Relaxation of capillary-gravity waves due to contact line nonlinearity: A projection method

We present a physics-inspired mathematical model based on successive linear eigenmode projections to solve the relaxation of small-amplitude and two-dimensional viscous capillary-gravity waves with a phenomenological nonlinear contact line model. We show that each projection eventually induces a rapid loss of total energy in the liquid motion and contributes to its nonlinear damping.

The comprehension of the role of wetting properties in the damping of liquid oscillations in confined basins is a long-standing problem in the hydrodynamics field and for which renewed interest has emerged in recent years. A series of careful lab-scale experiments have revealed that the damping of liquid natural small oscillations varies nonlinearly with the oscillation amplitude, in contrast with previous theoretical predictions, which prescribe a constant and unique value for the damping rate, thus indicating a dependence on the contact line behavior and hence on the solid substrate material. This effect has been tentatively attributed to a source of dissipation localized in the proximity of the air–liquid–solid triple line, which, during the dynamics, may exhibit a complex hysteretic behavior under the effect of a solid-like wall friction. In this work, assuming that the contact line behaves according to experimentally inspired phenomenological laws, we formalize a mathematical method based on successive linear eigenmode projections for solving numerically the nonlinear fluid motion in the limit of small oscillation amplitudes. This approach captures the transition from a contact line stick-slip (or nearly stick-slip) motion to a pinned (or nearly pinned) configuration, as well as the secondary fluid bulk motion following the arrest of the contact line, overlooked by previous asymptotic analyses.

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Caption: TOP-LEFT: Sketch of a two-dimensional rectangular container of width $2 l$ and filled to a depth $h$ (e.g., $h/l=3$, nearly deep water regime, without loss of generality) with a liquid of density $\rho$ and dynamic viscosity $\mu$. The air-liquid surface tension is $\gamma$. The origin of the Cartesian coordinate system is fixed at the center of the free liquid surface at rest, while the bottom is placed at $z=−h$. $\theta$ is the contact angle. The dashed-dotted line is the geometrical axis of symmetry. The dashed-dotted line is the geometrical axis of symmetry. $\Omega$ denotes the bulk domain, $\partial\Omega$ its solid boundaries, and $\eta$ denotes here the moving interface.

TOP-RIGHT: (a) Experimental contact line model proposed in Fig.3 of Cocciaro et al. (1993) for water oscillations in a cylindrical container. Here, the capillary number is defined as $Ca=\mu gR/\gamma$ , where $R$ ($\approx 5$ $cm$) is the container radius. (b) Damping rate vs the amplitude of the angle measured at the container axis. $\varphi^*$ indicates the value for which the contact line is pinned.

BOTTOM: (a) Contact line elevation versus time. The blue and green colors indicate the free and pinned phases, respectively. Parameters are set as in Fig. 4 of the paper, with a static hysteresis range $\Delta=20^{\circ}$. We initialize the problem setting the complex amplitude of the first antisymmetric mode in order to have an initial contact line elevation and speed equal to $0$ and $0.1$, respectively. The time instants indicated by the star symbol denote the final time of arrest for the contact line resulting from the weakly nonlinear calculation (WNL) and from the projection scheme WNL PROJ (PROJ). See also Integral Multimedia Movie 1 for a full free-surface dynamic representation. Multimedia view.