Oscillations of a viscous-free surface with a pinned contact line

We study the oscillations of a viscous liquid filling a two-dimensional rectangular container to the brim such that the free surface of the liquid is assumed to have a pinned contact line, by using the natural viscous complex eigenfunctions of the problem. By projecting the governing equations onto an appropriate basis, a nonlinear eigenvalue problem for the complex frequencies is obtained. This is then solved to obtain the modal frequencies as a function of the Reynolds and Bond numbers, Re and Bo. In the limit of an infinitely deep container, for fixed Bo, the oscillatory frequency Ωi decreases with decreasing Re, very slowly at first and very rapidly near a critical Reynolds number Recrit where it goes to, and below which it remains, zero. Recrit increases with increasing mode number. The damping rate, Ωr, increases for decreasing Re till Re=Recrit after which it decreases. For a fixed mode, Recrit decreases with decreasing Bo. For fixed Re, the damping rate Ωr is only weakly dependent on the Bond number. These results also hold in general for finite depth.