The paper deals with the numerical simulation of highly viscous, non-Newtonian fluids in apparatus of mechanical engineering. Differential constitutive equations are used to approximate real material behavior of technical fluids such as polymer melts. The models consider shear-thinning viscosities, normal stress differences and other elastic properties which affect the fluid flow. Some details of the basis of the numerical simulations are discussed. The influence of the viscoelastic behavior is demonstrated through examples of flow calculations for two and three dimensional geometries. Comparison of the present results with other experimental and numerical results from the literature shows good agreement.