Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: P2.67 - Dynamics of a bubble pair
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00502
 Material Information
Title: P2.67 - Dynamics of a bubble pair Particle Bubble and Drop Dynamics
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Stanovsky, P.
Ruzicka, M.C.
Drahoš, J.
Sato, A.
Shirota, M.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: bubble pair
in-line interaction
stability
 Notes
Abstract: This work is focused on the dynamics of a pair of gas bubbles rising in a liquid at intermediate bubble Reynolds numbers (Re ~ 101-102). Theoretical description of the bubble pair motion in a general position was derived by Kok (1993), but the experimental verification of the theory was made only for bubbles out of in-line arrangement. However, there is lack of experimental data on the stability of bubble pair and his dynamics. The experiments on early dynamics of bubble pairs were conducted and in ultrapure water with conductivity less then 0,1mS/cm and TOC (total organic carbon content) less than 10ppb. The bubble pair was produced using a device described in Shirota et al. (2008). Three-dimensional information about bubbles' positions was obtained using the system of mirrors and moving high-speed camera. It was found that pair of spherical bubbles remained aligned in line for Re < 24 and trailing bubble approach leading bubble of the pair. When Re exceeds value 54, trailing bubble migration sideways from the in-line alignment was observed. The minute difference in the sizes of the bubbles in the pair was found as the source of a scatter of the data.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Bibliographic ID: UF00102023
Volume ID: VID00502
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: P267-Stanovsky-ICMF2010.pdf

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7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Dynamics of a Bubble Pair


Petr Stanovsky*, Marek C. Ruzicka*, JiNi DrahoS*, Ayaka Satot and Minori ShirotaT

Institute of Chemical Process Fundamentals ASCR, Rozvojova 135, 16502 Prague 6, Czech Republic
t Faculty of Science and Technology, Hirosaki University, 3 Bunkyo-cho, Hirosaki-shi, Aomori, 036-8561 Japan

Graduate School of Science and Technology Research Division, Hirosaki University,
3 Bunkyo-cho, Hirosaki-shi, Aomori, 036-8561 Japan
stanovsky@icpf.cas.cz



Keywords: bubble pair, in-line interaction, stability



Abstract

This work is focused on the dynamics of a pair of gas bubbles rising in a liquid at intermediate bubble Reynolds numbers
(Re ~ 10'-102). Theoretical description of the bubble pair motion in a general position was derived by Kok (1993), but the
experimental verification of the theory was made only for bubbles out of in-line arrangement. However, there is lack of
experimental data on the stability of bubble pair and his dynamics. The experiments on early dynamics of bubble pairs were
conducted and in ultrapure water with conductivity less then 0, 1S/cm and TOC (total organic carbon content) less than l0ppb.
The bubble pair was produced using a device described in Shirota et al. (2008). Three-dimensional information about bubbles'
positions was obtained using the system of mirrors and moving high-speed camera. It was found that pair of spherical bubbles
remained aligned in line for Re < 24 and trailing bubble approach leading bubble of the pair. When Re exceeds value 54,
trailing bubble migration sideways from the in-line alignment was observed. The minute difference in the sizes of the bubbles
in the pair was found as the source of a scatter of the data.


Introduction

The bubble-bubble and bubble-wall interactions are still
subject of the research due to still incomplete understanding
of regime transitions and insufficient knowledge about
dense bubbly flows. Interest for research in this field is
stimulated also by the presence of bubbly flows at finite
Reynolds numbers in an industry or emergence of bubbly
mixtures in new fields as the medical sciences for example.
The interactions between particles and the behaviour of the
particle clusters represent a mezzo-scale part in multi-scale
methodology used nowadays for a description of the
two-phase or multiphase systems. The information about
interaction between particles can be obtained from smallest
structural unit of the system pair and further addition of
another particles or from advanced analysis (statistical, etc.)
of large groups of particles. In the case of bubbles, there is
three aspects making the research quite difficult inertia of
accelerated bubbles is concentrated in the surrounding liquid,
the bubble shape can deform from a sphere and finally
boundary condition on the bubble surface delays the onset
of the boundary layer separation and the appearance of a
stable wake.
The literature reviewed below is focused on clean bubbles
which posses the free-slip condition at their surfaces and
they are spherical or nearly spherical. The focus of our
interest is interaction of bubbles in vertical (in-line, tandem)
arrangement because the bubble wake contribution as
viscous contribution to the hydrodynamic interaction is


