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Interacting Bubbles
1
Outline
I. Interaction of Oscillating Bubbles
• Introduction & Relevance
• Experimental apparatus and results
• Mathematical model and computer
simulations
2
I. Interaction of Oscillating Bubbles




Acoustic forcing
pA(t)
Bubble pulsation
Secondary waves
Interaction forces
r(t)
p'1(x1,t)
R1(t)
x2
F12
R2(t)
F21
x1
p'2(x2,t)
pA(t)
3
Introduction
– Degassing of the melt in microgravity
– acoustic cavitation
– ultrasonics degassing and medical
ultrasonic diagnostics
– bubble sonoluminescence
4
RELEVANCE to Microgravity

De-gassing liquids and melts (NASA
Microgravity: Crystal growth
techniques)
5
RELEVANCE - others

Prevention of erosion due to cavitation
6
Experimental set up
Signal
Generator
Strobe
light
Levitator
Imager
Monitor
Keypad
Amplifier
Hydrophone
Processor
Oscilloscope
7
ACOUSTIC LEVITATION
zlev
zmin
zlev
Buoyancy
zmax
Buoyancy
FL
FL
Standing wave:
Levitation force:
p(z,t) = A sin
2z

sin t
2 2 R0 3 A2
4z
FL 
sin
3 Pz (1  q 2 )

8
Acoustic levitator

High-speed
video images of
bubbles
interacting under
acoustic forcing
 Forcing
frequency f :
22.3-22.5 kHz
 Variables: R01,
R02, A, d0, u0
Distilled
Water
Levitation
Chamber
Piezoelectric
Ceramic
Iron Base
9
PULSATION MODEL

Harmonic
Ri (t )  R0i [1  i cos(t  i )]
response

Frequency qi  f / f 0i   / 0i
ratio

  Pz




Resonance   3k 
2 2
3  2
3 
0i
R0i 
R0i 
frequency
  R0i
1/ 2
10
BUBBLE CLASSIFICATION
Resonance size R0*
 Bubble types
– A: Above resonance size, R0>R0* , q>1
– X: Close to resonance size, R0~R0* , q~1
– B: Below resonance size, R0<R0*, q<1

11
INTERACTION FORCE
In phase ~0 (AA, BB pairs): Attraction
 Opposite phase ~ (AB pairs): Repulsion
 Phase shift ~/2 (XB,XA pairs): Longrange attraction, short-range repulsion

F21  F12  2R R
3
01
3
02
12 (cos  12 )
r2
12
Harmonic response of a single bubble
under pressure oscillation

External pressure

P(t )  PA  A cos[t ]


Linear Response of
bubble shape

R(t )  R0 (1   cos[t   ]

response amplitude



A
R02 ( 2   02 ) 2  4  2 2

phase difference
  arctan
2 
 2   02

PA :constant
pressure (usually
air pressure)
R0: equilibrium size
A : oscillatory
pressure amplitude
 : radian
frequency
0 : bubble natural
frequency
 : damping
13
coefficient
Damping of shape oscillation

Viscous component
v 

c: sound velocity in the liquid; : surface tension;
Thermal
component
T 

2
R02
 : liquid viscosity; Im ( ): complex function;
P  2 / R0
Im( )
2R02
Acoustic
component
A 
 2 R0
2c
Adapted from Brennen
14
Attracting bubbles
166 ms
88 ms
233 ms
112 ms
266 ms
120 ms
299 ms
128 ms
333 ms
136 ms
366 ms
15
Relative velocity of two
attracting bubbles
Experiment
10

R10= 0.4167
mm,
R20= 0.4167 mm

Az = 4.2 Kpa
f = 22 kHz

v0 = 12.5 mm/s
at r0 =20 radii
Drag model
8
v/ v0
Conservative
Model
6
4
2
0
0
5
10
15
20
r /R 0
16
Outcome of attracting bubbles

coalesce instantly
(most cases for
bubble with close
sizes)

coalesce with time
lag
attraction
depletion of
liquids
(Ri/Rj>2) 65-70%

collect and coexist
(rare and only for big
bubble size ratio)
collapse
17
Investigation of coalesce lag (1)
14
instant coalesce
12
coalesce lag
acceleration (x 102 mm/s 2)
For equal size
bubbles
 Az= 2.7 KPa
 f = 22.5 kHz
 R1=R2 0.5 mm
 v0=11.6 mm/s at
r0=12 radii
10
8
6
4
2
0
0
2
4
6
8
10
12
r/R0
18
Investigation of coalesce lag (2)

For bubble size ratio  2
time lag ( 15 sec - 45 sec)
12
2
acceleration (x10 mm/s )
10
-2
8
6
4
2
0
0
2
4
6
8
10
12
14
16
r/R0
19
Near-resonance coupling model

