A model has been developed to simulate the nonlinear, unsteady evolution of a high-speed viscous liquid jet issuing from a circular orifice. The model is based on a zonal approach in which an integral method is utilized for a thin viscous region at the jet periphery, while a boundary-element method is used for the inviscid "core" flow. Results indicate that steady-state solutions are possible neglecting the presence of the gas. Under these conditions, the jet "swells" in diameter and the boundary layer thins to a shear layer over a length of about half an orifice radius. Because boundary-layer relaxation is occurring during these simulations in which steady-state solutions appear, atomization mechanisms relying on this process cannot explain the observed behavior. The swelling phenomenon has the potential to explain several fundamental experimental atomization observations regarding turbulence and orifice design.