substantial in this direction, but this not exclude vertical
(side-by-side, lateral) interaction because they are usually
coupled together.
The creping flow conditions (Re < 1, Re = DU/v) simplify a
theoretical approach and also experimental difficulties are
modest due to lower rising velocity of bubbles. Wacholder
and Weihs (1972) gave an analytical solution for the flow
around two fluid spheres of an arbitrary size rising at
creeping flow conditions (Re << 1). From their solution, it
results that a bubble pair rises faster than an isolate bubble
and a drag on the bubble decreases as the separation
between the bubbles shortens. A little bit easier solution for
creeping flow around a pair of in line rising bubbles is given
by Harper (1983). More general concept for numerical
computation of the flow around two or more fluid particles
at creeping flow conditions can be found in Kim and Karilla
(2005). An experimental observation of a pair of spherical
bubbles rising in line at creeping flow condition was done
by Kumagi et al. (2002). They measured an interaction of
two and four equal sized bubbles rising in glycerol at
Re < 0.5. In the case of a bubble pair, they coalesced in
contact, prior to contact a trailing bubble was elongated. In
the case of quadruplet, the second bubble was attracted to
the first and they coalesced. Then the third and the fourth
followed the same scenario. It should be noted that inertia is
not negligible and just the Oseen flow solution perfectly
matched the bubble dynamics. Contrary to the pair, in the
case of four bubbles the motion of the first and the second
bubble was greatly underestimated even for Oseen flow.









More important are hydrodynamic interactions at finite
Reynolds numbers, despite of neglecting a bubble
deformation with increasing Re. The equation of motion for
the pair of bubbles rising in a pure liquid in a general
position at the moderate Re was derived by Kok (1993). The
derivation assumed the irrotational flow and an indirect
calculation of drag forces by means of a dissipation method
as Levich did for a solitary bubble. He found that bubbles
are repulsed if they are vertical or slightly misaligned from
that position. Otherwise, the bubbles are attracted if they are
horizontally aligned or slightly misaligned from the
side-by-side position. Nevertheless, from all initial
orientations, the bubbles are shifted into the horizontal
arrangement. He made an experiment with the bubbles
rising side-by-side or in the inclined position (at least 600
inclination of bubbles' centreline from vertical) in the
ultrapure water. Bubbles with separation from 3 to 9 of
bubble radii at Re = 240 confirmed a theoretical prediction
of the proposed model. However, due to the bubble
production technique, the vertical and the nearly vertical
arrangements were not tested.
Their results are in contrast with the result of Yuan and
Prosperetti (1994), who solved an axisymmetric unsteady
Navier-Stokes equation for two equal-sized bubbles with a
fixed spherical shape at Re from 50 up to 200 and separation
from 2.6 to 20 of radii. The boundary-fitted calculation
revealed that the drag of a leading bubble was almost
unaffected by the trailing bubble even if the separation
S (= centre-to-centre distance / bubble radius) was very
small (~ 4). In contrast, the drag of trailing bubble was
lower in comparison with the drag of solitary sphere even at
very high separation S ~ 30. Consequently, the trailing
bubble tended to approach the leading bubble. At low
separation, a repulsive potential force became important and
that pair of bubbles rising in line reached an equilibrium
separation Seq= Seq(Re). From the prediction it results that
the vertical pair of spherical bubbles should collide for Re
below 28. Results of their simulation are accordance with
the analytical theory of Harper (1970) for a pair of spherical
bubbles rising in line at Re ~ 0(102). His model is based on
a viscous wake (as the boundary layer approximation)
behind the bubble in an irrotational surrounding flow. He
revealed that the drag of the trailing bubble is reduced by
the presence of a previous bubble, which counteracted
against the repulsive force due to a higher pressure between
the bubbles. This resulted in an existence of equilibrium
separation between the bubbles. However, he also showed
that the in-line arrangement is unstable to any lateral
perturbation (i.e. misalignment from the vertical
arrangement) in pure liquids. He mentioned that a very tiny
amount of surfactant that is present on the rear of the bubble
could stabilize this in-line arrangement, but this amount has
to be small enough to avoid the increase of the bubble drag.
The influence of surfactant on the bubble interactions and a
result of the contact of bubbles was experimentally studied
by Duineveld (1998). He found that bubbles coalesced in
the ultrapure water if the approach velocity was lower than a
certain limit. Above this limit the bubbles bounced. In the
presence of surfactant above a critical concentration, the
bubbles rising side-by-side repelled each other. He observed
in presence of the surfactants that the bubbles misaligned
from a vertical (~ 400 inclination of bubbles' centreline from
the vertical) were attracted into the wake of contaminated


