# Self-Propulsion of Two Contacting Bubbles Due to the Radiation Interaction Force

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Interaction Force between Two Bubbles in an Acoustic Field

#### 2.2. Linear Scattering Coefficients When Parametric Excitation Is Absent

#### 2.3. Net Force Experienced by Two Contacting Bubbles

## 3. Numerical Simulations

^{3}, $\eta =0.001$ Pa s, $\sigma =0.0727$ N/m, $\gamma =1.4$, ${P}_{0}=101.3$ kPa, ${P}_{a}=10$ kPa, $f=\omega /2\pi =30$ kHz. These parameters correspond to air bubbles in water.

## 4. Experiments

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mathematical Formulas Used in Calculations

## References

- Bunea, A.-I.; Taboryski, R. Recent advances in microswimmers for biomedical applications. Micromachines
**2020**, 11, 1048. [Google Scholar] [CrossRef] - Sridhar, V.; Park, B.-W.; Sitti, M. Light-driven Janus hollow mesoporous TiO2-Au microswimmers. Adv. Funct. Mater.
**2018**, 28, 1704902. [Google Scholar] [CrossRef] - Alapan, Y.; Yigit, B.; Beker, O.; Demirörs, A.F.; Sitti, M. Shape-encoded dynamic assembly of mobile micromachines. Nat. Mater.
**2019**, 18, 1244–1251. [Google Scholar] [CrossRef] - Ceylan, H.; Yasa, I.C.; Yasa, O.; Tabak, A.F.; Giltinan, J.; Sitti, M. 3D-printed biodegradable microswimmer for theranostic cargo delivery and release. ACS Nano
**2019**, 13, 3353–3362. [Google Scholar] [CrossRef] - Louf, J.F.; Bertin, N.; Dollet, B.; Stephan, O.; Marmottant, P. Hovering microswimmers exhibit ultrafast motion to navigate under acoustic forces. Adv. Mater. Interfaces
**2018**, 5, 1800425. [Google Scholar] [CrossRef] - Bertin, N.; Spelman, T.A.; Stephan, O.; Gredy, L.; Bouriau, M.; Lauga, E.; Marmottant, P. Propulsion of bubble-based acoustic microswimmers. Phys. Rev. Appl.
**2015**, 4, 064012. [Google Scholar] [CrossRef] - Feng, J.; Yuan, J.; Cho, S.K. Micropropulsion by an acoustic bubble for navigating microfluidic spaces. Lab Chip
**2015**, 15, 1554–1562. [Google Scholar] [CrossRef] - Ahmed, D.; Lu, M.; Nourhani, A.; Lammert, P.E.; Stratton, Z.; Muddana, H.S.; Crespi, V.H.; Huang, T.J. Selectively manipulable acoustic-powered microswimmers. Sci. Rep.
**2015**, 5, 9744. [Google Scholar] [CrossRef] - Jeong, J.; Jang, D.; Kim, D.; Lee, D.; Chung, S.K. Acoustic bubble-based drug manipulation: Carrying, releasing and penetrating for targeted drug delivery using an electromagnetically actuated microrobot. Sens. Actuators A
**2020**, 306, 111973. [Google Scholar] [CrossRef] - Ahmed, D.; Mao, X.; Shi, J.; Juluri, B.K.; Huang, T.J. A millisecond micromixer via single-bubble-based acoustic streaming. Lab Chip
**2009**, 9, 2738–2741. [Google Scholar] [CrossRef] - Gao, Y.; Wu, M.; Luan, Q.; Papautsky, I.; Xu, J. Acoustic bubble for spheroid trapping, rotation and culture: A tumor-on-a-chip platform (ABSTRACT platform). Lab Chip
**2022**, 22, 805–813. [Google Scholar] [CrossRef] - Luo, T.; Wu, M. Biologically inspired micro-robotic swimmers remotely controlled by ultrasound waves. Lab Chip
**2021**, 21, 4095–4103. [Google Scholar] [CrossRef] - Pak, O.S.; Zhu, L.; Brandt, L.; Lauga, E. Micropropulsion and microrheology in complex fluids via symmetry breaking. Phys. Fluids
**2012**, 24, 103102. [Google Scholar] [CrossRef] - Bjerknes, V.F.K. Fields of Force; Columbia U. P.: New York, NY, USA, 1906. [Google Scholar]
- Crum, L.A. Bjerknes forces on bubbles in a stationary sound field. J. Acoust. Soc. Am.
**1975**, 57, 1363–1370. [Google Scholar] [CrossRef] - Doinikov, A.A. Acoustic radiation forces: Classical theory and recent advances. In Recent Research Developments in Acoustics; Transworld Research Network: Trivandrum, India, 2003; Volume 1, pp. 39–67. [Google Scholar]
- Doinikov, A.A. Dissipative effects on Bjerknes forces between two bubbles. J. Acoust. Soc. Am.
**1997**, 102, 747–751. [Google Scholar] [CrossRef] - Doinikov, A.A. Bjerknes forces between two bubbles in a viscous fluid. J. Acoust. Soc. Am.
**1999**, 106, 3305–3312. [Google Scholar] [CrossRef] - Doinikov, A.A.; Regnault, G.; Mauger, C.; Blanc-Benon, P.; Inserra, C. Acoustic microstreaming produced by two interacting gas bubbles undergoing axisymmetric shape oscillations. J. Fluid Mech.
**2022**, 931, A19. [Google Scholar] [CrossRef] - Shaw, S.J. Translation and oscillation of a bubble under axisymmetric deformation. Phys. Fluids
**2006**, 18, 072104. [Google Scholar] [CrossRef] - Guédra, M.; Inserra, C. Bubble shape oscillations of finite amplitude. J. Fluid Mech.
**2018**, 857, 681–703. [Google Scholar] [CrossRef] - Doinikov, A.A. Acoustic radiation pressure on a compressible sphere in a viscous fluid. J. Fluid Mech.
**1994**, 267, 1–21. [Google Scholar] [CrossRef] - Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 10th ed.; Dover: New York, NY, USA, 1972. [Google Scholar]
- Varshalovich, D.A.; Moskalev, A.N.; Khersonskii, V.K. Quantum Theory of Angular Momentum; World Scientific: Teaneck, NJ, USA, 1988. [Google Scholar]
- Zwillinger, D. Standard Mathematical Tables and Formulae, 31st ed.; CRC: Boca Raton, FL, USA, 2003. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Fluid Mechanics, 2nd ed.; Pergamon Press: Oxford, UK, 1987. [Google Scholar]
- Doinikov, A.A.; Cleve, S.; Regnault, G.; Mauger, C.; Inserra, C. Acoustic microstreaming produced by nonspherical oscillations of a gas bubble. I. Case of modes 0 and m. Phys. Rev. E
**2019**, 100, 033104. [Google Scholar] [CrossRef] - Prosperetti, A. Viscous effects on perturbed spherical flows. Quart. Appl. Math.
**1977**, 34, 339–352. [Google Scholar] [CrossRef] - Cleve, S.; Guédra, M.; Inserra, C.; Mauger, C.; Blanc-Benon, P. Surface modes with controlled axisymmetry triggered by bubble coalescence in a high-amplitude acoustic field. Phys. Rev. E
**2018**, 98, 033115. [Google Scholar] [CrossRef]

