Damping of surface waves in a brimful circular cylinder with a contaminated free surface

We consider the effect of insoluble surfactants on the frequency and damping of surface waves on a viscous liquid in a circular cylinder. The contact line is assumed to be pinned and the surfactants are characterized by an elastic film of Marangoni elasticity gamma. The natural complex viscous eigenfunctions for cylindrical geometry are used to obtain a nonlinear eigenvalue problem for the complex frequencies by projecting the governing equations onto an appropriate basis. This is then solved to obtain the modal frequencies as a function of the Reynolds number Re, elasticity gamma, Bond number Bo and liquid depth h. Comparison with the theoretical results of Henderson and Miles (1994 J. Fluid Mech. 275 285) for the case of an inextensible film (gamma = infinity) showed that there is good agreement. Comparison with the experimental results of Henderson and Miles (1994 J. Fluid Mech. 275 285) and the semi-analytical results of Nicolas and Vega (2000 J. Fluid Mech. 410 367) for the (1, 0), (2, 0), (3, 0) and (4, 0) modes, with an assumed value of gamma = 100, also showed good agreement. Results are also presented for a four decadal range of gamma. For all modes, a maximum in the damping rate is observed at an intermediate gamma. The case of soluble surfactants is briefly considered as well; these results seem to be new. The method can be used for a wide range of parameters; results are presented here for the cases of low Re and shallow depth.