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

leading bubble followed by bouncing. Presence of
surfactants at the rear of the bubble increased the wake
strength; hence the attractive force was greater than
repulsive potential force.
The first experiments with pairs of clean bubbles aligned in
vertical line were performed by Sanada et al. (2006) using a
silicon oils in the range 5 < Re < 145 with separation
20 low Re 5 was approaching the leading one and finally
they collided without coalescence. In the rage 25 < Re < 54
bubbles were initially approached in a line and finally they
rose in a stable distance. However, the observed equilibrium
separation was roughly 20-times larger than the prediction
given by Yuan and Prosperetti. In addition, sometimes the
trailing bubble escaped from the vertical arrangement after
some time, when approaching the leading bubble. However,
after that, the bubbles remained in some equilibrium vertical
distance. It should be noted that the bubbles got the
spherical shape only at low Re, for higher Re the bubbles
were highly deformed. Another confirmation of the
existence of the stable separation was presented by
Bonometti (2007) with axisymmetric numerical simulation
of a pair of spherical bubbles rising at Re ~ 20 and 180
using an interface-capturing method. In the case of the low
Reynolds number, the bubbles approached each other and
they coalesced. In the case of the high Re, the bubbles were
deformed into an ellipsoidal shape shortly after the start
(with different deformation for each bubble) and finally the
bubbles rose with a stable separation, a bit lower than
predicted by Yuan and Prosperetti (1994).
It should be noted that for bubbles rising side-by-side was
also found an stable separation in 3D DNS simulation of
two bubbles with a fixed spherical shape rising in a
horizontal arrangement for 0.02 < Re < 500 and separation
2.25 < S, < 20 by Legendre et al. (2003). They found that
the bubbles are attracted/repulsed to/from each other, and
the sign of resulting interaction forces depends on Re and S,.
They concluded that the sign of interaction force is
governed by the competition between the irrotational
mechanism associated with the asymmetrical deflection of
the fluid between and around the bubbles and the
wake/boundary layer phenomena resulting from the
asymmetric generation and diffusion of vorticity around
them. They also concluded that the equilibrium separation
between the bubbles exists for Re > 28, what is in
remarkable corespondece with the Yuan and Prosperetti,
and for the initial separation lower than a certain value
S, which also depends on Re.
Another source of the information are carefully designed
experiments and numerical simulations with large number
of bubbles. Sangani and Didwania (1993) performed a
numerical simulation of many spherical bubbles based on an
irrotational flow with a suspended coalescence for Re =
200-500. They observed a formation of horizontal layers of
bubbles for a uniform and a non-uniform size distribution of
the bubbles. The formation of the horizontal layers of
bubbles was confirmed numerical simulation on the same
base done by Smereka (1993). However, in real systems the
horizontal bubble layers were not observed. Only weak
bubble clustering was detected in dilute bubble mixtures
with suppressed coalescence using electrolytes by Zenit
(2001).





