^{1,a)}, Olgert Lindau

^{2,b)}, John R. Blake

^{3}and Andrew J. Szeri

^{4,c)}

### Abstract

Acoustically driven bubbles can develop shape instabilities and, if forced sufficiently strongly, distort greatly and break up. Perturbation theory provides some insight as to how these nonspherical shape modes grow initially but loses validity for large deformations. To validate the perturbation theory, we use a numerical model based on the boundary integral method capable of simulating nonspherical, axisymmetric bubbles subject to acoustic driving. The results show that the perturbation theory compares well with numerical simulations in predicting bubble breakup and stability. Thereafter, we compare the peak temperatures and pressures of spherical to nonspherical bubble collapses by forcing them with standing waves and traveling waves, respectively. This comparison is made in parameter ranges of relevance to both single bubble sonoluminescence and multibubble sonoluminescence and sonochemistry. At moderate forcing, spherical and nonspherical collapses achieve similar peak temperatures and pressures but, as the forcing is increased, spherical collapses become much more intense. The reduced temperature of nonspherical collapses at high forcing is due to residual kinetic energy of a liquid jet that pierces the bubble near the time of minimum volume. This is clarified by a calculation of the (gas) thermal equivalent of this liquid kinetic energy.

The authors would like to acknowledge support from the National Science Foundation Program in Theoretical Physics, from the National Aeronautics and Space Administration Program in Microgravity Fluid Physics, and from the Applied Science and Technology Graduate Group at the University of California, Berkeley. O.L. was supported by a Feodor Lynen Postdoctoral Fellowship from the Alexander von Humboldt Foundation. The authors would also like to thank Dr. Antony Pearson, Dr. Anil Reddy, and Professor Brian Storey for helpful discussions, and Professor Stanley Berger and Professor Edgar Knobloch for reviewing early drafts of this study.

I. INTRODUCTION

II. FORMULATION OF THE MODEL

A. Basic fluid mechanics assumptions

B. Acoustic forcing

C. The perturbation model of bubble stability and the criterion for breakup

D. Numerical model based on the boundary integral method

1. Dimensionless variables

2. Boundary integral method

3. Mixture properties and equation of state

4. Kinetic energy and its thermal equivalent

5. Chemistry and thermodynamics of rapid attainment of chemical equilibrium

6. Slow and rapid dynamics

III. RESULTS AND DISCUSSION

A. Initial conditions

B. Validation of the ODE model of bubble stability

C. Energy focusing during collapse: Comparison of spherical to nonspherical collapses

1. SBSL versus MBSL cases

2. Comparison of peak temperatures and pressures

3. Bubble translation and breakup

D. Discussion

IV. CONCLUSIONS

### Key Topics

- Bubble dynamics
- 70.0
- Acoustic waves
- 27.0
- Numerical modeling
- 18.0
- Chemical reactions
- 16.0
- Surface dynamics
- 11.0

## Figures

A typical axisymmetric bubble prior to (a) and just after (b) jet impact (, , ). Note that the traveling acoustic wave propagates upward. The vertical line is the axis of cylindrical symmetry. The length is made dimensionless by .

A typical axisymmetric bubble prior to (a) and just after (b) jet impact (, , ). Note that the traveling acoustic wave propagates upward. The vertical line is the axis of cylindrical symmetry. The length is made dimensionless by .

Stability and breakup diagram for bubbles with initial radius of driven by a traveling wave of frequency and amplitude . Bubble stability as calculated by the ODE model is shown by the shading of the region (, ) whereas stability results from the BIM model are indicated by individual points (, ). The natural frequency of volume oscillations is .

Stability and breakup diagram for bubbles with initial radius of driven by a traveling wave of frequency and amplitude . Bubble stability as calculated by the ODE model is shown by the shading of the region (, ) whereas stability results from the BIM model are indicated by individual points (, ). The natural frequency of volume oscillations is .

Stability and breakup diagram for bubbles driven by a traveling wave of frequency with varying initial radius and amplitude . Bubble stability and breakup are indicated in the same manner as in Fig. 2.

Stability and breakup diagram for bubbles driven by a traveling wave of frequency with varying initial radius and amplitude . Bubble stability and breakup are indicated in the same manner as in Fig. 2.

Dimensionless volume vs dimensionless time for a typical spherical bubble collapse (, , ). The dot indicates the transition point from slow to rapid dynamics. Volume and time are made dimensionless by the scales discussed in Sec. ???.

Dimensionless volume vs dimensionless time for a typical spherical bubble collapse (, , ). The dot indicates the transition point from slow to rapid dynamics. Volume and time are made dimensionless by the scales discussed in Sec. ???.

Normalized temperature (solid) and volume (dashed) vs dimensionless time for the typical spherical bubble collapse shown in Fig. 4. Both temperature and volume are normalized so that their maximum is unity. Note that temperature is constant at until the bubble switches to rapid dynamics.

Normalized temperature (solid) and volume (dashed) vs dimensionless time for the typical spherical bubble collapse shown in Fig. 4. Both temperature and volume are normalized so that their maximum is unity. Note that temperature is constant at until the bubble switches to rapid dynamics.

