Profile development of the melt spinning process of poly(ethylene terephthalate) (PET) was simulated by a numerical method under the consideration of two constitutive equations of Newtonian and upper convected Maxwell (UCM) models. The viscoelastic characteristics of the polymer were considered via UCM constitutive equation that considered relaxation time as a function of temperature and molecular weight. The UCM model predicted the diameter profile better than the Newtonian, while velocity development was slower than the Newtonian model. Viscoelasticity played an important role in accurately predicting diameter profile. However, even though neck-like deformation was observed in the UCM model, the exact position of the deformation under high speed spinning was not obtained.