Dynamics of a clamped drop under translational vibrations

Abstract

The free and forced translational oscillations of an ideal liquid drop have been investigated. The drop was placed in a large-sized vessel filled with a liquid of different density. In equilibrium, the drop has the shape of a circular cylinder, and it is sandwiched between the lid and bottom of the vessel. The contact line-speed at the end plates is proportional to the deviation of the contact angle from its equilibrium value (the angle is formed by the corresponding plate and the undeformed cylindrical surface of the drop). The proportionality coefficient (the wetting or Hocking parameter) is different for each surface; it characterizes the force of interaction between the contact line and the solid surface, which leads to energy dissipation during its movement. This makes it possible to use the velocity potential to describe motion in the presence of a deformed interface between inviscid fluids. It is shown that the fundamental frequency of the translational mode of natural oscillations may not vanish in contrast to the case of equal wetting parameters. The energy dissipation is determined by the total contribution of these parameters, which makes it possible to vary the motion of the contact line over a wide range. The oscillation amplitude is proportional to the density difference of the liquids, i.e., at equal densities, the system moves as a whole. It has been found that both integer and non-integer harmonics of shape oscillations are excited due to different wetting parameters. The external vibrational force excites only even harmonics and, with identical surfaces, only they are present in the oscillation spectrum.