Translational oscillations of a cylindrical drop in a bounded volume of fluid

Abstract

We consider the eigen and forced translational oscillations of a fluid drop surrounded by another fluid in a cylindrical vessel of finite volume. The drop is cylindrical in equilibrium and bounded axially by two parallel solid planes. Our approach takes into account the contact line dynamics of three continuous media (drop-liquid-solid surface), namely the velocity of the contact line proportional to the deviation of the contact angle from its equilibrium value. The Hocking’s parameter (so-called wetting parameter) is the proportionality coefficient in this case. A completely pinned contact line (so-called pinned-end edge condition) corresponds to the limiting value of Hocking’s parameter, which tends to zero. Hocking’s parameter tends to infinity in the opposite case of the fixed contact angle. The solution of the boundary value problem is found using Fourier series of Laplace operator eigen functions. The vibration force is directed along the symmetry axis of the drop. It is shown that the fundamental frequency of the translational mode of free oscillations vanishes at a critical value of Hocking’s parameter. Increasing the density of the drop or the radius of the vessel leads to the growth of the frequencies of eigenoscillations. The frequencies of the heavy drop (i.e. the drop having the density greater than that of the surrounding fluid) also increase with increasing relative radius of the drop, and for the light drop they decrease. We have revealed the existence of resonance effects for forced oscillations. The oscillation amplitude of the contact line is always finite, but the oscillation amplitude of the drop lateral surface tends to infinity in the zero limit of Hocking’s parameter. There are “anti-resonant” frequencies at which no deviation of the contact line from the equilibrium value is observed for any values of Hocking's parameter.