In order to elucidate the mechanism of flow mark surface defects, the stability of injection molding flow is investigated numerically using a transient finite element method. Experiments performed by Schepens and Bulters [Bulters, M., and A. Schepens, "The origin of the surface defect 'slip-stick' on injection moulded products," Paper IL-3-2, in Proceedings of the 16th Annual Meeting of the Polymer Processing Society, Shenghai, China, 2000a, pp. 144-145] using a novel two color injection molding technique are summarized and they indicate that surface defects are caused by a flow instability near the free surface during filling of the mold. Steady finite element calculations of a model injection molding flow using a single mode, exponential Phan-Thien-Tanner constitutive equation supply information about the base state streamlines and polymer stresses. By varying the parameters of the model, the degree of strain hardening in the extensional viscosity can be controlled. Then a linear stability analysis is used to determine the most unstable eigenmode of the flow and the dependence on the extensional properties of the polymer. For strain softening materials, the injection molding flow is predicted to be stable up to a Weissenberg number of five. However, the most unstable disturbance is consistent with the swirling flow near the interface observed experimentally. For strain hardening rheologies, an instability is observed in the channel flow far from the interface, in agreement with calculations performed by Grillet et al. [Grillet, A. M., A. C. B. Bogaerds, G. W. M. Peters, and F. P. T. Baaijens, "Stability analysis of constitutive equations for polymer melts in viscometric flows," J. Non-Newt. Fluid Mech. (accepted, 2001)] on planar Poiseuille flow of a Phan-Thien-Tanner fluid.