Finite Element Discretization of the Giesekus Model for Polymer Flows

We consider the Giesekus model for steady flows of polymeric liquids. This model, characterized by the presence in the constitutive law of a quadratic term in the stress tensor, yields a realistic behavior for shear, elongational and mixed flows. Its numerical approximation is achieved by means of Crouzeix-Raviart non-conforming finite elements for the velocity and the pressure, respectively piecewise constant elements for the stress tensor. Appropriate upwind schemes are employed for the convective terms, and the nonlinear discrete problem is solved by Newton's method. We next investigate the positive definiteness of the discrete conformation tensor and show that under certain hypotheses, this property is preserved by Newton's method. This allows us to attain the convergence of the algorithm for rather large Weissenberg numbers. Numerical tests validating the code are presented.