Resonance oscillations of a drop or bubble in a viscous vibrating fluid

In the present work, we study the resonance oscillations of a single liquid drop or gas bubble relative to their equilibrium spherical shape, which arise as a result of harmonic, translational vibrations of a volume of liquid of different densities, containing such an inclusion. The parametric resonance of the forced translational mode with the main resonant frequency equal to the sum of two adjacent eigenfrequencies of capillary oscillations of the interface was previously discovered in an inviscid approximation [Lyubimov et al., “Resonance oscillations of a drop (bubble) in a vibrating fluid,” J. Fluid Mech. 909, A18 (2021)], also taking into account low viscosity using a phenomenological approach. Now, the effect of viscosity on the resonance is investigated by rigorous accounting for the viscosity in the dynamic boundary layer near the interface. The relations governing the evolution of the resonance modes near the threshold of instability excitation are determined. The equivalence of the two approaches taking viscosity into account is revealed: the rigorous one, built on a method of multiple scales, and the previously developed phenomenological approach, which is based on the artificial addition of the dissipative terms to the amplitude equations.