Capillary oscillations of a constrained liquid drop

An inviscid spherical liquid drop held by surface tension exhibits linear oscillations of a characteristic frequency and mode shape (Rayleigh oscillations). If the drop is pinned on a circle of contact the mode shapes change and the frequencies are shifted. The linear problem of inviscid, axisymmetric, volume-preserving oscillations of a liquid drop constrained by pinning along a latitude is solved here. The formulation gives rise to an integrodifferential boundary value problem, similar to that for Rayleigh oscillations, and for oscillations of a drop in contact with a spherical bowl [M. Strani and F. Sabetta, J. Fluid Mech. 141, 233 (1984)], only more constrained. A spectral method delivers a truncated solution to the eigenvalue problem. A numerical routine has been used to generate the eigenfrequencies/eigenmodes as a function of the location of the pinned circle of constraint. The effect of pinning the drop is to introduce a new low-frequency eigenmode. The center-of-mass motion, important in application, is partitioned among all the eigenmodes but the low-frequency mode is its principal carrier.