Stick-slip-to-stick transition of liquid oscillations in a U-shaped tube

The nonlinear decay of oscillations of a liquid column in a U-shaped tube is investigated within the theoretical framework of the projection method formalized by Bongarzone et al. (2021). Starting from the full hydrodynamic system supplemented by a phenomenological contact line model, this physics-inspired method uses successive linear eigenmode projections to simulate the relaxation dynamics of liquid oscillations in the presence of sliding triple lines. Each projection is shown to eventually induce a rapid loss of total energy in the liquid motion, thus contributing to its nonlinear damping. A thorough quantitative comparison with experiments by Dollet et al. (2020) demonstrates that, in contradistinction with their simplistic one-degree-of-freedom model, the present approach not only describes well the transient stick-slip dynamics, but it also correctly captures the global stick-slip to stick transition, as well as the residual exponentially decaying bulk motion following the arrest of the contact line, which has been so far overlooked by existing theoretical analyses but is clearly attested experimentally. This study offers a further contribution to rationalizing the impact of contact angle hysteresis and its associated solidlike friction on the decay of liquid oscillations in the presence of sliding triple lines.

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Caption: _ Left, Sketch of the U-tube configuration: two- dimensional (2D) view of the centerline plane. The tube radius is assumed constant and denoted by $a$. The length of the liquid column is $l$. $h$ indicates the height difference of the liquid column between the left and right straight channels. $g$ is the gravity acceleration. The advancing and receding dynamic contact angles are, respectively, $\theta_a$ and $\theta_r$, whereas the static contact angle is labelled as $\theta_s$ and it is in general $\ne 90^{\circ}$. Top-Right, Phenomenological law used in the present work to model the apparent dynamic contact angle, $\theta$, vs the non-dimensional contact line speed $Ca’ = Ca\,\partial\eta/\partial t$, with $Ca = \nu\rho\sqrt{gl/2}/\gamma$, $\nu$ the kinematic liquid viscosity, $\rho$ the liquid density and $\gamma$ the liquid-air surface tension. Bottom-Right, Contact line elevation versus time for different initial conditions. Blue solid line: predictions from the projection model. Markers: experiments by Dollet et al. (2020)._