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Портал Begell Электронная Бибилиотека e-Книги Журналы Справочники и Сборники статей Коллекции
International Journal of Fluid Mechanics Research
ESCI SJR: 0.206 SNIP: 0.446 CiteScore™: 0.5

ISSN Печать: 2152-5102
ISSN Онлайн: 2152-5110

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International Journal of Fluid Mechanics Research

DOI: 10.1615/InterJFluidMechRes.2018024983
pages 261-275


César Yepes
Departamento de Termofluidos, Facultad de Ingeniería, UNAM, México, 04510, Mexico
Jorge Naude
Departamento de Termofluidos, Facultad de Ingeniería, UNAM, México, 04510, Mexico
Federico Mendez
Departamento de Termofluidos, Facultad de Ingenieria, UNAM. México, D.F. 04510, Mexico
Faculty of Engineering
Margarita Navarrete
Instituto de Ingeniería, UNAM, México, 04510, Mexico
Fátima Moumtadi
Departamento de Ingeniería Eléctrica, UNAM, México, 04510, Mexico

Краткое описание

In this work, we have revisited theoretically the linear and nonlinear oscillations of a single bubble immersed in a Maxwell's fluid under the action of an acoustic pressure field. We adopt the above rheological model to identify in a simple manner the viscoelastic influence and the impact of the external acoustic field on the motion of the bubble. The governing equations are reduced to a modified Rayleigh–Plesset equation, which is solved together with an ordinary differential equation needed to predict the rheological influence of the normal stresses on the interface of the bubble. The resulting dimensionless governing equations were solved numerically; however, for small deviations from the equilibrium radius of the bubble, we add a frequency analysis by using the multiple-scale method to find the influence of the viscoelastic parameters for those conditions near to resonance, together with the estimation of the bending curve, which characterizes the well-known bending phenomenon. For a single value of the dimensionless Weber number, we have identified that for low Reynolds and Deborah numbers, the oscillations are periodic, with some harmonic ones well defined. However, as we increase the above dimensionless parameters, strong nonlinearities appear, and they are more notable when the effect of the viscous damping is reduced. Furthermore, the effect of the nonlinear terms of the governing equations depends strongly on the amplitudes of the oscillations: when the multiple scale analysis is used and we consider small deviations of the dimensionless equilibrium radius, we obtain that the resonance conditions for the amplitude of the bubble are reduced if the Deborah number is increased. On the other hand, for moderate values of the deviations of the equilibrium radius and retaining the validity of the multiple scale analysis, the foregoing behavior is also conserved. In this last case, stronger amplitudes of the radius of the bubble are obtained also for increasing values of the Deborah number.


  1. Allen, J.S. and Roy, R.A., Dynamics of Gas Bubbles in Viscoelastic Fluids. I. Linear Viscoelasticity, J. Acoust. Soc. Am., vol. 107, pp. 3167–3178, 2000a.

  2. Allen, J.S. and Roy, R.A., Dynamics of Gas Bubbles in Viscoelastic Fluids. II. Nonlinear Viscoelasticity, J. Acoust. Soc. Am., vol. 108, pp. 1640–1650, 2000b.

  3. Aris, R., Vectors, Tensors and the Basic Equations of Fluid Mechanics, 1st ed., New York: Dover, 1989.

  4. Brujan, E.A., A First-Order Model for Bubble Dynamics in a Compressible Viscoelastic Liquid, J. Non-Newtonian Fluid Mech., vol. 84, pp. 83–103, 1999.

  5. Brujan, E.A., Cavitation in Non-Newtonian Fluids, 1st ed., Heidelberg: Springer, 2011.

  6. Ellis, A.T., Some Effects of Macromolecules on Cavitation Inception and Noise, California Institute of Technology, Pasadena Division of Engineering and Applied Science, Pasadena, CA, Tech. Rep. TR-AD0666012, June 1967.

  7. Fogler, H.S. and Goddard, J.D., Collapse of Spherical Cavities in Viscoelastic Fluids, Phys. Fluids, vol. 13, pp. 1135–1141, 1970.

  8. Francescutto, A. and Nabergoj, R., Steady-State Oscillations of Gas Bubbles in Liquids: Explicit Formulas for Frequency- Response Curves, J. Acoust. Soc. Am., vol. 73, pp. 457–460, 1983.

