Dynamics of gas bubbles in viscoelastic fluids. II. Nonlinear viscoelasticity

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Abstract
The nonlinear oscillations of a spherical, acoustically forced gas bubble in nonlinear viscoelastic media are examined. The constitutive equation [Upper-Convective Maxwell (UCM)] used for the fluid is suitable for study of large-amplitude excursions of the bubble, in contrast to the previous work of the authors which focused on the smaller amplitude oscillations within a linear viscoelastic fluid [J. S. Allen and R. A. Roy, J. Acoust. Soc. Am. 107, 3167–3178 (2000)]. Assumptions concerning the trace of the stress tensor are addressed in light of the incorporation of viscoelastic constitutive equations into bubble dynamics equations. The numerical method used to solve the governing system of equations (one integrodifferential equation and two partial differential equations) is outlined. An energy balance relation is used to monitor the accuracy of the calculations and the formulation is compared with the previously developed linear viscoelastic model. Results are found to agree in the limit of small deformations; however, significant divergence for larger radial oscillations is noted. Furthermore, the inherent limitations of the linear viscoelastic approach are explored in light of the more complete nonlinear formulation. The relevance and importance of this approach to biomedical ultrasound applications are highlighted. Preliminary results indicate that tissue viscoelasticity may be an important consideration for the risk assessment of potential cavitation bioeffects.

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