The article is devoted to theoretical and experimental study of gas bubble behavior (compressible particle) in oscillating liquid. This problem is of essential interest with regard to a number of mineral-dressing processes, in particular, flotation process, therefore a considerable number of studies are dedicated to this issue. By the present time, two different mechanisms have been proposed, two explanations of an experimentally registered effect of gas bubbles sinking in vertical oscillating vessel with liquid — «wave» and «non-wave» («oscillating») mechanism. In the first case, the determining significance belongs to wave amplitude gradient, conditioned by bubble surrounding media compressibility, and in the second — to compressibility of bubble itself. The present work generalizes the authors' studies, performed over recent years, dedicated to investigation of this effect. Both proposed mechanisms of sinking are considered simultaneously — motion of gas bubble in oscillating vessel with liquid is studied, taking into account compressibilities of bubble and of gas-saturated surrounding layer that is formed near free surface of liquid through turbulization. Conditions are determined, under which bubble will sink in such compressible medium. An expression is derived for critical thickness of gas-saturated layer of liquid; when this thickness is exceeded, such layer starts expanding down into vessel, i.e., the condition of oscillatory instability of gas—liquid system separate state. It is shown, that bubble sinking condition is essentially affected by both its own compressibility and compressibility of its surrounding medium. An expression is obtained for average velocity of bubble motion in gas-saturated layer of liquid, essentially dependent on bubble sinking depth and oscillation parameters.

1. Blekhman I. I., Vaisberg L. A., Blekhman L. I., Vasilkov V. B., Yakimova K. S. Doklady Akademii Nauk, 2008, Vol. 422, No 4. pp. 470–474. (In Eng.: Doklady Physics, 2008, Vol. 53, No 10, pp. 520–524).
2. Crum L. A., Eller A. I. Motion of bubbles in a stationary sound field, The Journal of the Acoustical Society of America, 1970, Vol. 48, No 1(2), pp. 181–189.
3. Kremer E. B. Control of gas content in bubble media by vibration. IUTAM Symposium «The active control of vibration», 5–8 Sept. 1994, L., MEP, 1994, pp. 197–201.
4. Blekhman I. I. Voprosy matematicheskoy fiziki i prikladnoy matematiki (Problems of mathematical physics and applied mathematics), Ioffe Phisical Technical Institute, Saint Petersburg, 2007, pp. 241–248.
5. Ganiev R. F., Ukrainskiy L. Ye. Nelineynaya volnovaya mekhanika i tekhnologii (The nonlinear wave mechanics and technologies), Moscow, Scientific Publishing Centre «Regulyarnaya i khaoticheskaya dinamika», 2008, 712 p.
6. Buchanan R. H., Jameson G. J. Cycling migration of bubbles in vertically vibrating liquid columns, Ind. Eng. Chem. Fundamen, 1962, 1(2), pp. 82–86.
7. Bleich H. H. Effect of Vibrations on the motion of small gas bubbles in a liquid, jet propulsion, Journal of the American Rocket Society, 1956, Vol. 26, 11, 978, pp. 958–964.
8. Blekhman I. I., Vasilkov V. B., Sorokin V. S. Obogashchenie Rud, 2010, No 4, pp. 13–20.
9. Tatevosyan R. A. Teoreticheskie osnovy khimicheskoy tekhnologii — Theoretical Foundations of Chemical Engineering. 1977, Vol. 11, No 1, pp. 153–155.
10. Loytsyanskiy L. G. Mekhanika zhidkosti i gaza (Mechanics of liquid and gas). Moscow, Nauka, 1970.
11. Batchelor G. K. Mekhanika — Mechanics, 1968, No 3, pp. 65–84.
12. Blekhman I. I. Vibrational Mechanics, Singapore еt al., World Scientific, 2000, 510 p.
13. Kizevalter B. V. Teoreticheskie osnovy gravitatsionnykh protsessov obogashcheniya (Theoretical foundations of gravitational processes of mineral processing), Moscow, Nedra, 1979.
14. Spravochnik po obogashcheniyu rud. Podgotovitelnye protsessy (Reference Book on Mineral Processing. Preparatory processes), 2nd ed., Moscow, Nedra, 1982, 366 p.
15. Grigoryan S. S., Yakimov Yu. L., Apshteyn E. Z. Fluid dynamics transactions, Warszawa, 1967, Vol. 3, pp. 713–719.