Decoupled finite element method to compute stationary upper-convected Maxwell fluid flow in 2-D convergent geometry

In this study, the stationary flow of a polymeric fluid governed by the upper convected Maxwell law is computed in a 2-D convergent geometry. A finite element method is used to obtain non-linear discretized equations, solved by an iterative Picard (fixed point) algorithm. At each step, two sub-systems are successively solved. The first one represents a Newtonian fluid flow (Stokes equations) perturbed by known pseudo-body forces expressing fluid elasticity. It is solved by minimization of a functional of the velocity field, while the pressure is eliminated by penalization. The second sub-system reduces to the tensorial differential evolution equation of the extra-stress tensor for a given velocity field. It is solved by the so-called 'non-consistent Petrov-Galerkin streamline upwind' method. As with other decoupled techniques applied to this problem, our simulation fails for relatively low values of the Weissenberg viscoelastic number. The value of the numerical limit point depends on the mesh refinement. When convergence is reached, the numerical solutions for velocity, pressure and stress fields are similar to those obtained by other authors with very costly mixed methods.