Meniscus effects on the frequency and damping of capillary-gravity waves in a brimful circular cylinder

We study the effects of a meniscus on the oscillations of a viscous liquid filling a right circular cylindrical container by using the natural viscous complex eigenfunctions of the problem. The free surface of the liquid is assumed to have a pinned contact line. By projecting the governing equations onto an appropriate basis, a nonlinear eigenvalue problem for the complex frequencies is obtained. This is then solved to13; obtain the modal frequencies as a function of the contact angle $\theta_c$, the Reynolds and Bond numbers Re and Bo and the liquid depth h. At shallow depths, the13; effect of the meniscus is, in general, to increase the modal frequency and decrease the damping rate with increasing $\theta_c$. At large depths and for higher modes, the13; damping rate monotonically decreases with increasing $\theta_c$ while the frequency attains13; a maximum in the neighbourhood of $90^0$. However, there are parameter values for13; which the meniscus results in anomalous behaviours in the damping rate. Extensive13; comparison with experimental and computational results for the $\theta_c = 90^0$ case is13; very good; comparison with the one available experimental result for $\theta_c = 62^0$ is also13; very good. Extensive results for a variety of contact angles have been tabulated for13; Reynolds and Bond numbers for which experimental results exist in the flat interface13; case, for comparison with future experimental results that may be obtained with13; curved menisci