Condition


cos ~ O (12);
2; 1/2
Assumption

F12 (r )  2 2 R13 R231 2 (

2, i unchanged
1 (r )  1 (1 
K
)
r
Interaction force
Coupling coefficient
k 21 

m k 21n

)
r2 r3
 2 R23 2
A
coefficients m,n
Phenomenon


possible oscillation with
stable equilibrium spacing
requ
sign of force may change
during the motion
m  cos   1 2
n  21 2  cos 
20
Two bubble oscillation

8
model
experiment
6
separation (mm)
Bubble sizes
R1 = 0.161 mm
R2 = 0.151 mm
 Acoustic parameter
Az =1.26 KPa
f = 20.5 kHz
 Motion
pattern:oscillation
T = 0.86 s
amplitude = 3.15 mm
4
2
0
0
1
2
3
4
time (s)
21
Relative velocity versus time


Repulsion is much
violent than attraction
the motion of two
bubbles are generally
symmetric
highest velocity
around 20 mm/s
about 4-5 times as
large as that in
approach stage
left bubble
20
right bubble
10
v (mm/s)

30
0
-10
-20
-30
0
1
2
3
4
t (s)
22
Relative velocity versus separation



Model underpredict the velocity
in repulsion stage
out of balance
position in
levitation plane
loss of spherical
shape at small
spacing
model
simplification
40
30
20
v (mm/s)

10
0
-10
-20
model
experiment
-30
-40
1
2
3
4
5
6
r (mm)
23
Force field for the resonant couple

10
repulsion force
Force (x10-10 kg m/s 2)
Equilibrium separation
requ20 radii
 repulsion force have a
sharper change in
small spacing
 attraction force
increase from requ with
the increase of
separation then
decrease very slowly
8
attraction force
6
4
2
0
0
10
20
30
40
r/R0
24
Three Bubble Oscillation
Condition




R1=R20.133 mm
R0 = 0.146 mm
f = 22.5 kHz
Az = 1.34 Kpa
Model simplification




x-symmetry
bubble 0 motionless
interaction between
bubble 1 and bubble 2
ignored
coupling coefficient
2  2 R131
k0 
Az
25
History Location
of the bubbles
6
right bubble
r/R0

2
bound 3.64 - 4.8 radii
frequency 16.6 Hz
0


bound 3.68-4.82 radii
frequency 16.2 Hz
0.3
0.4
left bubble
bound 3.58 - 4.6 radii
frequency 16.5 Hz
Model
0.2
6
model
5
r/R0

0.1
t (s)
Left bubble

4
3
Right bubble

model
5
4
3
2
0
0.1
0.2
0.3
0.4
t (s)
26
Other experimental observation
A

C
Experiment A
small bubble oscillate with
big one and at the same
time has angular motion

Experiment B
five bubbles aligned with
oscillation, bubble in the
middle shift position

B
Experiment C
three bubbles in same
levitation plane perform
planar oscillation
27
Discussion
System of more than two bubbles
may display collective or evolution
motion
 Two-dimensional
is likely to
happen with more than two bubble
or given initial angular motion
 Group
oscillation
may
not
restricted to the condition for twobubble oscillation

28
Summary
Non-resonant pair






motion: attraction for R0>Rr
force ~ a/r2, sign unchanged
 and  not change
conservative model
drag force
outcome of two attracting
bubbles
Resonant pair






motion: possible
oscillation
force~ a/r2-b/r3, sign of
force may change
1 changed with separation
r
two bubble oscillation
repulsion violent than
approach
three bubble oscillation
more bubbles and 2-D
motion
29
BUBBLE CLOUD MODEL

Interaction
forces Fji
 Drag force
Di
 Resultant Fi
 Velocity Vi
n-1
rij
Fij
j
Fji
Di
n
i
Fi
Vi
2
3
1
30
BUBBLE CLOUD MODEL

Equations
of motion

 
n 

dVi
dri
mi
  F ji  Di ;
 Vi
dt
dt
j 1
j i

  (cos  ij   i j )
2 6 3 3 i j
F ji  2 R0 si s j
rij2

Di  6R0 k D f D (Re i )
si  R0i / R0
31
BUBBLE CLOUD MODEL

Coupling
equations
i 
 (qi )
2
i
s
 (qi ) 
n

j 1
j i
s3j cos  j
rij
A (qi )
j 
 2 R02 si2
2
i
q
2 2
q

1

4

(  i qi
2
i
2
i  1,2, n
32
EVOLUTION PATTERNS





Coalescence
Dispersion
Transition to equilibrium
Vibration
Combined patterns
33
CONCLUSIONS AND RECOMMENDATIONS
Desired Effect
Evolution
Pattern(s)
Range for
Forcing
Amplitude A
Frequency
Ratio
q = f/f0
Liquid degassing
Coalescence
Large
q 17
.
q  1

 q min  1
Uniform gas
concentration
Dispersion
Moderate
Regions of
high/low
concentration.
No motion
Equilibrium
Small to
moderate
1 q 17
.
Bubbles
vibration.
Continuous
stirring
Vibration
Moderate
1 q 17
.
34
Future work
Multi-bubble dynamics
 Two dimensional motion of the bubbles
 Bubble behavior in various acoustic
environment

