We present a computational study of planar extrudate swell flows of Newtonian liquids with a viscous liquid-gas interface. The model consists of the equations of motion coupled with the Boussinesq-Scriven constitutive equation for the interfacial stress tensor. The resulting set of equations is solved with the finite element method coupled with an elliptic mesh generation strategy to capture the free surface. The results show a detailed parametric study in terms of the capillary number and the Boussinesq number, a dimensionless parameter used to measure the ratio of viscous forces at the interface to viscous forces in the bulk liquid. The predictions reveal that the extrudate swells dramatically as the interfacial viscosity grows. The interfacial viscosity slows down the flow both in the bulk liquid and at the interface, and thus the extrudate size increases to conserve mass in the slow plug flow that develops under the free surface.