubbles. Therefore, the cha
boundary condition from freeslip to nosl
bubble motion from spiral to zigzag.
1J + ** *+
legion Zigzag region
D + **
O. 
o
0.6 I
S Zigzag 
I Straight n Spiral region
0.4 Spiral 0 
02
%
0.0 Straightregion j
200 400 600
Re
Fig. 11 Motion modes in Re v.s. CD
IO0
600,
400
200
0
0 10
0 10
y[mm] 10 10 x[mm]
800
the motion is
, the velocity
Conclusions
STriton X100 Surfactant effects on path instability is experimentally
radual decrease analyzed. Measurements of 3D trajectories, velocities, and
slip condition aspect ratios are carried out with a tank of 1.0m height filled
lip. The same with super purified water and a small amount of surfactant.
id Finch (2001) Due to Marangoni effect, surfactant concentration controls
nge of surface surface slip conditions of the bubble from freeslip to noslip.
ip changes the First, we discussed about bubbles in 1.92.1mm diameter.
Our result indicates that the slip condition decides not only the
drag force but also the wake structure. The maximum
amplitude of the horizontal projection of the spiral motion is
4.7 times as large as the minimum one. The maximum value
Corresponds to not the maximum rise velocity but the
maximum horizontal velocity. As a result of the comparison
with simple models, the mechanism of the trajectory change by
surfactant is explained by effects of drag force and the direction
of motion affected by nonaxisymmetric wake structure.
Second, all measured trajectories which are categorized
Sin straight, spiral or zigzag mode are plotted on two
S* dimensional field of Reynolds number Re and normalized
Sdrag coefficient CD*. While freeslip and noslip bubble
Zlgzagregion motions are strongly dependent on Re, halfslip bubble
*t motions of 0.4
800 1000
on Re. Bubbles in certain condition move along trajectories
changing from spiral to zigzag. These interesting motions
Smap are caused by surfactant accumulation on bubble surface
which change surface boundary condition from freeslip to
noslip. Comparing this transitional motion to spiral motion
of freeslip bubble in super purified water and to zigzag
motion of noslip one, we deduced that instantaneous slip
L condition on a bubble surface also decides bubble motion.
400
200
10
0 10
y[mm] 1
(a) Zigzag to spiral in super purified water
(b) Spiral to zigzag in 0.27 ppm Triton X
Fig.12 Transitional motion of a single
quiescent water
0 Spiral path
400 Super purified water
0  ......<... ........ ........ ....
S300 I si
I I. TritonX
S200 AAAA A 
100 A Zizag Path
TritonX100 2.7 ppm
0 100 200 300
Distance (mm)
Fig. 13 Velocity profiles of bubbles in Supe
Triton X100 0.27 ppm solution, and Triton
solution
Acknowledgements
This research is supported by GrantinAid for JSPS
Fellows.
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