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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**I. Interaction of Oscillating Bubbles**• Acoustic forcing pA(t) • Bubble pulsation • Secondary waves • Interaction forces**Introduction**• Degassing of the melt in microgravity • acoustic cavitation • ultrasonics degassing and medical ultrasonic diagnostics • bubble sonoluminescence**RELEVANCE to Microgravity**• De-gassing liquids and melts (NASA Microgravity: Crystal growth techniques)**RELEVANCE - others**• Prevention of erosion due to cavitation**Experimental set up**Imager Monitor Strobe light Levitator Signal Generator Keypad Hydrophone Amplifier Processor Oscilloscope**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**PULSATION MODEL**• Harmonic response • Frequency ratio • Resonance frequency**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**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**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**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**Attracting bubbles**166 ms 233 ms 266 ms 299 ms 333 ms 366 ms 88 ms 112 ms 120 ms 128 ms 136 ms**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**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**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**Investigation of coalesce lag (2)**r • For bubble size ratio 2 time lag ( 15 sec - 45 sec)**Interaction force**Coupling coefficient coefficients m,n Condition cos ~ O (12); 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**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**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**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**Force field for the resonant couple**• Equilibrium separation requ20 radii • repulsion force have a sharper change in small spacing • attraction force increase from requ with the increase of separation then decrease very slowly**Three Bubble Oscillation**Condition • R1=R20.133 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**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**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**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**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**BUBBLE CLOUD MODEL**• Interaction forces Fji • Drag force Di • Resultant Fi • Velocity Vi**BUBBLE CLOUD MODEL**• Equations of motion**BUBBLE CLOUD MODEL**• Coupling equations**Coalescence**Dispersion Transition to equilibrium Vibration Combined patterns EVOLUTION PATTERNS**Future work**• Multi-bubble dynamics • Two dimensional motion of the bubbles • Bubble behavior in various acoustic environment**History Location of the bubbles**set t=0 at r0=10 R0 set t=0 at r0=20 R0**Numerical solution for velocity and acceleration**I II III I II III**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**Boundary condition**• Parallel case for two attracting bubbles • use image source to replace the rigid wall • phase difference ignored (>>x) Physical condition Image geometry**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**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**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**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**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**Experiment methods and procedures**• Experimental apparatus and set up • Experimental methods • Forcing amplitude on the levitation plane