Simulation of interaction of gas bubbles in a liquid with allowing for their small asphericity

A. A. Aganin ; A. I. Davletshin

Matematičeskoe modelirovanie, Tome 21 (2009) no. 6, pp. 89-102

Résumé

A mathematical model of interaction of two gas bubbles in a liquid with allowing for the small distortions of the bubble surfaces is proposed. This model is a set of ordinary differential equations of the second order in the radii of the bubbles, the coordinates of their centers and the amplitudes of the deflections of the shape of the bubble surfaces from the spherical one. It is of the fourth order of accuracy relative to the ratio of the characteristic radius of the bubbles to the characteristic distance between them. In the model, the effects of the liquid viscosity and compressibility are taken into account, the gas in the bubbles is assumed homobaric. For its validation, the known solution of the problem of collapse of an empty cavity near a plane rigid wall, which was obtained by the boundary element method, has been used. To illustrate the applicability of the model, three problems of interaction of two bubbles are considered. In the first problem, the bubbles move away from one another, deflections of their surfaces from the spherical ones decrease. In the second problem, the bubbles approach each other and then create a bound pair, which then translates in the liquid as a unit. During the approach the deflections from the spherical shape of the bubbles slightly increase but keep small. In the third problem, the bubbles approach each other, the distortions of their spherical shape rapidly increase so that the amplitude of the deflections from the spherical shape of one of the bubbles soon becomes equal to its radius, which, according to the chosen criterion, means its destruction.

Bibliographie

[1] Putterman S. J., Weninger K. P., “Sonoluminescence: How Bubbles Turn Sound into Light”, Annu. Rev. Fluid Mech., 32 (2000), 445 | DOI | Zbl

[2] Taleyarkhan R. P., West C. D., Cho J. S., Lahey R. T. (jr.), Nigmatulin R. I., Block R. C., “Evidence for Nuclear Emissions During Acoustic Cavitation”, Science, 295 (2002), 1868 | DOI

[3] Taleyarkhan R. P., West C. D., Lahey R. T. (jr.), Nigmatulin R. I., Block R. C., Xu Y., “Nuclear emissions during self-nucleated acoustic cavitation”, Phys. Rev. Lett., 96 (2006), 034301 | DOI

[4] Parlitz U., Mettin R., Luther S., Akhatov I., Voss M., Lauterborn W., “Spatio-temporal dynamics of acoustic cavitation bubble clouds”, Phil. Trans. R. Soc. Lond. A, 357 (1999), 313–334 | DOI

[5] Bjerknes V. F. K., Field of Force, Columbia Univ. Press, New York, 1906

[6] Mettin R., Akhatov I., Parlitz U., Ohl C. D., Lauterborn W., “Bjerknes force between small cavitation bubbles in a strong acoustic field”, Phys. Rev. E, 56:3 (1997), 2924–2931 | DOI

[7] Konovalova S., Akhatov I., “Structure formation in acoustic cavitation”, Multiphase Science and Technology, 17:3 (2005), 343–371 | DOI

[8] Pelekasis N. A., Gaki A., Doinikov A., Tsamopoulos J. A., “Secondary Bjerknes forces between two bubbles and the phenomenon of acoustic streamers”, J. Fluid Mech., 500 (2004), 313–347 | DOI | MR | Zbl

[9] Doinikov A. A., “Mathematical model for collective bubble dynamics in strong ultrasound fields”, J. Acoust. Soc. Am., 116:2 (2004), 821–827 | DOI

[10] Margulis I. M., Margulis M. A., “Dinamika vzaimodeistviya puzyrkov v kavitatsionnom oblake”, Zhurnal fizicheskoi khimii, 78:7 (2004), 1326–1337

[11] Konovalova S. I., Translyatsionnye effekty i strukturoobrazovanie pri akusticheskoi kavitatsii, Dissertatsiya na soiskanie uchenoi stepeni k.f.-m.n., Ufa, 2006, 120 pp.

[12] Doinikov A. A., “Translational motion of two interacting bubbles in a strong acoustic field”, Phys. Rev. E, 64:2 (2001), 026301, 6 pp. | DOI

[13] Harkin A., Kaper T. J., Nadim A., “Pulsation and translation of two gas bubbles”, J. Fluid Mech., 445 (2001), 377–411 | DOI | MR | Zbl

[14] Kuznetsov G. N., Schukin I. E., “Vzaimodeistvie pulsiruyuschikh puzyrkov v vyazkoi zhidkosti”, Akust. zhurnal, 18 (1972), 565–570

[15] Reddy A. J., Szeri A. J., “Shape stability of unsteadily translating bubbles”, Phys. Fluids, 14:7 (2002), 2216–2224 | DOI

[16] Plesset M. S., Chapman R. B., “Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary”, J. Fluid Mech., 47 (1971), 283–290 | DOI

[17] Lauterborn W., Bolle H., “Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary”, J. Fluid Mech., 72 (1975), 391–399 | DOI

[18] Gavrilyuk S. L., Teshukov V. M., “Drag force acting on a bubble in a cloud of compressible spherical bubbles at large Reynolds number”, European Journal of Mechanics B Fluids, 24:4 (2005), 468–477 | DOI | MR | Zbl

[19] Doinikov A. A., “Equations of coupled radial and translational motions of a bubble in a weakly compressible liquid”, Phys. Fluids, 17:12 (2005), 128101, 4 pp. | DOI

[20] Khairer E., Nersett S., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Nezhestkie zadachi, Mir, M., 1990, 512 pp. | MR