Theoretical framework to analyze the combined effect of surface tension and viscosity on the damping rate of sloshing waves

Gravity force, surface tension, oscillating boundary layers, viscous dissipation, static meniscus, contact angle hysteresis, and moving contact lines are typical features of sloshing dynamics. All these ingredients influence capillary-gravity waves in a container and have been investigated theoretically, usually building on the standard potential flow solution [Lamb, Hydrodynamics (Cambridge University Press, Cambridge, 1932)], without dealing with their combined effects. We propose here a theoretical framework to study viscous sloshing waves in a circular cylinder incorporating a realistic contact angle model observed experimentally [Dussan, Annu. Rev. Fluid Mech. 11, 371 (1979)] as a boundary condition at the contact line. The resulting nonlinear system of equations, which accounts for contact angle hysteresis, is then solved asymptotically in order to determine the effect of the nonlinear relation between the contact line velocity and the dynamic contact angle on the viscous dissipation and, as a consequence, on the wave-damping rate and frequency.