7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


More close to a real bubbly flows was using a numerical
simulation using front tracking method performed by
Bunnner and Tryggvason (2002). They modelled a
periodical group of 27-216 spherical bubbles at Re 20-30
(depending on void fraction from 2-24%) initially. The
examination of the pair distribution function for the bubbles
showed a preference for a horizontal alignment of bubble
pairs but the distribution remained nearly uniform. The
probability of finding the horizontal alignment grew with
the increasing Re. Further Bunner and Tryggvason (2003)
examined an effect of deformation with the simulation of
initially ellipsoidal bubbles at Re 26. After some travelled
distance, the initial uniform distribution of ellipsoidal
bubbles began accumulating into vertical streams, rising
much faster than in the uniform distribution. This effect was
attributed to a change of the lift force in the shear flow (in
the wake of the preceding bubble) due to the deformability
of the bubble. They also described interactions of the
bubbles in the pair observed in the simulation. For the
spherical bubbles initially in line, the trailing bubble was
accelerated in the wake of leading bubble, but they rotated
before they collided and after they repulsed one another.
This occurs only in a dilute case. In the case of the
deformable bubbles, they remained in the line until the
collision, then they rotated as one body and they repulsed
one another when they reached their horizontal position.
Some of these results are in accordance with the
experiments of Kitagawa et al. (21"'14), who studied
bubble-bubble interactions in the two-dimensional swarm of
spherical bubbles sliding along the wall at 0.1 < Re < 16 in
the silicon oil. Using a statistic sampling from PTV data
(Particle Tracking Velocimetry), assuming a
nearest-neighbour interaction as the leading order
interaction, they revealed that the bubbles aligned a
vertically or horizontally were rising faster than a solitary
bubble. Bubbles rising in a vertical arrangement were faster
than those rising side-by-side. In all experimental cases, the
bubbles aligned vertically (side-by-side) were repulsing
each other. In the case of Re < 3 the nearest bubbles were
frequently in a vertical direction. In contrast, the nearest
bubbles for Re 15 were frequently distributed in the
horizontal direction. Murai at al. (2006) measured bubble
interactions in the bubbly flow with void fraction up to 3%
using stereo-matching PTV for 5 < Re < 75 in the silicon oil.
They extended the previously mentioned method for the
nearest-neighbour approximation into a 3D using a
distribution of relative vectors of the nearest-neighbours.
From their analysis, it results that bubbles rising in the
horizontal alignment repelled each other and the bubbles
rising in the vertical were attracted at large spacing.
Nevertheless, the bubbles rising at Re > 30 in a close
vertical arrangement were repulsed and the bubbles
frequently created a slightly inclined arrangement. In
addition, the bubbles rising in a close horizontal alignment
at high Re were unstable and the bubbles created the already
mentioned inclined arrangement.
It's clear from presented results that our picture about
bubble-bubble interactions is not complete. This work is
another contribution into this field focusing on bubble pair
interactions at moderate values of Re.


Nomenclature


bubble diameter (mm)
bubble velocity (mm.s 1)
dimensionless bubble centre to centre spacing (-)
gravity acceleration (m.s 2))


Greek letters
a surface tension (N.m 1)
v kinematic viscosity (Pa.s)
p Density (kg.m-3)

Subsripts
i initial
bubble free-slip condition
Otsu Otsu's method
rigid no-slip condition


Experimental Facility

The hydrodynamic interaction of two initially rising air
bubbles rising in ultrapure water was studied in specially
designed closed tank. The experimental tank, made of
Plexiglas, had a size 0.14 x 0.14 x 0.48 m3 and it had both
ends opened. The tank was fixed between two blocks made
of polytetrafluoroethylene (PTFE) by steel bolts. A
polyamide tube in horizontal position with a drilled orifice
of 0.3 mm in diameter was mounted in the bottom block
using PTFE fittings. An ultrapure water feed and a water
drain were connected to the bottom block. A mixed
borosilicate/active carbon filter as the connection of the tank
to the atmosphere was mounted in the upper block.


2

3 4


6
= 1111


Figure 1: The experimental set-up for experiments with
the bubble generator with an acoustic wave.
1-pulse/delay generator, 2-multifunction synthesizer,
3-amplifier, 4-bubble generator, 5-traversing device, 6-high
speed camera, 7- system of mirrors, 8-experimental tank,
9-flat cold light.

