## Flow-fiber interactions, turbulence modeling

Interactions of fluid flows with short or long fibers are of great interest in industry. One important step in the manufacturing of nonwoven materials is, for example, the entanglement of filaments by a highly turbulent air flow before their deposition on a conveyor belt decisively shapes the end product. This way the air-fiber interactions influence the micro-structure of the generated material and thus the functional properties of the final product. The mathematical research in this field makes it possible to optimize the corresponding production chains and contributes this way to quality improvements, saving of resources and to other innovations. The challenge is to simulate the proper behavior of thousands of fibers in a turbulent air flow, where every fiber is described by a complex internal dynamics. The direct numerical simulation of such a multi-scale problem is nearly impossible in general. Possible modeling approaches are homogenization or kinetic descriptions. Both reduce the degrees of freedom by replacing the original problem with many fibers by a homogeneous surrogate problem. This is reached by two-scale convergence in the case of homogenization and by deriving some macroscopic properties of a flow-fiber suspension with the help of ensemble averages. Another problem which needs to be considered in this context is the appropriate modeling of turbulence as random processes. This leads to models with stochastic partial differential equations. Finally the correct simulation of problems like the mentioned production of nonwovens requires an accurate modeling of the deposition. This can also be achieved by stochastic surrogate models.

Contact: Alexander Vibe

## Asymptotics and numerics for Cosserat rod models

This research area is placed around the asymptotic derivation of Cosserat string and rod models for curved slender objects and their numerical handling as transient problems of partial differential-algebraic systems of equations. The equations have an hyperbolic character that might change with the considered material law (elastic, viscoelastic, viscous). Different numerical approaches can be pursued to discretize and solve the arising problems. The main focus lies on Finite-Volume methods for a spatial semi-discretization (key issue conservation) and stiffly accurate Runge-Kutta type methods for the time- integration. An alternative are discontinuous Galerkin methods which combine features of the Finite-Element and Finite-Volume framework. Challenges for the numerical solver with these types of equations arise from very large gradients in the solution. They are handled by grid adaptivity strategies with a main focus on moving mesh methods (r-refinement).

Contact: Manuel Wieland, Nicole Marheineke