Abstract
A mathematical model of interaction of weakly non-spherical gas bubbles in a liquid with the centers in a straight line is developed. This model is a system of ordinary differential equations of the second order in the radii of the bubbles, the spatial coordinates of their centers, and the amplitudes of their deviations from the spherical ones. It is of the sixth order of accuracy in terms of the ratio of the characteristic radius of the bubbles to the characteristic distance between them, which exceeds the accuracy of currently existing analogs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
V. F. K. Bjerknes, Field of Force (Columbia Univ. Press, New York, 1906).
R. Mettin, I. Akhatov, U. Parlitz, C. D. Ohl, and W. Lauterborn, ‘‘Bjerknes force between small cavitation bubbles in a strong acoustic field,’’ Phys. Rev. E 56, 2924–2931 (1997).
B. Kieser, R. Phillion, S. Smith, and T. McCartney ‘‘The application of industrial scale ultrasonic cleaning to heat exchangers,’’ in Proceedings of International Conference on Heat Exchanger Fouling and Cleaning (2011), pp. 336–366.
K. S. Suslick, ‘‘Sonochemistry,’’ Science (Washington, DC, U. S.) 247 (4949), 1439–1445 (1990).
D. L. Miller and J. Quddus, ‘‘Diagnostic ultrasound activation of contrast agent gas bodies induces capillary rupture in mice,’’ Proc. Natl. Acad. Sci. U. S. A. 97, 10179–10184 (2000).
G. N. Kuznetsov and I. E. Shchekin, ‘‘Interaction of pulsating bubbles in a viscous liquid,’’ Ultrasonics 18, 466тAY469 (1973).
A. A. Doinikov, ‘‘Translational motion of two interacting bubbles in a strong acoustic field,’’ Phys. Rev. E 64, 026301 (2001).
S. Konovalova and I. Akhatov, ‘‘Structure formation in acoustic cavitation,’’ Multiphase Sci. Technol. 17, 343–371 (2005).
J. F. Liang, W. Z. Chen, W. H. Shao, and S. B. Qi, ‘‘Aspherical oscillation of two interacting bubbles in an ultrasound field,’’ Chin. Phys. Lett. 29, 074701 (2012).
A. Harkin, T. J. Kaper, and A. Nadim, ‘‘Coupled pulsation and translation of two gas bubbles in a liquid,’’ J. Fluid Mech. 445, 377–411 (2001).
A. A. Aganin and A. I. Davletshin, ‘‘Simulation of interaction of gas bubbles in a liquid with allowing for their small asphericity,’’ Mat. Model. 21 (6), 89–102 (2009).
A. A. Aganin, A. I. Davletshin, and T. F. Khalitova ‘‘Expansion and collapse of bubbles in the central region of a streamer,’’ Lobachevskii J. Math. 42, 15–23 (2021).
S. Fujikawa and H. Takahira, ‘‘Dynamics of two nonspherical cavitation bubbles in liquids,’’ Fluid Dyn. Res. 4, 179–194 (1988).
A. A. Aganin and A. I. Davletshin, ‘‘A refined model of interaction of spherical gas bubbles in a liquid,’’ Mat. Model. 21 (9), 89–98 (2009).
J. R. Blake, G. S. Keen, R. P. Tong, and M. Wilson, ‘‘Acoustic cavitation: The fluid dynamics of nonтAYspherical bubbles,’’ Philos. Trans. R. Soc. A 357 (1751), 251–267 (1999).
A. A. Aganin, L. A. Kosolapova, and V. G. Malakhov ‘‘Numerical simulation of the evolution of a gas bubble in a liquid near a wall,’’ Math. Models Comput. Simul. 10, 89–98 (2018).
E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge Univ. Press, Cambridge, 2012).
Funding
The study was supported by a grant from the Russian Science Foundation no. 21-11-00100, https://rscf.ru/en/project/21-11-00100/.
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by D. A. Gubaidullin)
Rights and permissions
About this article
Cite this article
Davletshin, A.I. Refined Model of Dynamics of Weakly Non-Spherical Bubbles with the Centers in a Straight Line. Lobachevskii J Math 43, 1082–1087 (2022). https://doi.org/10.1134/S1995080222080054
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080222080054