35
History Location of the bubbles
20
Experiment
10
Experiment
Drag Model
Drag Model
Conservative
Model
16
8
12
r/ R0
r / R0
Conservative
Model
6
8
4
4
2
0
0
0.2
0.4
t (s)
set t=0 at r0=20 R0
0.6
0.8
0
0.02
0.04
0.06
0.08
0.1
0.12
t (s)
set t=0 at r0=10 R0
36
Numerical solution for velocity
and acceleration
120
4
100
acceleration (mm /s)
3
v (mm/s)
2
80
I
60
II
III
40
20
2
I
II
III
1
0
-1
0
0
0.2
0.4
t (s)
0.6
0.8
0
0.2
0.4
0.6
0.8
t (s)
37
Secondary Bjerknes force
& drag force
2nd Bjerknes
force
drag force
40
30
force ( x 10
-7
kg m/s 2)
50
20
10
0
0
0.2
0.4
0.6
0.8
time (s)
38
Velocity ratio

3
2

1
ave (r<=12 radii)
.4
18 0
.6
16 0
.9
15 5
.0
7
12
.9
10 7
.7
2
9.
34
7.
57
6.
25
4.
87
3.
03
0
20
Ve / Vm
0.23
r/R0

Ratio of
experimental
velocity to the
velocity of model
prediction
ratio approach 1
with the decrease of
spacing
high ratio in the
large spacing
caused by the
pressure gradient in
the levitation plane
39
Error Analysis
1.5
0.
98
1.
04
0.5
0
2
3
4
5
6
7
Experiments
8
average velocity
ratio (Ve/Vm)
Standard
Deviation 
1
0.99
0.15
2
0.99
0.18
3
1.31
0.23
4
1.16
0.23
5
1.11
0.34
6
1.25
0.25
7
1.14
0.21
8
1.09
0.24
9
1.22
0.37
10
1.17
0.36
11
1.04
0.26
12
0.98
0.26
1.
17
1.
22
1.
09
1.
14
1.
11
1.
16
1.
25
1.
31
0.
99
0.
99
velocity ratio (average)
1
1
r<10 radii
Experiment
9 10 11 12
40
Boundary condition

Parallel case for
two attracting
bubbles
 use image source
to replace the
rigid wall
 phase difference
ignored (>>x)
Physical condition
Image geometry
41
Mathematical model

The reflected force

FR  2 21 2 R13 R23 cos 

2 cos 
y2
Total force

1
2
FR  2   R R cos  2 [1 
]
r
(1  4 x 2 / r 2 )3 / 2

2
3
1 2 1
3
2


 is the pressure
reflection
coefficient
reflection angle
 =arccos (r/y)
glass=2300 kg/m3
velocity in glass
c = 5200 m/s
x is the distance
between bubble
and boundary
42
Model prediction of relative
velocities

R1 = R2 = 0.45 mm
70

Az= 3 Kpa
x= 2 mm
x = 1mm ~ 20 mm

v0 = 0 at r0 = 6 mm
x= 5 mm
50
f = 22.5 kHz

x= 1 mm
60
x=20 mm
v (mm/s)

40
30
20
10
0
0.2
0.4
0.6
0.8
r/r0
43
Relative velocities in experiments



Bubble sizes
R1 = 0.455 mm
R2 = 0.355 mm
forcing amplitude
Az = 2.55Kpa
f = 22.5 kHz
v0= 11 mm/s at r0=14
R1
experiment data
30
velocity (mm/s)

