The extrudate swell singularity of Phan-Thien-Tanner and Giesekus fluids

The stress singularity for Phan-Thien-Tanner (PTT) and Giesekus viscoelastic fluids is determined for extrudate swell (commonly termed die swell). In the presence of a Newtonian solvent viscosity, the solvent stress dominates the polymer stresses local to the contact point between the solid (no-slip) surface inside the die and the free (slip) surface outside the die. The velocity field thus vanishes like rλ0, where r is the radial distance from the contact point and λ₀ is the smallest Newtonian eigenvalue (dependent upon the angle of separation between the solid and free surfaces). The solvent stress thus behaves like r-(1-λ0) and dominates the polymer stresses, which are like r-4(1-λ0)/(5+λ0) for PTT and r-(1-λ0)(3-λ0)/4 for Giesekus. The polymer stresses require boundary layers at both the solid and free surfaces, the thicknesses of which are derived. These results do not hold for the Oldroyd-B fluid.