Unsteady free surface flow phenomena of a non-Newtonian fluid in a two-dimensional rectangular cavity are studied by numerically solving equations of continuity and momentum using the power-law fluid model. The marker and cell method and the DuFont-Frankel finite difference approximation are utilized in the present numerical experiment. The iniial condition is simply that the cavity is empty. A no slip boundary condition is applied at the walls, while at the free surface zero tangential stress is assumed. Physical properties of rigid PVC melt are used and computations are carried out for two different average inlet velocities. The numerical stability problem associated with high viscosity flows can be overcome by the use of the DuFont-Frankel finite difference method. The calculated results provide not only pressure and viscosity field but also a visual observation of the flow pattern by means of marker particles.