Free vibrations of a spherical drop constrained at an azimuth

Abstract

Two droplets coupled through a liquid filled (a) hole in a plate or (b) tube is referred to as a double droplet system (DDS) or a capillary switch. Such capillary systems are gaining increasing attention due to their utility in applications. A particularly exciting application is one where a DDS is employed as a liquid lens, one flavor of which entails using a DDS as a variable focus lens by keeping it under sustained oscillations at its natural frequencies. The natural modes of oscillation of a DDS are determined analytically here in the limit in which the plate thickness (or tube length) is vanishingly small and when the effect of gravity is negligible compared to that of surface tension. In this limit, a DDS at rest reduces to two spherical caps that are pinned to and coupled along a common circular ring of contact of negligible thickness. Here, the caps are taken to be complementary pieces of a sphere so that the equilibrium state of the system is a sphere that is constrained by a ring of negligible thickness at an azimuthal angle with respect to the center of the sphere. Both the constrained drop and the fluid exterior to it are taken to be inviscid fluids undergoing irrotational flow. Similar to the linear oscillations of a free drop first studied by Rayleigh, the analytical formulation of the linear oscillations of the constrained drop results in a linear operator eigenvalue problem but with one additional boundary condition, i.e., that which accounts for zero shape perturbation along the circle of contact. Exploiting properties of linear operators, an implicit expression is obtained for the frequency of each mode of oscillation, a feat that appears not to have been accomplished to date in any problem involving oscillations of constrained drops. An extension of a method based on Green's functions that was developed to analyze the linear oscillations of a drop in contact with a spherical bowl [M. Strani and F. Sabetta, “Free-vibrations of a drop in partial contact with a solid support,” J. Fluid Mech. 141, 233–247 (1984)]10.1017/S0022112084000811 is also employed to verify the aforementioned results. Results obtained from these two approaches are then compared to those reported by Bostwick and Steen [“Capillary oscillations of a constrained liquid drop,” Phys. Fluids 21, 032108 (2009)]10.1063/1.3103344. Careful examination of flow fields within drops reveals that by pinning a drop, it should be possible to selectively excite just a portion of a drop's surface.