Skip to main content
SpringerLink
Log in

Dynamics of Gas Bubbles in a Spherical Cluster under the Increase of Their Pressure

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

The dynamics of gas (air) bubbles in a spherical cluster under an increase of their pressure is studied. The dynamics of the bubbles is considered only until one of the bubbles becomes destroyed because of large deformations or comes into contact with any of its neighbors. The cluster is comprised of 123 bubbles located at the nodes of a cubic mesh (one bubble is at the center of the cluster). The liquid (water) pressure is 1 bar. Initially the bubbles and the liquid are at rest, all the bubbles are spherical with the radius 0.1 mm, the cluster radius is approximately equal to 30 times the initial radius of the bubbles. Main attention is focused on the maximum pressures attained inside the bubbles. The model of the joint dynamics of bubbles is a variant of the particle-models. It takes into account the translations and deformations of the bubbles. It has been found that the maximum value of the pressure in the bubbles is of the order of 80 times the liquid pressure and is achieved when the initial pressure in the bubbles is increased by about 3.4 times the liquid pressure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

REFERENCES

  1. G. L. Chahine, ‘‘Pressure generated be a bubble cloud collapse,’’ Chem. Eng. Commun. 28, 355–367 (1984).

    Article  Google Scholar 

  2. Y. Matsumoto, ‘‘Bubble and bubble cloud dynamics,’’ AIP Conf. Proc. 524, 65–74 (2000).

    Article  Google Scholar 

  3. C. E. Brennen, ‘‘Bubbly cloud dynamics and cavitation,’’ Invited Lecture at the Acoustical Society of America Meeting, Salt Lake City, Utah, June 2007.

  4. B. Cui, B. Ni, and Q. Wu, ‘‘Bubble-bubble interaction effects on dynamics of multiple bubbles in a vortical flow field,’’ Adv. Mech. Eng. 8, 168781401663170 (2016).

    Article  Google Scholar 

  5. M. Kornfeld and L. Suvorov, ‘‘On the destructive action of cavitation,’’ J. Appl. Phys. 15, 495–506 (1944).

    Article  Google Scholar 

  6. E. Lauer, X. Y. Hu, S. Hickel, and N. A. Adams, ‘‘Numerical investigation of collapsing cavity arrays,’’ Phys. Fluids 24, 052104 (2012).

    Article  Google Scholar 

  7. I. Hansson and K. A. Mørch, ‘‘The dynamics of cavity clusters in ultrasonic (vibratory) cavitation erosion,’’ J. Appl. Phys. 51, 4651–4658 (1980).

    Article  Google Scholar 

  8. W. Lauterborn and T. Kurz, ‘‘Physics of bubble oscillations,’’ Rep. Prog. Phys. 73, 106501 (2010).

    Article  Google Scholar 

  9. A. J. Coleman, J. E. Saunders, L. A. Crum, and M. Dyson, ‘‘Acoustic cavitation generated by an extracorporeal shockwave lithotripter,’’ Ultrasound Med. Biol. 13, 69–76 (1987).

    Article  Google Scholar 

  10. Y. A. Pishchalnikov, O. A. Sapozhnikov, M. R. Bailey, J. C. Williams, R. O. Cleveland, T. Colonius, L. A. Crum, A. P. Evan, and J. A. McAteer, ‘‘Cavitation bubble cluster activity in the breakage of kidney stones by lithotripter shockwaves,’’ J. Endourol. 17, 435–446 (2003).

    Article  Google Scholar 

  11. Y. Lu, J. Katz, and A. Prosperetti, ‘‘Dynamics of cavitation clouds within a high-intensity focused ultrasonic beam,’’ Phys. Fluids 25, 073301 (2013).

    Article  Google Scholar 

  12. O. V. Rudenko, ‘‘Nonlinear acoustics in medicine: A review,’’ Phys. Wave Phenom. 30, 73–85 (2022).

    Article  Google Scholar 

  13. K. S. Suslick and D. J. Flannigan, ‘‘Inside a collapsing bubble: Sonoluminescence and the conditions during cavitation,’’ Ann. Rev. Phys. Chem. 59, 659–683 (2008).

    Article  Google Scholar 

  14. S. W. Fong, D. Adhikari, E. Klaseboer, and B. C. Khoo, ‘‘Interactions of multiple spark-generated bubbles with phase differences,’’ Exp. Fluids 46, 705–724 (2009).

    Article  Google Scholar 

  15. G. L. Chahine, ‘‘Dynamics of the interaction of nonspherical cavities,’’ Math. Approach. Hydrodyn. 4, 51–66 (1991).

    Google Scholar 

  16. G. L. Chahine and R. Duraiswami, ‘‘Dynamical interactions in a multi-bubble cloud,’’ J. Fluids Eng. 114, 680–686 (1992).

