Interacting Bubbles

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Interacting Bubbles
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Outline

• I. Interaction of Oscillating Bubbles
• Introduction Relevance
• Experimental apparatus and results
• Mathematical model and computer simulations

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I. Interaction of Oscillating Bubbles

• Acoustic forcing pA(t)
• Bubble pulsation
• Secondary waves
• Interaction forces

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Introduction

• Degassing of the melt in microgravity
• acoustic cavitation
• ultrasonics degassing and medical ultrasonic
  diagnostics
• bubble sonoluminescence

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RELEVANCE to Microgravity

• De-gassing liquids and melts (NASA Microgravity
  Crystal growth techniques)

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RELEVANCE - others

• Prevention of erosion due to cavitation

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Experimental set up
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ACOUSTIC LEVITATION
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Acoustic levitator
Distilled Water

• High-speed video images of bubbles interacting
  under acoustic forcing
• Forcing frequency f 22.3-22.5 kHz
• Variables R01, R02, A, d0, u0

Levitation Chamber
Piezoelectric Ceramic
Iron Base
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PULSATION MODEL

• Harmonic response
• Frequency ratio
• Resonance frequency

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BUBBLE CLASSIFICATION

• Resonance size R0
• Bubble types
• A Above resonance size, R0gtR0 , qgt1
• X Close to resonance size, R0R0 , q1
• B Below resonance size, R0ltR0, qlt1

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INTERACTION FORCE

• In phase j0 (AA, BB pairs) Attraction
• Opposite phase jp (AB pairs) Repulsion
• Phase shift jp/2 (XB,XA pairs) Long-range
  attraction, short-range repulsion

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Harmonic response of a single bubble under
pressure oscillation

• External pressure
• Linear Response of bubble shape
• response amplitude
• phase difference

• PA constant pressure (usually air pressure)
• R0 equilibrium size
• A oscillatory pressure amplitude
• ? radian frequency
• ?0 bubble natural frequency
• ? damping coefficient

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Damping of shape oscillation
? liquid viscosity Im (? ) complex
function c sound velocity in the liquid ?
surface tension

• Viscous component
• Thermal component
• Acoustic component

Adapted from Brennen
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Attracting bubbles
166 ms 233 ms 266 ms 299 ms 333 ms 366 ms
88 ms 112 ms 120 ms 128 ms 136 ms
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Relative velocity of two attracting bubbles

• R10 0.4167 mm,
• R20 0.4167 mm
• Az 4.2 Kpa
• f 22 kHz
• v0 12.5 mm/s
• at r0 20 radii

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Outcome of attracting bubbles

• coalesce instantly
• (most cases for bubble with close sizes)
• coalesce with time lag
• (Ri/Rjgt2) 65-70
• collect and co-exist
• (rare and only for big bubble size ratio)

collapse
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Investigation of coalesce lag (1)

• For equal size bubbles
• Az 2.7 KPa
• f 22.5 kHz
• R1R2? 0.5 mm
• v011.6 mm/s at r012 radii

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Investigation of coalesce lag (2)

• For bubble size ratio ? 2
• time lag ( 15 sec - 45 sec)

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Near-resonance coupling model

• Interaction force
• Coupling coefficient
• coefficients m,n

• Condition
• cos? O (?1?2)
• ?2?? ?1??/2
• Assumption
• ?2, ?i unchanged
• Phenomenon
• possible oscillation with stable equilibrium
  spacing requ
• sign of force may change during the motion

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Two bubble oscillation

• Bubble sizes
• R1 0.161 mm
• R2 0.151 mm
• Acoustic parameter
• Az 1.26 KPa
• f 20.5 kHz
• Motion patternoscillation
• T 0.86 s
• amplitude 3.15 mm

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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

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Relative velocity versus separation

• Model under-predict the velocity in repulsion
  stage
• out of balance position in levitation plane
• loss of spherical shape at small spacing
• model simplification

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Force field for the resonant couple

• 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

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Three Bubble Oscillation

• Condition
• R1R2?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

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History Location of the bubbles

• Right bubble
• bound 3.64 - 4.8 radii
• frequency 16.6 Hz
• Left bubble
• bound 3.58 - 4.6 radii
• frequency 16.5 Hz
• Model
• bound 3.68-4.82 radii
• frequency 16.2 Hz

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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
• Experiment C
• three bubbles in same levitation plane perform
  planar oscillation

B
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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 two-bubble oscillation

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Summary

• Non-resonant pair
• motion attraction for R0gtRr
• 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

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BUBBLE CLOUD MODEL

• Interaction forces Fji
• Drag force Di
• Resultant Fi
• Velocity Vi

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BUBBLE CLOUD MODEL

• Equations of motion

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BUBBLE CLOUD MODEL

• Coupling equations

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EVOLUTION PATTERNS

• Coalescence
• Dispersion
• Transition to equilibrium
• Vibration
• Combined patterns

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CONCLUSIONS AND RECOMMENDATIONS
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Future work

• Multi-bubble dynamics
• Two dimensional motion of the bubbles
• Bubble behavior in various acoustic environment

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History Location of the bubbles
set t0 at r010 R0
set t0 at r020 R0
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Numerical solution for velocity and acceleration
II
III
I
II
III
I
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Secondary Bjerknes force drag force
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Velocity ratio

• 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

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Error Analysis
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Boundary condition

• Parallel case for two attracting bubbles
• use image source to replace the rigid wall
• phase difference ignored (?gtgtx)

Physical condition
Image geometry
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Mathematical model

• The reflected force
• Total force

• ? is the pressure reflection coefficient
• reflection angle
• ? arccos (r/y)
• ?glass2300 kg/m3
• velocity in glass
• c 5200 m/s
• x is the distance between bubble and boundary

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Model prediction of relative velocities

• R1 R2 0.45 mm
• Az 3 Kpa
• f 22.5 kHz
• x 1mm 20 mm
• v0 0 at r0 6 mm

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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 r014 R1

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Boundary effect compared between two experiments
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Error Analysis (1)
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Error Analysis (2)
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Primary Bjerknes force

• lt gt time average
• P(r,t) time-and-spacing-varying pressure field
• A amplitude of the stationary wave
• kz?/c wave number
• k gas polytropic number

• General form
• Force in a stationary sound field
• sinusoidal pressure variation

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Secondary pressure radiation

• Function ?
• phase difference between two pulsation ?
• F12(r) F21(r)

• Secondary wave emitted by the bubble
• Secondary Bjerknes force

Flt0, attraction Fgt0, repulsion
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Experiment methods and procedures

• Experimental apparatus and set up
• Experimental methods
• Forcing amplitude on the levitation plane

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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 (FBFp) to obtain the
  forcing amplitude on the levitation plane
• Check the wave with oscilloscope and hydrophone

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Forcing amplitude

• Buoyancy force
• Primary Bjerknes force
• Forcing Amplitude
• Forcing Amplitude on
• the levitation plane

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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

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Mathematical models for non-resonant pair

• Virtual mass of each bubble
• Secondary Bjerknes force
• Drag force
• Initial condition vv0 at rr0