model with
b oundary
model without
b oundary
20
10
0
2
4
6
8
10
12
r/R0
44
Boundary effect compared between
two experiments
35
with b oundary
30
without b oundary
v (mm/s)
25
20
15
10
5
0
0
5
10
15
r/R0
45
Error Analysis (1)
Cycle
Experiment separation (mm)
1
upper
lower
amplitude
error
2
upper
lower
amplitude
error
3
upper
lower
amplitude
error
4
upper
lower
amplitude
error
1
5.357
2.536
2.821
13.5%
1.780
0.702
1.078
2.0%
3.000
1.320
1.680
18.0%
2.900
1.100
1.800
1.6%
2
4.714
2.286
2.428
25.5%
1.800
0.810
0.990
10.0%
3.375
1.356
2.019
1.5%
2.680
1.180
1.500
18.0%
3
4.857
2.25
2.607
20.0%
1.530
0.720
0.810
26.4%
3.333
1.425
1.908
6.9%
2.650
1.210
1.440
21.3%
4
4.786
2.286
2.500
23.3%
1.620
0.720
0.900
18.2%
2.970
1.300
1.670
18.5%
2.830
1.050
1.780
2.7%
5 Average
5.143
4.971
2.214
2.314
2.929
2.657
10.2%
18.5%
1.714
1.689
0.720
0.734
0.994
0.954
9.6%
13.2%
/
3.170
/
1.350
/
1.819
/
11.3%
/
2.765
/
1.135
/
1.630
/
10.9%
46
Error Analysis (2)
Cycle
Experiment time (s)
1
repulsion
approach
ratio
total
error
2
repulsion
approach
ratio
total
error
3
repulsion
approach
ratio
total
error
4
repulsion
approach
ratio
total
error
1
0.200
0.800
4.00
1.000
11.1%
0.032
0.104
3.25
0.136
6.6%
0.075
0.283
3.77
0.358
7.5%
0.079
0.183
2.32
0.262
7.7%
2
0.183
0.600
3.28
0.783
13.0%
0.032
0.128
4.00
0.160
9.9%
0.050
0.275
5.50
0.325
2.4%
0.083
0.221
2.66
0.304
7.0%
3
0.200
0.600
3.00
0.800
11.1%
0.032
0.096
3.00
0.128
12.1%
0.033
0.267
8.09
0.300
9.9%
0.050
0.200
4.00
0.250
12.0%
4
0.200
0.683
3.42
0.883
1.9%
0.032
0.112
3.50
0.144
1.1%
0.034
0.283
8.32
0.317
4.8%
0.067
0.233
3.48
0.300
5.6%
5 Average
0.150
0.187
0.680
0.673
4.53
3.60
0.830
0.859
7.8%
4.5%
0.040
0.034
0.104
0.109
2.60
3.24
0.144
0.142
1.1%
2.2%
/
0.048
/
0.277
/
6.42
/
0.325
/
2.4%
/
0.070
/
0.209
/
3.11
/
0.279
/
1.8%
47
Primary Bjerknes force


General form


F (r , t )    V (t )P(r , t ) 

Force in a
stationary sound
field
k R A
F
sin[ 2k z ]
z
3
0
2
3kP (1   2 /  02 )

z
< >: time average
 P(r,t) : time-and-spacingvarying pressure field
 A : amplitude of the
stationary wave
 kz=/c : wave number
 k : gas polytropic
number
sinusoidal

pressure
variation
P
(r , t )  P  A  sin[
k z] sin[ t ]

z
48
Secondary pressure radiation

Secondary wave
emitted by the
bubble
p' (r , t )  

 2 R03
r
cos(t   )

1 2 12   22
  1

 21 2 cos   O(1i 2j )
cos 
4

Secondary
Bjerknes force
phase difference
between two
pulsation 
  1  2

F12 (r )    V2 (t )p1' (r , t ) 
2 2 R13 R231 2 cos 

 (1 ,  2 ,  )
r2
Function 

F12(r) = F21(r)
F<0, attraction
F>0, repulsion
49
Experiment methods and
procedures
Experimental apparatus and set up
 Experimental methods
 Forcing amplitude on the levitation plane

50
Experimental methods

Adjust the frequency and water level to
make one full wave length of standing
wave generated by the acoustic levitator
 Use high-speed camera to capture the
motion of bubbles and measure the
size/location of the bubble frame by frame
using the movable reticle
 Balance the buoyancy force (FB=Fp) to
obtain the forcing amplitude on the
levitation plane
 Check the wave with oscilloscope and
hydrophone
51
Forcing amplitude

Buoyancy force
FB 



4
gR03
3
Primary Bjerknes force
k z R03 A2
Fp 
sin[ 2k z z ]
3kP (1   2 /  02 )
Forcing Amplitude
4 gP (1   2 /  02)
A
sin[ 2k z z ]
Forcing Amplitude on
the levitation plane
Az  A  sin[ k z z]
52
Experimental study and analysis of
bubble dynamics
Non-resonant bubble dynamics



Conservative model and drag model
Outcome of two attracting bubbles
Boundary effects
Resonant bubble interaction



2 bubble oscillation
3 bubble oscillation
other observations
53
Mathematical models for
non-resonant pair

Virtual mass of each bubble

Secondary Bjerknes force
m
2
R03
3
3
F12  22 R103 R20
1 2 cos  / r 2
D  4R0 uˆ

Drag force

Initial condition v=v0 at r=r0
54