## Section: Application Domains

### Viscoelastic flows

Polymeric fluids are, from a rheological point of view, viscoelastic non-Newtonian fluids, see Figure 1 . Their non-Newtonian behavior can be observed in a variety of physical phenomena, which are unseen with Newtonian liquids and which cannot be predicted by the Navier-Stokes equations. The better known examples include the rod climbing Weissenberg effect, die swell and extrusion instabilities (cf. fig. 1). The rheological behavior of polymers is so complex that many different constitutive equations have been proposed in the literature in order to describe these phenomena, see for instance [62] . The choice of an appropriate constitutive law is still a central problem. We consider realistic constitutive equations such as the Giesekus model. In comparison to the classical models used in CFD, such as UCM or Oldroyd B fluids, the Giesekus model is characterized by a quadratic stress term.

Our aim is to develop new algorithms for the discretization of polymer
models, which should be efficient and robust for We>10 . For this
purpose, we will develop a mathematical approach based on recent ideas
on discretizations preserving the positivity of the conformation
tensor. This property is believed to be crucial in order to avoid
numerical instabilities associated with large Weissenberg numbers. In
order to develop monotone numerical schemes, we shall use recent
discretization techniques such as stabilized finite element and
discontinuous Galerkin methods. We intend to validate the codes to be
developed at hand of academic benchmark problems in comparison with
the commercial code PolyFlow^{®}.