The three-dimensional view of the the bubble pair behaviour
was achieved using the set of four mirrors (see Fig. 2) and a
digital high-speed camera mounted on the traversing device
(THK, type KT30A-B06-060B), which allowed the camera
to move vertically together with the rising bubble pair. The
movement of the camera was controlled using a software
HAS-220 and an internal PC data card (THK). The start of
the camera record and its movement together with the
emission of bubbles were triggered using a digital


*=*









pulse/delay generator (DG 535, Stenford Research Systems).
The images were captured using a high speed digital camera
Phantom v4.2 with a mounted pair of 1,5x teleconvertor
(Kenko) and lens Micro-Nikkor 105mm/2.8D. A flat cold
light without flickering (Sakai Glass Sci.; HF-SLA214-LC)
was used as a backlight illumination.

S4



2


1 *



Figure 2: The top view on the system of mirrors.
1- high-speed camera, 2-system of mirrors, 3-tube with
orifice, 4-flat cold light.

The bubbles were created by using a bubble generator with
pulsed acoustic pressure wave, which is able to create a
bubble of desired size with good reproducibility. The
principle of the bubble generator operation is explained by
Shirota et al. (2008). Two bubbles of different separation
were created doubling of a signal packet for the acoustic
wave. The signal packet was of similar shape as in formerly
mentioned article it was created using Arbitrary Waveform
Editor software (ver. 0105, NF) and multifunction
synthesizer (WaveFactory-47-WF 1946A, NF). The size of
the bubbles was controlled by a combination of the signal
amplification (Kenwood KAF 3030R) and air pressure
adjustment in the inlet tube.
The ultrapure water used in the experiments had parameters
(the conductivity and Total Organic Carbon content) as a
Type I water according to ASTM specification, which is
mostly considered as an ultra-clean water standard as there
is no ISO standard available. This water was produced using
combination of Millipore Elix and Mili-Q Gradient A10
purification system and it was supplied in closed loop into
the experimental tank. The conductivity of water at the
output from the purification system was 0.055 gS/cm and
total organic carbon (TOC) content was 4 ppb at 250C
according the built-in measurement.

K TOC
Number of rinsing [S/cm] [b]

1st 0,16 37

2nd 0,09 19
3rd 0,06 11


after experiment 0,11 21


Figure 3: Measurement of
experimental tank.


the water purity in the


The conductivity and the TOC measurement at the output
from the experimental tank was placed in the draining line
(Thornton 2000 Metler-Toledo Conductivity Monitor and


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

Millipore A10 TOC Monitor). An example of measurements
is given in Figure 3. The theoretical conductivity of water is
K = 0.055 pS/cm at 250C. The temperature of water in the
experiments was 27C, thus the theoretical conductivity has
to be also slightly higher. It is visible that even after several
hours of experiments quality of the water did not get
considerably worse. The water level was kept 0.3 m above
the orifice. An air from laboratory clean air distribution was
used in the experiments as the working gas.

Data Treatment

Original images were of size 512x512 pixel. The images
were processed in MATLAB 2007 with Image Processing
Toolbox. As the basic length scale was used a diameter of
polyamide tube with an orifice (6 mm), recorded after each
experiment. An accuracy of the evaluation of the bubble
diameter (thus bubble axes) is limited due to the low
resolution of the digital high-speed camera and also due to
restricted optical zoom caused by the experimentally
available bubble spacing, which had to fit into a picture. In
these experiments, the ratio between the bubble spacing and
the bubble size was usually 15:1 and the length scale were
around 40 pm/pixel in all experiments.
The bubble diameter evaluated using the image analysis
varied from 10 to 20 pixels and the visible pixilation of the
edge of the bubbles made the direct calculation of bubble
size questionable. Therefore, the correctness of the bubble
size evaluation using the image analysis based on Otsu's
method (Otsu, 1979) was tested by a single bubble motion
recorded with a not moving camera at the time as of the
experiments with the bubble pair. It was found that the
bubble centre velocity was smaller in comparison to the
theoretical velocity of the spherical clean bubble, based on
the bubble diameter acquired by image analysis with Otsu's
method. The example of the evaluation is depicted on Fig. 4
The bubble velocity U was calculated from time difference
of the bubble centre evaluated by image analysis as a
centroid (centre of mass) of given area and it is depicted as
black circles. It should be noted that the variation in the
bubble velocity after 60ms was due to a finite pixel size.
The bubble diameter Dotsu,bubble was calculated as a sphere
equivalent to the bubble axes length a,b evaluated by image
analysis. The corresponding velocity of a clean spherical
bubble with such size was calculated using Mei at al. (1994)
equation and the velocity evolution is presented as blue
dashed line in the Fig. 5 (DOtsu.,bbble = 0.710 mm).
The bubble diameter corresponding the measured bubble
velocity was calculated using Mei et al. (1994) equation for
the last ten points of the bubble centre velocity
(De, = 0.594 mm). Bubble acceleration to their terminal
value for this diameter from the rest is depicted as red line
in the Fig. 4. It should be noted that variation in the terminal
velocity value refers to the variation about 0.3 pixel in the
travelled distance (during the 2,5 ms interval between two
points).
The overprediction was for the all tested bubble sizes
around 2,5 pixels. It is natural that an accuracy of evaluation
for a centroid of well-illuminated spherically symmetric
body is much more grater then body dimensions. The
threshold value in the image analysis routine shrinks or
increases the bubble diameter, but distribution of mass is
still symmetric around the centroid.