**Figure 2.**The magnitudes of the mode amplitudes ${s}_{n}^{(j)}$ for a bubbleman with ${R}_{10}=20$ μm and ${R}_{20}=50$ μm in the case that parametric excitation is absent. The calculation was carried out by the equations derived in Section 2.2.

**Figure 3.**Net force experienced by a bubbleman in the case that parametric excitation is absent. The equilibrium radius of bubble 2 is kept fixed, ${R}_{20}=50$ μm, while the equilibrium radius of bubble 1 is varied.

**Figure 4.**The magnitude of the shape modes developing on bubble 1 as a function of ${R}_{10}$. The equilibrium radius of bubble 2 is kept fixed, ${R}_{20}=50$ μm, $\eta =0.001$ Pa s, and the other parameters are as in Figure 3.

**Figure 5.**The contributions of the force components, given by Equations (14), (15) and (17), to the net force ${F}_{1}+{F}_{2}$ on the bubbleman as a function of the acoustic pressure amplitude ${P}_{a}$ for (

**a**) $\eta =0.001$ Pa s and (

**b**) $\eta =0.004$ Pa s. ${R}_{10}=20$ μm, ${R}_{20}=50$ μm, $f=30$ kHz.

**Figure 6.**Schematic of the experimental setup used for the creation of a two-bubble microswimmer (bubbleman) and for the capture of its oscillation and translation dynamics.

**Figure 7.**(

**a**) Bubble trajectories prior to the contact. A first bubble (right side) is trapped at a stable location in the acoustic standing wave. Once a second bubble is nucleated (left side), it moves due to the primary radiation force towards the same stable location. When the bubbles come to proximity, the radiation interaction force makes them come into contact. (

**b**,

**c**) The last moments prior to the contact. (

**d**) The bubbles come into contact and remain in this state. (

**e**) As soon as the bubbles come into contact, the bubbleman moves to a new stable location.

**Figure 8.**(

**a**) Trajectory of the bubbleman captured from above. The diameter of the quasi-circular trajectory can reach 4 mm. (

**b**) A zoom of the trajectory captured from the side. The bubbleman is clearly visible, showing that both bubbles remain spherical during the motion.

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**MDPI and ACS Style**

Doinikov, A.A.; Micol, T.; Mauger, C.; Blanc-Benon, P.; Inserra, C.
Self-Propulsion of Two Contacting Bubbles Due to the Radiation Interaction Force. *Micromachines* **2023**, *14*, 1615.
https://doi.org/10.3390/mi14081615

**AMA Style**

Doinikov AA, Micol T, Mauger C, Blanc-Benon P, Inserra C.
Self-Propulsion of Two Contacting Bubbles Due to the Radiation Interaction Force. *Micromachines*. 2023; 14(8):1615.
https://doi.org/10.3390/mi14081615

**Chicago/Turabian Style**

Doinikov, Alexander A., Thomas Micol, Cyril Mauger, Philippe Blanc-Benon, and Claude Inserra.
2023. "Self-Propulsion of Two Contacting Bubbles Due to the Radiation Interaction Force" *Micromachines* 14, no. 8: 1615.
https://doi.org/10.3390/mi14081615