Effect of liquid viscosity on the resonant oscillations of a gas bubble in an acoustic field

We investigate parametrically excited shape perturbations of a spherical gas bubble oscillating in a sound wave field within a liquid. The parameters of the acoustic excitation are chosen such that the sound wavelength significantly exceeds the bubble size, allowing the pressure field far from the bubble to be considered uniform. Starting from first principles and employing the method of multiple scales together with a boundary-layer-resolving coordinate transformation, we investigate the effect of weak liquid viscosity on the parametric instability of shape modes driven by the forced monopole mode. In the first parametric resonance zone, the acoustic excitation frequency must be close to half the natural frequency of one of the capillary modes. The equations governing the evolution of the resonant mode near the threshold of parametric instability excitation are determined. The dependence of the threshold amplitude of the acoustic pressure on the problem parameters and the frequency detuning from the resonant value in the first resonant zone is found. For each mode number, the threshold is minimized at a specific bubble radius at which the resonance frequency considered here approaches the natural frequency of the bubble's monopole mode. Additionally, under the assumption of large surface shear viscosity, the effect of a surfactant layer adsorbed onto the bubble surface is analyzed. It is found that the surfactant effect is non-monotonic and size-dependent. Within the present model and parameter ranges examined, the surfactant layer can reduce the instability threshold for smaller bubble radii (R0≲1 mm), but tends to increase the threshold for larger bubbles. Furthermore, it is shown that the presence of a surfactant leads to a measurable decrease in the resonant frequency. In the clean-bubble case, the viscous damping decrement and the threshold scale linearly with the kinematic viscosity ν, whereas for the surfactant-laden bubble both the damping decrement and the threshold scale as ν, reflecting the transition from bulk viscous dissipation to boundary-layer-dominated dissipation.