7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

The second interaction regime (Regime B) represents the
laterally unstable arrangement of the bubble pair. Bubbles
with Re > 54 deflected out of the vertical arrangement a
short time after the release.


t [ms]
Figure 4: Evaluation of the bubble size from its velocity.

The additional magenta dashed line in the Fig. 4 shows the
terminal velocity of the sphere with rigid surface with radius
Dotsurgid as the argument for rejection of a conjecture that
the decrease in velocity was due to a minute amount of
impurities. Moreover, the latest conjecture can be also
rejected on the basis of result of conductivity and TOC
measurement of water in the tank before and after the
experiments.
With respect to these results, the bubble size D was
calculated from the bubble terminal velocity using Mei et al.
(1994) equation for all the data. So, in the case of pair of
bubbles the diameter was calculated only for the first bubble
because the second bubble was already influenced by the
wake of the previous bubble. In addition, the calculation of
travelled distance of the second bubble was inaccurate
because the camera started to move upwards and this
movement was coupled with unpredictable camera
oscillations during a starting period.
The bubble spacing S between the bubbles during their rise
was evaluated from their centroids using the Matlab Image
Processing Toolbox only for the selected time. The reason
for this simple procedure was an insufficient result of the
threshold routine in some parts of the movies with the
record of bubble pair rise. In the middle part of the movies,
the bubble was visible as the shadowy shape in the left side
view. During the measurement, the corrections were done to
have all the parts of the mirrors system and the experimental
vessel in a plumb line. However, none of this correction
improved the quality of movies and it should be only
concluded that optical imperfections of the mirrors system
were probably the source of this problem.

Results and Discussion

Two interaction regimes were found in the experiments:
Regime A: Vertical in-line motion of the pair with the
coalescence during the contact
Regime B: Deflection of the bubble pair from the
vertical line
The visual examples of these two regimes are given on
Fig. 5. and Fig. 6.
The first interaction regime (Regime A) represents the
laterally stable arrangement of the bubble pair. It was found
that the bubbles in the pair with Re < 24 approached in line
and finally they coalesced on the contact


Figure 5: The laterally stable pair of the bubbles. The
bubbles coalesced on the contact. Re = 16, S, = 66. The
time interval between the frames is 250 ms and 10ms in the
upper and bottom row, respectively.


Figure 6: The laterally unstable pair of the bubbles.
Re = 103, S, = 29. The time interval between the frames is
35 ms and 850ms in the upper and bottom row, respectively.
The white dashed line is guide for eye representing the
vertical line.


left righ t
side side
view view









The examples of the trajectory of the bubble pairs during
rise are presented on the Fig. 8 and Fig. 9 for the interaction
regime A and regime B, respectively. Their vertical and
lateral dimensionless separation Sx and Sz (defined by
scheme on Fig. 7) is plotted versus dimensionless time t =
2tUr/D.


