Statics and dynamics of liquid barrels in wedge geometries

We present a theoretical study of the statics and dynamics of a partially wetting liquid droplet, of equilibrium contact angle $\theta_{\rm e}$, confined in a solid wedge geometry of opening angle $\beta$. We focus on a mostly non-wetting regime, given by the condition $\theta_{\rm e} - \beta > 90^\circ$, where the droplet forms a liquid barrel -- a closed shape of positive mean curvature. Using a quasi-equilibrium assumption for the shape of the liquid-gas interface, we compute the surface energy landscapes experienced by the liquid upon translations along the symmetry plane of the wedge. Close to equilibrium, our model is in good agreement with numerical calculations of the surface energy minimisation subject to a constrained position of the centre of mass of the liquid. Beyond the statics, we put forward a Lagrangian description for the droplet dynamics. We focus on the the over-damped limit, where the driving capillary force is balanced by the frictional forces arising from the bulk hydrodynamics, the corner flow near the contact lines and the contact line friction. Our results provide a theoretical framework to describe the motion of partially wetting liquids in confinement, and can be used to gain further understanding on the relative importance of dissipative processes that span from microscopic to macroscopic length scales.