Stability of a liquid ring on a substrate

We study the stability of a viscous incompressible fluid ring on a partially wetting substrate within the framework of long-wave theory. We discuss the conditions under which a static equilibrium of the ring is possible in the presence of contact angle hysteresis. A linear stability analysis (LSA) of this equilibrium solution is carried out by using a slip model to account for the contact line divergence. The LSA provides specific predictions regarding the evolution of unstable modes. In order to describe the evolution of the ring for longer times, a quasi-static approximation is implemented. This approach assumes a quasi-static evolution and takes into account the concomitant variation of the instantaneous growth rates of the modes responsible for either collapse of the ring into a single central drop or breakup into a number of droplets along the ring periphery. We compare the results of the LSA and the quasi-static model approach with those obtained from nonlinear numerical simulations using a complementary disjoining pressure model. We find remarkably good agreement between the predictions of the two models regarding the expected number of drops forming during the breakup process.