Influence of viscosity on nonlinear resonance of forced oscillations of a bubble in a vibrating liquid

Abstract

The paper presents the results of the study on dynamics of a gas bubble in an oscillating viscous fluid. The focus is on the case when the frequency of external vibrations is close to half the natural frequency of the quadrupole mode of free oscillations, which corresponds to the condition of the nonlinear resonance previously discovered in the inviscid approximation. A viscous potential flow model was used to take into account dissipation. In the framework of this model, the analytical expressions of the main process characteristics were obtained. The solution was expressed as an asymptotic series in a small parameter - the ratio of vibration amplitude to the bubble radius. The first-order theory shows that the bubble oscillates as a whole without deformation. The dependence of the amplitude and the phase shift of these oscillations on viscosity was established. The second-order theory investigates the nonlinear resonance associated with the excitation of the quadrupole deformation mode. It is shown that viscosity does not shift the resonant frequency, which remains half the natural frequency of the quadrupole mode, as in an ideal fluid, but significantly limits the amplitude of resonant oscillations. The results of numerical modeling of 3D axisymmetric bubble oscillations at the resonant frequency show good agreement with the analytical data. These findings are important for understanding and controlling processes in such applications as acoustic flotation, chemical reactors, and medical ultrasound diagnostics.