Normalized pressure (solid) and volume (dashed) vs dimensionless time for the typical spherical bubble collapse shown in Fig. 4. Both pressure and volume are normalized so that their maximum is unity. Note that pressure is *not* constant during the slow phase but rather the magnitude of change during this phase is small compared to the relatively large pressure increase after the bubble switches to rapid dynamics.

Normalized pressure (solid) and volume (dashed) vs dimensionless time for the typical spherical bubble collapse shown in Fig. 4. Both pressure and volume are normalized so that their maximum is unity. Note that pressure is *not* constant during the slow phase but rather the magnitude of change during this phase is small compared to the relatively large pressure increase after the bubble switches to rapid dynamics.

Normalized kinetic energy (solid) and volume (dashed) vs dimensionless time for the typical spherical bubble collapse shown in Fig. 4. Both kinetic energy and volume are normalized so that their maximum is unity. Note that the kinetic energy is zero at both the volume maximum and minimum when the bubble motion is instantaneously at rest.

Normalized kinetic energy (solid) and volume (dashed) vs dimensionless time for the typical spherical bubble collapse shown in Fig. 4. Both kinetic energy and volume are normalized so that their maximum is unity. Note that the kinetic energy is zero at both the volume maximum and minimum when the bubble motion is instantaneously at rest.

SBSL case : Peak temperature vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. The sum is also shown .

SBSL case : Peak temperature vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. The sum is also shown .

SBSL case : Peak pressure vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. (Note the log scale for the vertical axis.)

SBSL case : Peak pressure vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. (Note the log scale for the vertical axis.)

SBSL case : Minimum volume vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. (Note the log scale for the vertical axis.)

SBSL case : Minimum volume vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. (Note the log scale for the vertical axis.)

MBSL case : Peak temperature vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. The sum is also shown .

MBSL case : Peak temperature vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. The sum is also shown .

MBSL case : Peak pressure vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. (Note the log scale for the vertical axis.)

MBSL case : Peak pressure vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. (Note the log scale for the vertical axis.)

MBSL case : Minimum volume vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. (Note the log scale for the vertical axis.)

MBSL case : Minimum volume vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles. (Note the log scale for the vertical axis.)

SBSL case : For a given value of acoustic forcing parameter , the nonspherical bubble shape at minimum volume and (black dot) are shown together. Note that the last three bubbles achieve their volume minimum during the toroidal stage.

SBSL case : For a given value of acoustic forcing parameter , the nonspherical bubble shape at minimum volume and (black dot) are shown together. Note that the last three bubbles achieve their volume minimum during the toroidal stage.

MBSL case : For a given value of acoustic forcing parameter , the nonspherical bubble shape at minimum volume and (black dot) are shown together. Note that the last two bubbles achieve their volume minimum during the toroidal stage.

MBSL case : For a given value of acoustic forcing parameter , the nonspherical bubble shape at minimum volume and (black dot) are shown together. Note that the last two bubbles achieve their volume minimum during the toroidal stage.

SBSL case : Peak temperature vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles *without chemistry*. The sum is also shown .

SBSL case : Peak temperature vs acoustic forcing parameter for spherical () and nonspherical (*) bubbles *without chemistry*. The sum is also shown .

Centroid displacement and velocity vs time for a typical nonspherical bubble collapse (, , ): (a) Centroid displacement (solid) and volume (dashed) normalized to maximum values of unity vs dimensionless time; (b) dimensionless centroid velocity vs dimensionless time near the end of collapse (near minimum volume). Note that the centroid velocity prior to this time is negligible. The time and the centroid velocity in (b) are made dimensionless by the scales discussed in Sec. ???.

Centroid displacement and velocity vs time for a typical nonspherical bubble collapse (, , ): (a) Centroid displacement (solid) and volume (dashed) normalized to maximum values of unity vs dimensionless time; (b) dimensionless centroid velocity vs dimensionless time near the end of collapse (near minimum volume). Note that the centroid velocity prior to this time is negligible. The time and the centroid velocity in (b) are made dimensionless by the scales discussed in Sec. ???.

SBSL case : (a) Net centroid displacement vs acoustic forcing parameter ; (b) maximum centroid velocity vs acoustic forcing parameter . The quantities shown are for a single expansion and collapse cycle as shown in Fig. 17.

SBSL case : (a) Net centroid displacement vs acoustic forcing parameter ; (b) maximum centroid velocity vs acoustic forcing parameter . The quantities shown are for a single expansion and collapse cycle as shown in Fig. 17.

MBSL case : (a) Net centroid displacement vs acoustic forcing parameter ; (b) maximum centroid velocity vs acoustic forcing parameter . The quantities shown are for a single expansion and collapse cycle as shown in Fig. 17.

MBSL case : (a) Net centroid displacement vs acoustic forcing parameter ; (b) maximum centroid velocity vs acoustic forcing parameter . The quantities shown are for a single expansion and collapse cycle as shown in Fig. 17.

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