The extrudate swell singularity of viscoelastic fluids

Abstract

We consider the planar extrudate swell flow of Giesekus and PTT fluids. The separation angleis denoted by α and takes the values between 129^o and 270^o.

Local to the exit singularity, a class of self-similar solutions has been identified with stream function behaviour O(rλ0+1) and polymer stress singularities of O(r−4(1−λ0)/(λ0+5)) for PTT and O(r−(1−λ0)(3−λ0)/4) for Giesekus, where r is the radial distance from the die wall at the exit. The lead eigenvalue λ0 is related to α through a transcendental equation and takes values between 1/3 and 1. These behaviours transpire in a core region of the flow set away from both the die wall and the exiting free-surface of the flow. Within this region the solvent stress dominates the polymeric stress and the momentum equation reduces to the Stokes flow equation. Using the method of matched asymptotics, the core region is reconciled with boundary layers at the stick surface of the die wall and the exiting slip or free-surface of the flow. The analysis benefits from the representation of the stress in both Cartesian and natural stress formulations, and is implemented when the Weissenberg number (the dimensionless relaxation time) is O(1). These results hold for all values of the retardation parameter β ∈ (0, 1], but breakdown in the limit β → 0.

These theoretical behaviours are numerically verified locally along the streamlines of the flow and globally through simulations of the full extrudate swell problem. The streamline integration benefits from representing the stress in a polar coordinate system while a Cartesian system used for the full numerical solution. For the full numerical simulations, a finite-volume scheme for planar extrusion is implemented from the rheoTool toolbox in OpenFOAM. The Eulerian free-surface solver of rheoInterFoam is chosen which tracks the free-surface boundary with a volume of fluid (VOF) surface-capturing algorithm.