  In this paper we study the dynamics of an incompressible viscous fluid
evolving in an open-top container in two dimensions. The fluid mechanics are
dictated by the Navier-Stokes equations. The upper boundary of the fluid is
free and evolves within the container. The fluid is acted upon by a uniform
gravitational field, and capillary forces are accounted for along the free
boundary. The triple-phase interfaces where the fluid, air above the vessel,
and solid vessel wall come in contact are called contact points, and the angles
formed at the contact point are called contact angles. The model that we
consider integrates boundary conditions that allow for full motion of the
contact points and angles. Equilibrium configurations consist of quiescent
fluid within a domain whose upper boundary is given as the graph of a function
minimizing a gravity-capillary energy functional, subject to a fixed mass
constraint. The equilibrium contact angles can take on any values between $0$
and $\pi$ depending on the choice of capillary parameters. The main thrust of
the paper is the development of a scheme of a priori estimates that show that
solutions emanating from data sufficiently close to the equilibrium exist
globally in time and decay to equilibrium at an exponential rate.