Lx


R.


Q I

Figure 7: Scheme of the studied bubble configuration.

The overview of the successfully created bubble pairs is
reported on Fig. 10. The each point usually corresponds to
five experimental runs. The data covers Re = 16 124 for
initial separation S, = 11 71. Using other common
dimensionless variables this corresponds to Weber number
We= 0,1 0,24 (We = DU2p /c) or Bond number
Bo = 0,01 0,06 (Bo = D2p g/() or to Galilei number range
Ga = 315 3477 (Ga D3g/v2).
The image post-processing revealed that the bubbles were
sometimes slightly misaligned in a lateral direction; their
lateral offset had values Sx 0.1 0.8. In the case of the
bubbles coalescing in the vertical line (Regime A) there was
a non-zero lateral separation as well, but it did not change or
it decreased in certain cases. Hence, for these cases, the
charts with the lateral separation Sx versus time are omitted.

An explanation for the scatter of the evolution of the bubble
pair spacing could be the difference between the size of
leading and trailing bubbles. This hypothesis was tested by
solving the simplified equations of motion for the bubble
pair with unequally sized bubbles for in-line arrangement.
The drag reduction of a trailing bubble was given by
far-wake approximation given by Batchelor (1967).


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

The solution for two equally sized bubbles is shown as the
red line. The trailing bubble was slightly greater than the
leading bubble in the most cases. The lower bound, drawn
as the black line, represents difference about 7tm. For a
single case only, the upper bound (yellow line) represents a
smaller trailing bubble about 6tm.
The comparison of the model predictions with the
experimental results in case of the pair of the bubbles
coalescing in line has shown that the scatter in the time
evolution of the bubble spacing is caused by a slight
difference in their size. An imperfect fit to the experimental
data can be attributed to the negligence of a history force
and to other simplifications in the model. Forasmuch as
derivation of the equation of motion for the bubble pair that
deflects out of the in-line arrangement is difficult, further it
is supposed that bubble size difference is responsible for the
scatter.


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. ......:-.. :Z Z.. : :::::SZ '.. ..


*ig ,


t' [-]
Figure 9: Evolution of bubble pair spacing during rise.
Regime B, Re. = 124, S,.= 29. Points with dotted line
represent the experimental data.


Figure 8: Evolution of bubble pair spacing during rise.
Regime A, Re. = 124, S,.= 29. Solid lines are model for
bubbles of different sizes. Points with dotted line represent
the experimental data. The subscripts 1 and 2 refer to the
leading and the trailing bubble










0 coalescence in line, Water
82
deflection out of line, Water
72 -
stable vertical separation
2 (Sanada,2005)
62 stable vertical separation
(Yuan and Prosperetti, 1993)
52
0
42
-4--2 - - - - - -
32 -

22 - - -
12 -

2
0 20 40 60 80 100 120 140
Re [-]
Figure 10: The parametric map Re- S, of the lateral stability
of the bubble pair.


Conclusions

Two qualitatively different interaction regimes were
observed for the pair of in line aligned bubbles:

A. In line motion of the bubble pair with the coalescence
B. Deflection of the bubble pair out of the line

The interaction regime A was observed for bubbles with low
Reynolds numbers Re < 24. The regime B occurred at
moderate Re numbers Re > 54. The lack of other
experimental data did not allow to determine transition
value of Re for loose of lateral stability of the bubble pair. It
can be speculated that value is close to Re = 28 suggested by
numerical simulation of Yuan and Prosperetti (1994) and
Legendre at al. (2003). The experiments were carried out in
the ultrapure water with measured purity to assure free-slip
condition at bubble surface.

The comparison of the experimental data for Regime A with
the model based on the far-wake approximation reveal that
minute difference in the sizes of the bubbles in the pairs was
found as the source of a scatter of the data. That the
production of the perfectly arranged bubble pairs is a highly
difficult task.

Acknowledgements

This work was supported by the Grant Agency of Academy
of Sciences of CR under grant KJB 200720901. Also
support of the Grant Agency of the Czech Republic (under
grant No. 104/07/1110) in the specific tasks of the work is
gratefully acknowledged.

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