A new finite element technique has been developed for using integral‐type constitutive equations in viscoelastic flow simulations with recirculating regions. The present method is relatively inexpensive and in the meantime can reach higher elasticity levels without numerical instability, compared with the best available similar calculations in the literature. The new method has been used to simulate entry flows of polymer melts in circular contractions using the K‐BKZ integral model. The numerical results show that LDPE melts exhibit a rapid corner vortex growth as the flow rate [or a dimensionless recoverable shear (SR) value] is increased, while HDPE melts do not produce a significant corner vortex growth even at very high flow rates, confirming existing numerical and experimental data. A comprehensive parametric study of the effects of elongational viscosity on vortex growth has also been carried out. It has been found that a combination of LDPE shear properties including the normal stresses with an HDPE‐like (strain‐thinning) elongational viscosity gives performance similar to an HDPE melt (in terms of the corner vortex growth), while a combination of HDPE shear properties with an LDPE‐like (strain‐thickening) elongational viscosity gives performance similar to an LDPE melt. This finding convincingly shows that the strain‐thickening elongational viscosity is primarily responsible for the rapid vortex growth exhibited by LDPE melts, while the strain‐thinning elongational viscosity of HDPE melts is responsible for their slow vortex growth. In other words, a strain‐thickening elongational viscosity enhances vortex growth withSR, while a strain‐thinning elongational viscosity stunts vortex growth withSR.