Biannual Report - Fachbereich Mathematik - Technische Universität ...

Project: L p -Theory for Incompressible Newtonian Flows subject to Energy Preserving
Boundary Conditions
This project aimed at a rigorous derivation as well as an analysis of a large class of boundary
conditions for the Navier-Stokes equations, which contains classical boundary conditions
for fixed walls (e. g. no-slip, Navier, perfect slip conditions), classical boundary
conditions, which arise in model problems for free boundary problems (e. g. Neumann
conditions), as well as numerous artificial boundary conditions, which already proved to
be useful for direct numerical simulations. The obtained results are available in [1, 2].
Partner: J. Prüß, Universität Halle-Wittenberg
Support: Center of Smart Interfaces (DFG Cluster of Excellence 259)
Contact: D. Bothe, M. Köhne
References
[1] D. Bothe, M. Köhne, and J. Prüß. On a Class of Energy Preserving Boundary Conditions
for Incompressible Newtonian Flows. Submitted to SIAM Journal on Mathematical Analysis,
Preprint: http://arxiv.org/abs/1207.0707, 2012.
[2] M. Köhne. L p -Theory for Incompressible Newtonian Flows. Energy Preserving Boundary Conditions,
Weakly Singular Domains. Springer Spektrum, Wiesbaden, 2013.
Project: L p -Theory for Two-Phase Flows with Soluble Surfactant
The presence of surfactants has a pronounced effect on the surface tension and, hence, on
the stress balance at the phase separating interface of two-phase flows. The transport of
momentum induced by the local variations of the capillary forces are known as Marangoni
effects. The aim of this project is to study a model which assumes the surfactant to be
soluble in one of the adjacent bulk phases and which represents a generalization of the
two-phase Navier-Stokes equations. The obtained results are available in [1].
Partner: J. Prüß, Universität Halle-Wittenberg
Support: Center of Smart Interfaces (DFG Cluster of Excellence 259)
Contact: D. Bothe, M. Köhne
References
[1] D. Bothe, M. Köhne, and J. Prüß. On Two-Phase Flows with Soluble Surfactant. Preprint:
http://arxiv.org/abs/1210.8131, 2012.
Project: Direct Numerical Simulation of binary collisions of viscous and non-
Newtonian droplets
Direct Numerical Simulations based on an extended Volume of Fluid (VOF) method are
used to investigate binary droplet collisions. During collisions, extremely thin fluid lamellas
appear especially in case of a shear-thinning rheology. These have to be accounted for
within the numerical simulation in a physically sound way. One major finding is that an
effective constant viscosity can be calculated which leads to the same collision dynamics.
In order to simulate viscoelastic two-phase flow, the VOF method has been extended to
the Oldroyd-B model. The energy balance and the elongation of the polymer molecules is
studied numerically during droplet collision.
Partner: M. Sommerfeld, Universität Halle-Wittenberg
22 1 Research