Nonlinear Dynamics

, Volume 74, Issue 3, pp 559–570 | Cite as

Chaotic behavior of gas bubble in non-Newtonian fluid: a numerical study

  • S. Behnia
  • F. Mobadersani
  • M. Yahyavi
  • A. Rezavand
Original Paper


In the present paper, the nonlinear behavior of bubble growth under the excitation of an acoustic pressure pulse in non-Newtonian fluid domain has been investigated. Due to the importance of the bubble in the medical applications such as drug, protein or gene delivery, blood is assumed to be the reference fluid. Effects of viscoelasticity term, Deborah number, amplitude and frequency of the acoustic pulse are studied. We have studied the dynamic behavior of the radial response of bubble using Lyapunov exponent spectra, bifurcation diagrams, time series and phase diagram. A period-doubling bifurcation structure is predicted to occur for certain values of the effects of parameters. The results show that by increasing the elasticity of the fluid, the growth phenomenon will be unstable. On the other hand, when the frequency of the external pulse increases the bubble growth experiences more stable condition. It is shown that the results are in good agreement with the previous studies.


Bubble dynamics Non-Newtonian fluids Chaotic oscillations Deborah number Bifurcation diagrams Lyapunov spectrum 


  1. 1.
    Suzuki, R., Takizawa, T., Negishi, Y., Utoguchi, N., Maruyama, K.: Effective gene delivery with novel liposomal bubbles and ultrasonic destruction technology. Int. J. Pharm. 354, 49–55 (2008) CrossRefGoogle Scholar
  2. 2.
    Hernot, S., Klibanov, A.L.: Microbubbles in ultrasound-triggered drug and gene delivery. Adv. Drug Deliv. Rev. 60, 1153–1166 (2008) CrossRefGoogle Scholar
  3. 3.
    Ibsen, S., Benchimol, M., Simberg, D., Schutt, C., Steiner, J., Esener, S.: A novel nested liposome drug delivery vehicle capable of ultrasound triggered release of its payload. J. Control. Release 155, 0168 (2011) CrossRefGoogle Scholar
  4. 4.
    Husseini, G.A., Diaz de la Rosa, M.A., Richardson, E.S., Christensen, D.A., Pitt, W.G.: The role of cavitation in acoustically activated drug delivery. J. Control. Release 107, 253–261 (2005) CrossRefGoogle Scholar
  5. 5.
    Frenkel, V.: Ultrasound mediated delivery of drugs and genes to solid tumors. Adv. Drug Deliv. Rev. 60, 1193–1208 (2008) CrossRefGoogle Scholar
  6. 6.
    Hynynen, K.: Ultrasound for drug and gene delivery to the brain. Adv. Drug Deliv. Rev. 60, 1209–1217 (2008) CrossRefGoogle Scholar
  7. 7.
    Suzuki, R., Namai, E., Oda, Y., et al.: Cancer gene therapy by IL-12 gene delivery using liposomal bubbles and tumoral ultrasound exposure. J. Control. Release 142, 245–250 (2010) CrossRefGoogle Scholar
  8. 8.
    Johnston, B.M., Johnston, P.R., Corney, S., Kilpatrick, D.: Non-Newtonian blood flow in human right coronary arteries: transient simulations. J. Biomech. 39, 1116–1128 (2006) CrossRefGoogle Scholar
  9. 9.
    Janela, J., Moura, A., Sequeira, A.: A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries. J. Comput. Appl. Math. 234, 2783–2791 (2010) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Razavi, A., Shirani, E., Sadeghi, M.R.: Numerical simulation of blood pulsatile flow in a stenosed carotid artery using different rheological models. J. Biomech. 44, 2021–2030 (2011) CrossRefGoogle Scholar
  11. 11.
    Ashrafizaadeh, M., Bakhshaei, H.: A comparison of non-Newtonian models for lattice Boltzmann blood flow simulations. Comput. Math. Appl. 58, 1045–1054 (2009) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Shaw, S., Murthy, P.V.S.N.: Magnetic targeting in the impermeable microvessel with two-phase fluid model non-Newtonian characteristics of blood. Microvasc. Res. 80, 209–220 (2010) CrossRefGoogle Scholar
  13. 13.
