Free vibrations of a spherical drop constrained at an azimuth

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 t...