    Article  Google Scholar 

  17. J. R. Blake, G. S. Keen, R. P. Tong, and M. Wilson, ‘‘Acoustic cavitation: The fluid dynamics of non-spherical bubbles,’’ Phil. Trans. R. Soc. London, Ser. A 357 (1751), 251–267 (1999).

  18. N. Bremond, M. Arora, S. M. Dammer, and D. Lohse, ‘‘Interaction of cavitation bubbles on a wall,’’ Phys. Fluids 18, 121505 (2006).

    Article  MATH  Google Scholar 

  19. N. Bremond, M. Arora, C.-D. Ohl, and D. Lohse, ‘‘Controlled multibubble surface cavitation,’’ Phys. Rev. Lett. 96, 224501 (2006).

    Article  Google Scholar 

  20. M. Ida, T. Naoe, and M. Futakawa, ‘‘Suppression of cavitation inception by gas bubble injection: Anumerical study focusing on bubble-bubble interaction,’’ Phys. Rev. E 76, 046309 (2007).

    Article  Google Scholar 

  21. T. T. Bui, E. T. Ong, B. C. Khoo, E. Klaseboer, and K. C. Hung, ‘‘A fast algorithm for modeling multiple bubbles dynamics,’’ J. Comput. Phys. 216, 430–453 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  22. Z. Fu and V. Popov, ‘‘The ACA-BEM approach with a binary-keymosaic partitioning for modelling multiple bubble dynamics,’’ Eng. Anal. Bound. Elem. 50, 169–179 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Tiwari, C. Pantano, and J. B. Freund, ‘‘Growth-and-collapse dynamics of small bubble clusters near a wall,’’ J. Fluid Mech. 775, 1–23 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  24. X. Huang, A. M. Zhang, and Y. L. Liu, ‘‘Investigation on the dynamics of air-gun array bubbles based on the dual fast multipole boundary element method,’’ Ocean Eng. 124, 157–167 (2016)

    Article  Google Scholar 

  25. A. A. Aganin and A. I. Davletshin, ‘‘Equations of interaction of weakly non-spherical gas bubbles in liquid,’’ Lobachevskii J. Math. 39, 1047–1052 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  26. A. A. Doinikov, ‘‘Mathematical model for collective bubble dynamics in strong ultrasound fields,’’ J. Am. Stat. Assoc. 116, 821–827 (2004).

    Google Scholar 

  27. O. V. Voinov and A. M. Golovin, ‘‘Lagrange equations for a system of bubbles of varying radii in a liquid of small viscosity,’’ Fluid Dyn. 5, 458–464 (1970).

    Article  Google Scholar 

  28. H. Takahira, T. Akamatsu, and S. Fujikawa, ‘‘Dynamics of a cluster of bubbles in a liquid (theoretical analysis),’’ JSME Int. J., Ser. B 37, 297–305 (1994).

    Google Scholar 

  29. S. Luther, R. Mettin, and W. Lauterborn, ‘‘Modelling acoustic cavitation by a Lagrangian approach,’’ AIP Conf. Proc. 524, 351–354 (2000).

    Article  Google Scholar 

  30. V. M. Teshukov and S. L. Gavrilyuk, ‘‘Kinetic model for the motion of compressible bubbles in a perfect liquid,’’ Eur. J. Mech. B: Fluids 21, 469–491 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  31. Y. A. Ilinskii, M. F. Hamilton, and E. A. Zabolotskaya, ‘‘Bubble interaction dynamics in Lagrangian and Hamiltonian mechanics,’’ J. Am. Stat. Assoc. 121, 786–795 (2007).

    Google Scholar 

Download references

Funding

This study was supported by the Russian Science Foundation (project no. 21-11-00100).

Author information

Authors and Affiliations

  1. Institute of Mechanics and Engineering, Kazan Scientific Center, Russian Academy of Sciences, 420111, Kazan, Russia

    I. A. Aganin & A. I. Davletshin

Authors
  1. I. A. Aganin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. I. Davletshin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to I. A. Aganin or A. I. Davletshin.

Additional information

(Submitted by D. A. Gubaidullin)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aganin, I.A., Davletshin, A.I. Dynamics of Gas Bubbles in a Spherical Cluster under the Increase of Their Pressure. Lobachevskii J Math 44, 1538–1547 (2023). https://doi.org/10.1134/S1995080223050037

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080223050037

Keywords:

Search

Navigation