Longitudinal oscillations of a droplet in a vessel of finite volume with inhomogeneous walls

Abstract

This article examines how the properties of substrate surfaces influence the oscillations of a liquid droplet clamped between the ends of a cylindrical vessel of finite volume filled with another liquid. At equilibrium, the droplet is cylindrical and confined axially by the ends of the vessel. The end surfaces are non-uniform and may differ in their wetting properties. The dynamics of the contact line between three media (droplet–liquid–solid surface) is considered: the contact line velocity is proportional to the deviation of the contact angle from its equilibrium value. The proportionality coefficient (wetting parameter) is a function of the substrate surface coordinates, which allows this surface to be considered non-uniform. The vessel is subject to a vibrational force directed along the vessel's axis of symmetry and perpendicular to its ends. Such vibrations excite only odd harmonics of axisymmetric droplet oscillations; but due to inhomogeneity and different surface properties, there will be excited both even harmonics and azimuthal modes, the spectrum of which is determined by the type of inhomogeneity. The dependence of the frequencies and damping factors of natural oscillations on the problem parameters is investigated. It is shown that inhomogeneity changes the effective wetting parameter, decreasing it, and qualitatively alters the dependence of the damping factors on the Hawking parameter for elongated droplets. When studying forced oscillations, clearly visible resonance effects were discovered. It is shown that resonant frequencies of azimuthal modes are present.