Flow singularity and slip velocity in plane extrudate swell computations

Abstract

It is common knowledge that flows of viscoelastic liquids with stress singularities, like the extrudate swell flow, pose formidable obstacles to numerical computations at relatively low Weissenberg number. This paper describes an effort toward alleviating the stress singularity by means of a slip boundary condition at the die wall. The Oldoyd-B and the upper-convected Maxwell differential constitutive equations were used for simplicity and computational efficiency. With a no-slip boundary condition it was found that for Newtonian, upper-convected Maxwell and Oldroyd-B liquids the global solution was always mesh-dependent until the Newton iteration diverged at very fine tessellations in the vicinity of the static contact line. With a natural slip boundary condition the global solution became mesh-independent at the same tessellations. Moreover, the macroscopic predictions became independent of the amount of slip in a relatively broad region of slip coefficient. The Newton iteration converged up to Weissenberg number 0.6 with a no-slip boundary condition and up to 1.7 with a lip boundary condition for the upper-convected Maxwell liquid. For the Oldroyd-B liquid the maximum Weissenberg number was 0.85 without slip and 1.866 with slip. Although slip velocity, surface tension and Newtonian viscosity (or retardation time) enhanced some numerical stability in general, it appears unlikely that they could advance viscoelastic computations significantly. In the limiting case of no swelling, at infinitely large surface tension, the analytical solution for Newtonian and, a second order fluid showed: (a) elasticity increases the strength of the singularity that exists for Newtonian liquid at the contact line, and thus Newton iteration is expected to diverge at coarser and coarser tessellations as the elasticity increases in agreement with the finite element findings. (b) Finite element predictions for the same flow agreed with the analytical solution in the vicinity of the singularity only when a slip boundary condition was employed. (c) Slip boundary condition in the vicinity of the contact line alleviates the stress singularity. However, it forces the stress to go through a maximum which is equally catastrophic of the Newton iteration convergence.