Axisymmetric oscillations of a cylindrical drop in the final volume of fluid

The eigen and forced oscillations of a fluid drop surrounded by an incompressible fluid in a cylindrical container of a final volume are considered. The drop has a cylindrical shape in equilibrium and is bounded axially by two parallel solid surfaces. The equilibrium contact angle is a right angle. Dynamics of the contact line is taken into account by setting an effective boundary condition derived by Hocking: velocity of the contact line is assumed to be proportional to deviation of the contact angle from the equilibrium value. This condition leads to oscillation damping, which arises from the interaction of the contact line with a solid surface. Hocking’s parameter (wetting parameter) is the proportionality coefficient in this condition. A completely pinned contact line (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 eigenfunctions. Dependence of the eigenfrequency and damping rates on the problem parameters is investigated. It has been established that the main frequency of free oscillations can vanish at a certain value of Hocking’s parameter (so-called wetting parameter). The length of this interval depends on the ratio of height to radius of the drop. Other frequencies decrease monotonically with increasing Hocking’s parameter. The values of all frequencies increase with increasing relative radius of the drop or the radius of the vessel. Well-marked resonance effects are found in the study of forced oscillations. For the case of a pinned contact line or a fixed contact angle, the amplitude of forced oscillations grows without bound near the eigenfrequency. In other cases, the amplitude is finite. There are “anti-resonant” frequencies at which no deviation of the contact line from the equilibrium value is observed at any values of Hocking's parameter.