The stability of translational oscillations of a clamped drop of a low-viscosity liquid

Abstract

The study investigates oscillatory dynamics of a drop of a low-viscosity liquid surrounded by another liquid under translational low-amplitude vibration influence. A drop of equilibrium cylindrical shape is sandwiched between parallel solid planes. The contact angles are straight and constant, the contact lines of the three media slide freely along the surface of the plates. At the droplet–surrounding liquid interface, a thin viscous boundary layer is taken into account. Natural and forced oscillations of the drop are considered. In the main order of expansion in terms of small amplitude of vibrations, the natural frequencies of oscillations of an inviscid cylindrical drop were obtained. In the first order of expansion, a correction to the frequency was found, caused by energy dissipation in the viscous boundary layer. The stability of forced oscillations with respect to small disturbances was studied. Parametric resonance occurred when the synchronism condition was met: the vibration frequency was equal to the sum of the frequencies of two adjacent modes of natural oscillations. An expression describing the resonant regions was found. It is shown that low viscosity leads to the appearance of a vibration amplitude threshold and a shift in the instability region when compared with zero viscosity.