A numerical method has been developed for solving two-dimensional generalized Newtonian fluid flow with multiple free surfaces. It is an extension of the GENSMAC code which solves the time-dependent Navier-Stokes equations for the primitive variables of velocity and pressure in an arbitrary domain. Like GENSMAC, it is a finite-difference technique, based on staggered grids, using (virtual) marker particles as a means of flow visualization. The code has been employed to solve three time-dependent problems: extrudate die swell. viscous jet buckling, and injection moulding in complex cavities. Both Newtonian and non-Newtonian results are displayed.