Dynamics of gas bubbles in viscoelastic fluids. I. Linear viscoelasticity

The nonlinear oscillations of spherical gas bubbles in linear viscoelastic fluids are studied. A novel approach is implemented to derive a governing system of nonlinear ordinary differential equations. The linear Maxwell and Jeffreys models are chosen as the fluid constitutive equations. An advantage of this new formulation is that, when compared with previous approaches, it facilitates perturbation methods and numerical investigations. Analytical solutions are obtained using a multiple scale perturbation method and compared with the Newtonian results for various Deborah numbers. Numerical analysis of the full equations supports the perturbation analysis, and further reveals significant differences between the viscoelastic and Newtonian cases. Differences in the oscillation phase and harmonic structure characterize some of the viscoelastic effects. Subharmonic excitations at particular fluid parameters lead to a discrete group modulation of the radial excursions; this appears to be a unique, previously undiscovered phenomenon. Implications for medical ultrasound applications are discussed in light of these current findings.