The competing role of shear and extension-induced first normal stress differences within a mixed flow for a viscoelastic fluid

Interested in the previous work of Walters et al. (Korea Aust Rheol J 21:225–233, 2009) regarding the competing roles of extensional viscosity and normal stress differences in complex flows of elastic liquids, rheological studies rarely discuss the relationship between the shear and extension-induced first normal stress differences (N1S and N1E) within a mixed flow for a viscoelastic fluid. One, therefore, derives N1S and N1E related to Weissenberg’s number and Trouton’s ratio. The classic White–Metzner viscoelastic constitutive equation coupled with the recent GNF-X (Generalized Newtonian Fluid eXtended) weighted shear/extension viscosity has the potential to show the typical vortex growth in entry flow simulations. Based on the improved White–Metzner model, demonstrating the opposite effect of N1S and N1E with respect to strain rates is evident. N1S mainly dominates the shell layer near the wall boundary at high strain rates, whereas N1E controls the center core at low strain rates. In contraction flow simulations, the predicted slit-die velocity profile is in good agreement with experimental data. It is significant to conclude that N1E hinders flow and N1S facilitates flow. In addition, a comparison of extensional-thickening and extensional-thinning viscosity curves for the velocity profile is discussed herein.