Abstract
Axial flows in generalized ducts are studied for viscoelastic materials including a linear low-density polyethylene (LLDPE) melt. Viscoelasticity is described by an integral constitutive equation of the K-BKZ type with a spectrum of relaxation times, which fits well experimental data for the shear and elongational viscosities and the normal stresses as measured in shear flow. The K-BKZ model can be reduced to the Newtonian and Maxwell models with appropriate choice of the parameters. A new technique is developed where the Finger and Cauchy-Green tensors are simplified for axial flows, since particle tracking is only required in the flow z-direction (not required in the x-y coordinate plane). Numerical solutions are presented in two-dimensional cross-sectional geometries, namely square, concave square, and eccentric annulus, for different flow rate and pressure drop changes. For the Maxwell model, the dimensionless pressure drop is independent of the Weissenberg number and a function only of the geometry. For the K-BKZ model representing the LLDPE melt, the dimensionless pressure drop is reduced with increasing flow rate, hence Weissenberg number. The present results are offered as benchmark solutions for the imposition of entry velocity and stress profiles in three-dimensional ducts, when secondary flows are not present.