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Interacting Bubbles
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  1. Interacting Bubbles

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

  3. I. Interaction of Oscillating Bubbles • Acoustic forcing pA(t) • Bubble pulsation • Secondary waves • Interaction forces

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

  5. RELEVANCE to Microgravity • De-gassing liquids and melts (NASA Microgravity: Crystal growth techniques)

  6. RELEVANCE - others • Prevention of erosion due to cavitation

  7. Experimental set up Imager Monitor Strobe light Levitator Signal Generator Keypad Hydrophone Amplifier Processor Oscilloscope

  8. ACOUSTIC LEVITATION

  9. 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

  10. PULSATION MODEL • Harmonic response • Frequency ratio • Resonance frequency

  11. 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

  12. INTERACTION FORCE • In phase j~0 (AA, BB pairs): Attraction • Opposite phase j~p (AB pairs): Repulsion • Phase shift j~p/2 (XB,XA pairs): Long-range attraction, short-range repulsion

  13. 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 Harmonic response of a single bubble under pressure oscillation

  14. Damping of shape oscillation • Viscous component • Thermal component • Acoustic component  : liquid viscosity; Im ( ): complex function; c: sound velocity in the liquid; : surface tension; Adapted from Brennen

  15. Attracting bubbles 166 ms 233 ms 266 ms 299 ms 333 ms 366 ms 88 ms 112 ms 120 ms 128 ms 136 ms

  16. 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

  17. Outcome of attracting bubbles attraction depletion of liquids • coalesce instantly (most cases for bubble with close sizes) • coalesce with time lag (Ri/Rj>2) 65-70% • collect and co-exist (rare and only for big bubble size ratio) collapse

  18. Investigation of coalesce lag (1) 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

  19. Investigation of coalesce lag (2) r • For bubble size ratio  2 time lag ( 15 sec - 45 sec)

  20. 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 Near-resonance coupling model

  21. Two bubble oscillation • Bubble sizes R1 = 0.161 mm R2 = 0.151 mm • Acoustic parameter Az =1.26 KPa f = 20.5kHz • Motion pattern:oscillation T = 0.86 s amplitude = 3.15 mm

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. Other experimental observation • 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 A C B

  28. 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

  29. 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 Summary

  30. BUBBLE CLOUD MODEL • Interaction forces Fji • Drag force Di • Resultant Fi • Velocity Vi

  31. BUBBLE CLOUD MODEL • Equations of motion

  32. BUBBLE CLOUD MODEL • Coupling equations

  33. Coalescence Dispersion Transition to equilibrium Vibration Combined patterns EVOLUTION PATTERNS

  34. CONCLUSIONS AND RECOMMENDATIONS

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

  36. History Location of the bubbles set t=0 at r0=10 R0 set t=0 at r0=20 R0

  37. Numerical solution for velocity and acceleration I II III I II III

  38. Secondary Bjerknes force & drag force

  39. 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

  40. Error Analysis

  41. Boundary condition • Parallel case for two attracting bubbles • use image source to replace the rigid wall • phase difference ignored (>>x) Physical condition Image geometry

  42. The reflected force Total force  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 Mathematical model

  43. 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

  44. 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

  45. Boundary effect compared between two experiments

  46. Error Analysis (1)

  47. Error Analysis (2)

  48. General form Force in a stationary sound field sinusoidal pressure variation < >: 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 Primary Bjerknes force

  49. Secondary wave emitted by the bubble Secondary Bjerknes force Function  phase difference between two pulsation  F12(r) = F21(r) Secondary pressure radiation F<0, attraction F>0, repulsion

  50. Experiment methods and procedures • Experimental apparatus and set up • Experimental methods • Forcing amplitude on the levitation plane