    Chen, J., Lu, X.-Y.: Numerical investigation of the non-Newtonian blood flow in a bifurcation model with a non-planar branch. J. Biomech. 37, 1899–1911 (2004) CrossRefGoogle Scholar
  14. 14.
    Wang, C., Ho, J.-R.: A lattice Boltzmann approach for the non-Newtonian effect in the blood flow. Comput. Math. Appl. 62, 0898 (2011) MathSciNetGoogle Scholar
  15. 15.
    Favelukis, M., Albalak, R.J.: Bubble growth in viscous Newtonian and non-Newtonian liquids. Chem. Eng. J. 63, 149–155 (1996) Google Scholar
  16. 16.
    Jiang, S., Ma, Y., Fan, W., Yang, K., Li, H.: Chaotic characteristics of bubbles rising with coalescences in pseudoplastic fluid. Chin. J. Chem. Eng. 18, 18–26 (2010) CrossRefGoogle Scholar
  17. 17.
    Schembri, F., Sapuppo, F., Bucolo, M.: Experimental classification of nonlinear dynamics in microfluidic bubbles flow. Nonlinear Dyn. 67, 2807–2819 (2012) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ichihara, M., Ohkunitani, H., Ida, Y., Kameda, M.: Dynamics of bubble oscillation and wave propagation in viscoelastic liquids. J. Volcanol. Geotherm. Res. 129, 37–60 (2004) CrossRefGoogle Scholar
  19. 19.
    Fu, T., Ma, Y., Funfschilling, D., Li, H.Z.: Bubble formation in non-Newtonian fluids in a microfluidic T-junction. Chem. Eng. Process. 50, 438–442 (2011) CrossRefGoogle Scholar
  20. 20.
    Frank, X., Dietrich, N., Wu, J., Barraud, R., Li, H.Z.: Bubble nucleation and growth in fluids. Chem. Eng. Sci. 62, 7090–7097 (2007) CrossRefGoogle Scholar
  21. 21.
    Shaokun, J., Youguang, M., Wenyuan, F., Ke, Y., Huaizhi, L.: Chaotic characteristics of bubbles rising with coalescences in pseudoplastic fluid. Chin. J. Chem. Eng. 18, 18–26 (2010) Google Scholar
  22. 22.
    Kafiabad, H.A., Sadeghy, K.: Chaotic behavior of a single spherical gas bubble surrounded by a Giesekus liquid: a numerical study. J. Non-Newton. Fluid Mech. 165, 800–811 (2010) CrossRefMATHGoogle Scholar
  23. 23.
    Li, H.Z., Mouline, Y., Midoux, N.: Modelling the bubble formation dynamics in non-Newtonian fluids. Chem. Eng. Sci. 57, 339–346 (2002) CrossRefGoogle Scholar
  24. 24.
    Jiménez-Fernández, J., Crespo, A.: The collapse of gas bubbles and cavities in a viscoelastic fluid. Int. J. Multiph. Flow 32, 1294–1299 (2006) CrossRefMATHGoogle Scholar
  25. 25.
    Li, H.Z., Frank, X., Funfschilling, D., Mouline, Y.: Towards the understanding of bubble interactions and coalescence in non-Newtonian fluids: a cognitive approach. Chem. Eng. Sci. 56, 6419–6425 (2001) CrossRefGoogle Scholar
  26. 26.
    Bloom, F.: Bubble stability in a class of non-Newtonian fluids with shear dependent viscosities. Int. J. Non-Linear Mech. 37, 527–539 (2002) CrossRefMATHGoogle Scholar
  27. 27.
    Wang, H., Jiang, X., Ma, J., Zhang, W.: Vibration of a single protein bubble in Bingham liquid. J. Hydrodyn., Ser. B 21, 658–668 (2009) CrossRefGoogle Scholar
  28. 28.
    Allen, J.S., Roy, R.A.: Dynamics of gas bubbles in viscoelastic fluids. I. Linear viscoelasticity. J. Acoust. Soc. Am. 107, 3167–3178 (2000) CrossRefGoogle Scholar
  29. 29.
    Allen, J.S., Roy, R.A.: Dynamics of gas bubbles in viscoelastic fluids. II. Non-linear viscoelasticity. J. Acoust. Soc. Am. 108, 1640–1650 (2000) CrossRefGoogle Scholar
  30. 30.
    Jiménez-Fernández, J., Crespo, A.: Bubble oscillation and inertial cavitation in viscoelastic fluids. Ultrasonics 43, 643–651 (2005) CrossRefGoogle Scholar
  31. 31.
    Lind, S.J., Phillips, T.N.: Spherical bubble collapse in viscoelastic fluids. J. Non-Newton. Fluid Mech. 165, 56–64 (2010) CrossRefMATHGoogle Scholar
  32. 32.
    Brujan, E.A.: A first-order model for bubble dynamics in a compressible viscoelastic liquid. J. Non-Newton. Fluid Mech. 84, 83–103 (1999) CrossRefMATHGoogle Scholar
  33. 33.
