Developing viscoelastic flow in annuli arises in extrusion, coating, and biofluid transport, yet the combined effects of elasticity, wall slip, and geometric confinement on velocity and stress development have not been systematically quantified. Using finite-element simulations of exponential Phan–Thien–Tanner fluids with a Navier slip, we examine how six parameters—Reynolds number, Weissenberg number, slip coefficient, solvent–viscosity ratio, extensibility, and radius ratio—govern the axial equilibration of velocity and normal stress. The results show that slip shortens the kinematic entrance length while extending the stress-relaxation distance, creating a clear separation of timescales even though the constitutive coupling remains intact. Increasing elasticity transfers the slowest-developing region from the walls to the core, whereas inertia, solvent fraction, and geometry modulate this behavior without removing it. Among the available development measures, the global velocity length is the most reliable kinematic criterion, while the area-averaged normal-stress development length captures the delayed elastic equilibration that persists far downstream. These findings provide a unified framework for predicting entrance effects in viscoelastic annular flows and clarify when velocity-based or stress-based criteria must be used in the design of polymer and bio-processing systems.
Taha Rezaee (Mon,) studied this question.