A time dependent free boundary governed by the Navier-Stokes equations

A two dimensional linearized nonstationary capillary free boundary problem governed by the Navier-Stokes equations is studied. A liquid in a smooth walled container subject to gravity is considered. The fluid is isothermal, incompressible, newtonian, with constant density, viscosity, and surface tension coefficient. Under these assumptions the equations of nonstationary motion, which are the equations for conservation of momemtum and mass, reduce to the Navier-Stokes equations for the velocity and pressure. The effects of small perturbations are analyzed.