On the stick-slip flow from slit and cylindrical dies of a Phan-Thien and Tanner fluid model. I. Steady state

Abstract

The steady planar and cylindrical stick-slip flows for a viscoelastic fluid are computed using the Phan-Thien and Tanner (PTT) constitutive model. The mixed finite element method is used in combination with the elastic-viscous stress-splitting technique and the streamline upwind Petrov–Galerkin discretization for the constitutive equation. This combination of methods when applied to the PTT constitutive model allows us to compute steady state solutions up to high Weissenberg numbers; practically without an upper limit. Equally important, the global Jacobian matrix is generated in order to be able to perform a linear stability analysis of the computed steady state. The dependence of the steady solutions on all the problem parameters is examined. In the limit of a Newtonian fluid, the expansion coefficients near the singularity are computed with comparable accuracy to those from previous analytical and numerical studies, which include the singular finite element method. In the case of a viscoelastic liquid, it is shown that the computed solutions converge quadratically with mesh refinement even at the exit plane of the die and also locally very close to the singularity. The form of the converged solution near the singularity is examined as well as its dependence on various rheological parameters. It is shown that the singularity at the die exit is a logarithmic one and always integrable. Under such conditions our calculations can be extended to determine the linear stability of the herein computed steady states.