Capillary hysteresis in sloshing dynamics: a weakly nonlinear analysis

The sloshing of water waves in a vertical cylindrical tank is an archetypal damped oscillator in fluid mechanics. The wave frequency is traditionally derived in the potential flow limit (Lamb, Hydrodynamics, Cambridge University Press, 1932), and the damping rate results from the combined effects of the viscous dissipation at the wall, in the bulk and at the free surface (Case & Parkinson, J. Fluid Mech., vol. 2, 1957, pp. 172–184). Still, the classic theoretical prediction accounting for these effects significantly underestimates the damping rate when compared to careful laboratory experiments (Cocciaro et al., J. Fluid Mech., vol. 246, 1993, pp. 43–66). Specifically, theory provides a unique value for the damping rate, while experiments reveal that the damping increases as the sloshing amplitude decreases. Here, we investigate theoretically the effects of capillarity at the contact line on the decay time of capillary–gravity waves. To this end, we marry a model for the inviscid waves to a nonlinear empiric law for the contact line that incorporates contact angle hysteresis. The resulting system of equations is solved by means of a weakly nonlinear analysis using the method of multiple scales. Capillary effects have a dramatic influence on the calculated damping rate, especially when the sloshing amplitude gets small: this nonlinear interfacial term increases in the limit of zero wave amplitude. In contrast to viscous damping, where the wave motion decays exponentially, the contact angle hysteresis can act as Coulomb solid friction, thus yielding the arrest of the contact line in a finite time.