  9. Francescutto, A. and Nabergoj, R., A Multiscale Analysis of Gas Bubble Oscillations: Transient and Steady-State Solutions, Acustica, vol. 56, pp. 12–22, 1984.

  10. Jimenez-Fernandez, J. and Crespo, A., Bubble Oscillation and Inertial Cavitation in Viscoelastic Fluids, Ultrasonics, vol. 43, pp. 643–651, 2005.

  11. Khismatullin, D.B. and Nadim, A., Radial Oscillations of Encapsulated Micro Bubbles in Viscoelastic Liquids, Phys. Fluids, vol. 14, pp. 3534–3557, 2002.

  12. Levitskii, S.P. and Listrov, A.T., Effect of Viscoelastic Properties of a Liquid on the Dynamics of Small Oscillations of a Gas Bubble, J. Appl. Mech. Tech. Phys., vol. 17, pp. 363–366, 1976.

  13. Levitsky, S.P. and Shulman, Z.P., Bubbles in Polymeric Liquids: Dynamics and Heat-Mass Transfer, Lancaster, PA: Technomic, 1995.

  14. Naude, J. and Mendez, F., Periodic and Chaotic Acoustic Oscillations of a Bubble Gas Immersed in an upper Convective Maxwell Fluid, J. Non-Newtonian Fluid Mech., vol. 155, pp. 30–38, 2008.

  15. Nayfeh, A.H. and Mook, D.K., Nonlinear Oscillations, New York: John Wiley, 1979.

  16. Nayfeh, A.H. and Saric, W.S., Nonlinear Acoustic Response of a Spherical Bubble, J. Sound Vib., vol. 30, pp. 445–453, 1973.

  17. Ohl, C.D., Arora, M., Ikink, R., de Jong, N., Versluis, M., Delius, M., and Lohse, D., Sonoporation from Jetting Cavitation Bubbles, Biophys. J., vol. 91, pp. 4285–4295, 2006.

  18. Prosperetti, A., Nonlinear Oscillations of Gas Bubbles in Liquids: Steady State Solutions, J. Acoust. Soc. Am., vol. 56, pp. 878–885, 1974.

  19. Prosperetti, A., The Thermal Behavior of Oscillating Gas Bubbles, J. Fluid Mech., vol. 222, pp. 587–616, 1991.

  20. Sarkar, K., Shi, W.T., Chatterjee, D., and Forsberg, F., Characterization of Ultrasound Contrast Microbubbles using in Vitro Experiments and Viscous and Viscoelastic Interface Models for Encapsulation, J. Acoust. Soc. Am., vol. 118, pp. 539–550, 2005.

  21. Scriven, L.E., Dynamics of a Fluid Interface, Chem. Eng. Sci., vol. 12, pp. 98–108, 1960.

  22. Shima, A., Tsujino, T., and Nanjo, H., Nonlinear Oscillations of Gas Bubbles in Viscoelastic Fluids, Ultrasonics, vol. 24, pp. 142–147, 1986.

  23. Zhang, Y. and Li, S., Mass Transfer during Radial Oscillations of Gas Bubbles in Viscoelastic Mediums under Acoustic Excitation, Int. J. Heat Mass Transf., vol. 69, pp. 106–116, 2014.

  24. Zhang, Y. and Zhang, Y., Chaotic Oscillations of Gas Bubbles under Dual-Frequency Acoustic Excitation, Ultrasonics Sonochem., vol. 40, pp. 151–157, 2018.

  25. Zhang, Y., Du, X., Xian, H., and Wu, Y., Instability of Interfaces of Gas Bubbles in Liquids under Acoustic Excitation with Dual Frequency, Ultrasonics Sonochem., vol. 23, pp. 16–20, 2015.

  26. Zhang, Y., Gao, Y., and Du, X., Stability Mechanisms of Oscillating Vapor Bubbles in Acoustic Fields, Ultrasonics Sonochem., vol. 40, Part A, pp. 808–814, 2018.

  27. Zhao, S., Kruse, D.E., Ferrara, K.W., and Dayton, P.A., Selective Imaging of Adherent Targeted Ultrasound Contrast Agents, Phys. Med. Biol., vol. 52, pp. 2055–2072, 2007.