J. Non-Newtonian Fluid Mech. 165 (2010) 800–811 Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: www.elsevier.com/locate/jnnfm Chaotic behavior of a single spherical gas bubble surrounded by a Giesekus liquid: A numerical study H. Amini Kafiabad, K. Sadeghy ∗ University of Tehran, College of Engineering, Department of Mechanical Engineering, P.O. Box 11155-4563, Tehran, Iran article info Article history: Received 13 December 2009 Received in revised form 16 March 2010 Accepted 20 April 2010 Keywords: Bubble dynamics Giesekus model Ultrasound applications Deborah number abstract In the present work, nonlinear oscillations of a spherical, acoustically driven gas bubble in a Giesekus liq- uid are examined numerically. A novel approach based on the Gauss–Laguerre quadrature (GLQ) method is implemented to solve the integro-differential equation governing bubble dynamics in a Giesekus liquid. It is shown that, using this robust method, numerical results could be obtained at very high amplitudes and frequencies typical of ultrasound applications. The GLQ method also enabled obtaining results at very high Deborah and Reynolds numbers over prolonged dimensionless times not reported previously. Based on the results obtained in this work, it is concluded that the GLQ method is well suited for bub- ble dynamics studies in viscoelastic liquids. It is also concluded that the extensional-flow behavior of the liquid surrounding the bubble (as represented by the mobility factor in the Giesekus model) has a strong effect on the chaotic behavior of the bubble, and this is particularly so at high Deborah numbers, high amplitudes and/or high frequencies of the acoustic field. A period-doubling bifurcation structure is predicted to occur for certain values of the mobility factor. © 2010 Elsevier B.V. All rights reserved. 1. Introduction The growth/collapse of gas bubbles in viscoelastic liquids has been the subject of intense investigations over the years [1]. The interest in this field of study stems primarily from its application in cavitation suppression using polymeric additives [2–12]. In addi- tion, certain polymer processing operations (e.g., film blowing and foam production) involve bubbles growing and/or collapsing in vis- coelastic media [13]. In rheometry, a single gas bubble growing or collapsing in a large expanse of a viscoelastic fluid has successfully been used to measure the extensional viscosity of a liquid [14]. The discovery that gas bubbles can be used as contrast agents in medical ultrasound has significantly increased the interest in this topic [15–17]. Unfortunately, bubbles subjected to acoustic pres- sure fields may attain very large sizes during operation, and this can be very damaging to living tissues [18,19]. Studies carried out in recent years suggest that the viscoelasticity of the surrounding media (e.g., physiological fluids and body tissues) may affect bubble growth and collapse in acoustic fields [20–24]. For example, using linear and nonlinear Maxwell models, Allen and Roy [20–22] pre- dicted that the maximum bubble size is increased in viscoelastic fluids. Their numerical results also suggest that subharmonics may be seen in viscoelastic fluids for certain parameter values. From ∗ Corresponding author. E-mail address: sadeghy@ut.ac.ir (K. Sadeghy). their numerical results it can be concluded that the rheology of the surrounding medium should always be considered when address- ing ultrasound-induced cavitation bioeffects. An important finding of the work carried out by Allen and Roy [20–22] is that use should preferably be made of nonlinear vis- coelastic fluid models when studying bubble oscillations in acoustic forcing. For instance, whereas the linear Maxwell model predicts an explosive growth, the nonlinear Maxwell model (also referred to as UCM model) predicts a bounded growth [20–22]. Having said this, it should be conceded that the UCM model is not very suitable for bubble dynamic studies in polymeric liquids. That is to say that, it predicts a constant shear viscosity and an unbounded extensional viscosity—effects which are both known to be uncommon among polymeric liquids. In the UCM model, nonlinearity has been incorporated through invoking the upper-convected time derivative only. Rheologi- cal models such as Giesekus and PTT incorporate nonlinearity through invoking nonlinear stress terms in addition to using a con- vected time derivative. As a result, their predictions in shear and extensional flows better comply with experimental data, say, for polymeric liquids. Allen and Roy [20] tried to obtain numerical results using the Phan–Thien Tanner (PTT) model. But, for the range of the parameters tried, they could not detect any major differ- ence between PTT and UCM results. As a matter of fact, limitations in the ability to properly resolve the region near the bubble wall did not allow Allen and Roy [20–22] to obtain numerical results at dimensionless times longer than 10, Deborah numbers larger than 0377-0257/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2010.04.010