J. Non-Newtonian Fluid Mech. 155 (2008) 30–38 Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: www.elsevier.com/locate/jnnfm Periodic and chaotic acoustic oscillations of a bubble gas immersed in an Upper Convective Maxwell fluid J. Naude, F. M´ endez ∗ Facultad de Ingenier´ ıa, Universidad Nacional Aut´ onoma de M´ exico, 04510 M´ exico DF, Mexico article info Article history: Received 9 January 2007 Received in revised form 29 February 2008 Accepted 14 April 2008 Keywords: Chaotic behavior Oscillations Viscoelastic fluid Non-linear inertial cavitation abstract In the present work, the non-linear dynamics of a spherical gas bubble oscillating in a viscoelastic liquid is investigated numerically. The radial oscillations of the bubble are governed by a modified Rayleigh–Plesset equation due to the viscoelastic behavior of the liquid. For simplicity, the Upper Convective Maxwell model is chosen as the fluid constitutive equation. In addition, thermal dissipation effects within the gas bubble are taken into account and comparing with previous approaches, we consider the nearly isothermal model for the compression gas. The purely adiabatic case is also included. The resulting nondimensional governing equations depend on different dimensionless parameters; however, the rheological character is directly dictated by the Deborah number, De. The numerical results for the radial oscillations predict periodic solutions for values of De between 1 and 4 and showing a clear chaotic behavior for De ∼ 4.4, independently of the intensity of the thermal damping. In addition, the physical influence of other relevant parameters, like the characteristic Reynolds number, is also clarified. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Nowadays the theoretical and experimental analysis to treat the inertial cavitation in viscoelastic fluids has been widely recog- nized in the specialized literature due to its practical importance. In the past, the usage of a small amount of water-soluble poly- mer in order to suppress or control this type of cavitation was successfully applied in different fluid-dynamic configurations. In this direction, the pioneer works of Ellis and Hoyt [1] and Fogler and Goddard [2] are fundamental contributions that define the role of the viscoelasticity on the study of bubble dynamics. In the last years, the inertial cavitation has been intensively analyzed to char- acterize the growth and collapse of gas bubbles in non-Newtonian fluids. An excellent theoretical and experimental revision on the subject may be found in Ref. [3]. This author applying a singular perturbation method studied the dynamics of a spherical bubble in a compressible viscoelastic liquid. The main conclusion reported by Brujan is based on the fact that the rheological properties of the liquid have a profound effect on the dynamics of the bubble only for a set of values well defined of the Reynolds number. On the other hand, the importance of the inertial cavitation has also appeared in other areas. For example, in some medical ultrasonic applications the estimation of the threshold values of the pressure ∗ Corresponding author. Tel.: +52 55 56228103; fax: +52 55 56228106. E-mail address: fmendez@servidor.unam.mx (F. M ´ endez). amplitude is required to avoid damages in living tissues. It is well known that the human tissue can be conceptually treated as a vis- coelastic medium and the influence of this medium is crucial to study some applications related with medical ultrasonic diagnosis [4–6]. However, there are many situations for which the sudden growth or collapse of gas bubbles is not sufficient to describe effi- ciently the non-linear cavitation effects. In order to have a general perspective, we must include those cases for which the forced oscil- lations control the inertial cavitation. In this direction, the most relevant contributions distinguish between linear and non-linear approaches. In the first case, Levitskii and Lystrov [7] clarified the physical influence of the rheological fluid parameters on the natu- ral and forced frequencies of an oscillating gas bubble in the limit of small amplitudes of oscillation. For the non-linear analysis, Shima et al. [8] investigated theoretically the non-linear oscillations of gas bubble in a viscoelastic fluid of a three-constant Oldroyd model. In particular, these authors clarified the effects of relaxation and retar- dation times on frequency responses curves, including the relation between the maximum pressure at the bubble wall and the initial radius of the bubble. On the other hand, Allen and Roy [9] indicated that the tissue viscoelasticity may be an important consideration for the risk assessment of potential cavitation bioeffects. Recently, Jimenez-Fern ´ andez and Crespo [10] developed a theoretical analy- sis for the non-linear acoustic oscillations of a gas bubble immersed in viscoelastic fluid. They used a generalized Oldroyd rheological model, which includes the Oldroyd-B and the Upper Convected Maxwell models as particular cases. These authors confirmed that 0377-0257/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2008.04.003