    Sorokin, V.S., Blekhman, I.I., Thomsen, J.J.: Motions of elastic solids in fluids under vibration. Nonlinear Dyn. 60, 639–650 (2010) CrossRefMATHGoogle Scholar
  34. 34.
    Sorokin, V.S., Blekhman, I.I., Vasilkov, V.B.: Motion of a gas bubble in fluid under vibration. Nonlinear Dyn. 67, 147–158 (2012) MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Siewe Siewe, M., Yamgou, S.B., Moukam Kakmeni, F.M., Tchawoua, C.: Chaos controlling self-sustained electromechanical seismograph system based on the Melnikov theory. Nonlinear Dyn. 62, 379–389 (2010) CrossRefMATHGoogle Scholar
  36. 36.
    Gao, Q., Ma, J.: Chaos and Hopf bifurcation of a finance system. Nonlinear Dyn. 58, 209–216 (2009) MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Chen, H., Zuo, D., Zhang, Z., Xu, Q.: Bifurcations and chaotic dynamics in suspended cables under simultaneous parametric and external excitations. Nonlinear Dyn. 62, 623–646 (2010) MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Dorfman, J.R.: An Introduction to Chaos in Nonequilibrium Statistical Mechanics. Cambridge University Press, New York (1999) CrossRefMATHGoogle Scholar
  39. 39.
    Ott, E.: Chaos in Dynamical System. Cambridge University Press, New York (2002) CrossRefGoogle Scholar
  40. 40.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, A.: Determining Lyapunov exponents from a time series. Physica D 16D, 285–317 (1985) MathSciNetCrossRefGoogle Scholar
  41. 41.
    Behnia, S., Yahyavi, M.: Characterization of intermittency in hierarchy of chaotic maps with invariant measure. J. Phys. Soc. Jpn. 81, 124008-8 (2012) CrossRefGoogle Scholar
  42. 42.
    Simon, G., Cvitanovic, P., Levinsen, M.T., Csabai, I., Horath, A.: Periodic orbit theory applied to a chaotically oscillating gas bubble in water. Nonlinearity 15, 25–43 (2002) MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Parlitz, U., Englisch, V., Scheffczyk, C., Lauterborn, W.: Bifurcation structure of bubble oscillators. J. Acoust. Soc. Am. 88, 1061–1077 (1990) MathSciNetCrossRefGoogle Scholar
  44. 44.
    Lauterborn, W., Parlitz, U.: Methods of chaos physics and their application to acoustics p bifurcation structure of bubble oscillators. J. Acoust. Soc. Am. 84, 1975–1993 (1988) MathSciNetCrossRefGoogle Scholar
  45. 45.
    Albernaz, D.L., Cunha, F.R.: Bubble dynamics in a maxwell fluid with extensional viscosity. Mech. Res. Commun. 38, 255–260 (2011) CrossRefMATHGoogle Scholar
  46. 46.
    Behnia, S., Jafari Sojahrood, A., Soltanpoor, W., Jahanbakhsh, O.: Suppressing chaotic oscillations of a spherical cavitation bubble through applying a periodic perturbation. Ultrason. Sonochem. 16, 502–511 (2009) CrossRefGoogle Scholar
  47. 47.
    Macdonald, C.A., Gomatam, J.: Chaotic dynamics of microbubbles in ultrasonic fields. Proc. - Inst. Mech. Eng., 220, 333–343 (2006) Google Scholar
  48. 48.
    Behnia, S., Jafari, A., Soltanpoor, W., Jahanbakhsh, O.: Nonlinear transitions of a spherical cavitation bubble. Chaos Solitons Fractals 41, 818–828 (2009) CrossRefGoogle Scholar
  49. 49.
    Yasui, Y., Iida, K., Tuziuti, T., Kozuka, T., Towata, A.: Strongly interacting bubbles under an ultrasonic horn. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 77, 016609-016619 (2008) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • S. Behnia
    • 1
  • F. Mobadersani
    • 2
    • 3
  • M. Yahyavi
    • 4
  • A. Rezavand
    • 5
  1. 1.Department of PhysicsUrmia University of TechnologyUrmiaIran
  2. 2.Department of Mechanical EngineeringUrmia University of TechnologyUrmiaIran
  3. 3.Department of Mechanical engineeringUrmia UniversityUrmiaIran
  4. 4.Department of PhysicsBilkent UniversityBilkent, AnkaraTurkey
  5. 5.Department of Mechanical engineeringIran University of Science and TechnologyTehranIran

Personalised recommendations