Section: Scientific Foundations
Challenges related to numerical simulations of viscoelastic flows
Despite numerous efforts, numerical simulation of polymer flows is still a very challenging research area. There exist only relatively few commercial codes for the simulation of these flows (PolyFlow^{®}, Flow3D or Rem3D). Two reasons seem to be responsible for this :

First, the intrinsic properties of the polymeric liquids : nonlinear viscoelastic rheological behavior, dominant viscosity (of about a million times higher than water's viscosity) and small thermal conductivity (of about a hundred times smaller than steel's conductivity).

Second, it is still an open problem how to compute the internal coupling in realistic situations. The internal coupling between the viscoelasticity of the liquid and the flow is quantified by the dimensionless Weissenberg number We , defined as the product between the relaxation time and the shear rate. Note that the relaxation time increases with the elastic character of the polymer, whereas the shear rate translates the intensity of the flow.
A major issue to be addressed is the breakdown in convergence of the algorithms at critical values of the Weissenberg number. The commercial codes are only able to deal with Weissenberg numbers up to 10, which means that the behavior of the fluid is not very elastic. This limit is too low to describe the polymer flow in a processing machine and is often explained by difficulties related to the numerical schemes. In particular, it has been widely believed that the high Weissenberg number problem is attributed to the loss of the positivity of the conformation tensor C at the discrete level. Note that even if the positivedefiniteness of C is always true at the continuous level, it is very difficult to extend it to the discrete counterpart since the conformation tensor is not a direct unknown of the approximated problem.
Recently, different new attempts heave been made to overcome these difficulties. Lee and Xu [59] made an important step in the comprehension of how discretizations preserving the positivedefiniteness of the conformation tensor C can be derived. The idea is to write the constitutive law in terms of the conformation tensor and to recast it (by using certain Lie derivatives) into the formulation of symmetric Riccati differential equations. It is then natural to use the well developed theory for the approximation of Riccati equations, which arise in many other fields of applied mathematics such as optimal control, differential geometry or singular perturbation theory. However, one drawback of their approach is that it relies on the use of the characteristic method for the treatment of the convection term, which, in turn, presents several known drawbacks. So far, it has not yet been shown that this approach leads to a robust method for practical problems.
Another approach, which has attracted much attention, is the introduction of the logarithm of the conformation tensor by Fattal & Kupferman [52] . The main idea is, that the matrixexponential will always yield a symmetric positivedefinite matrix, even if a nonmonotone scheme is used for its approximation. However, in order to do so, the constitutive law has first to be expressed in terms of the logarithm of the conformation tensor. Although preliminary computational studies indicate a gain in stability, it is not so clear, what the impact of this nonlinear transformation, which can be viewed as a scale compression, on the numerical approximation is.
Besides this fundamental difficulty related to the Weissenberg number, several other aspects are to be taken into account when simulating polymer flows: large number of unknowns (pressure, velocity and stress at least), nonlinear character, treatment of the convection terms (especially in the constitutive law), strong thermomechanical coupling, and threedimensionality of flows. We note that most studies on numerical tools for viscoelastic flows deal with isothermal twodimensional flow (planar or axisymmetric). It remains a major issue to find stable and robust numerical methods capable to deal with 3D anisothermal flows at Weissenberg numbers greater than 10, in the frame of realistic models.