A complex molecular orientation distribution is observed in the injection molding of amorphous polymers. A major contribution is due to the flow‐induced deformation of the polymer molecules during mold filling and incomplete relaxation during cooling, after cessation of flow. In this work a numerical study is undertaken in an effort to resolve the impact of the fluid mechanics of mold filling on the molecular orientation of the final part, as reflected in available birefringence measurements. Viscoelasticity of the polymer melt is accounted for with the multimode Leonov model. A finite‐element solution of the fountain flow problem is obtained and the effect of the flow field on the deformation experienced by the fluid is elucidated. Frozen‐in stresses are calculated based only on material parameters and processing conditions. Predicted birefringence distributions agree well with published experimental results.