Prof. Dr. Günther Grün

Topological transitions like droplet coalescence or droplet break-up are fundamental features of two-phase flows. In recent years, diffuse interface models turned out to be a promising approach to describe such phenomena. Species transport across fluidic interfaces and the effects exerted by soluble and insoluble surfactants are additional issues of still increasing technological importance.
For those phenomena, novel thermodynamically consistent diffuse interface models shall be developed taking in particular general densities into account. Based on rigorous mathematical analysis, existence and qualitative behaviour of solutions will be investigated, this way enhancing the understanding of the fundamental model properties. Starting from energy and entropy inequalities, stable and convergent numerical schemes shall be formulated and implemented in two and three spatial dimensions. By numerical simulations, the models shall be validated and further improved.

The goal of the project is to analyse, to evaluate, and to improve mathematical models for the dewetting of thin liquid films on homogeneous surfaces and the formation of fluid structures on inhomogeneous substrates. During the second period of the Schwerpunktprogramm, investigations on evaporation and condensation processes will be included. The proposed project consists of three main parts which can be summarized as follows:Modelling: Based on lubrication approximation, evolution equations for the height of condensing fluids on inhomogeneous surfaces shall be derived.Analysis: Methods from the calculus of variations and the theory of partial differential equations shall be used to obtain results on existence and qualitative behaviour of solutions to the corresponding evolution equations for film height h and pressure p. Besides their obvious importance for a better understanding of the asymptotic behaviour of solutions, these theoretical results are also the key ingredient to formulate fast and reliable algorithms for numerical sumulations.Numerical simulations: During the last two years, G. Grün and M. Rumpf succeded in developing and analysing a finite-element/finite-volume scheme which drastically reduces the computation time for the simulation of spreading phenomena. Based on this scheme, a general finite-element solver shall be designed to enable numerical simulations of all the phenomena mentioned above. The evaluation will be performed by comparison with experimental data; it strongly depends on an intense cooperation with experimentalists inside the Schwerpunktprogramm. Wetting Kaleidoscope>>

Electric fields may influence the wetting behaviour of charged droplets. This effect may be used to manipulate droplet motion and to enforce droplet coalescence or break-up. It is summarized under the notion of electrowetting. First studied in the late 19th century by Lippmann and others, very recently this effect gained the interest of physicists and engineers on account of a variety of applications in micro-fluidics („lab-on-a-chip“). However, mathematical modeling of this effect is still in its childhood — the dependency of contact angles on the applied voltage is modeled by the so called Lippmann formula which may hold true at most in a small voltage regime. In this project, new thermodynamically consistent models shall be derived to describe the evolution of droplet, voltage and thereby to shed light on the mechanisms underlying electrowetting. Existence of solutions shall be proven and efficient numerical schemes shall be developed as well. Poster>>

In this project, models for complex flow phenomena are investigated, for instance pattern formation in sheared suspensions, moving contact lines or Rayleigh-Bénard convection. By means of rigorous analysis and numerical simulations, properties of the models will be predicted. The aforementioned specific models may be used to develop new methods for PDE in general.

• G. Grün, K. Mecke, M. Rauscher, Thin-film flow influenced by thermal noise, J.Stat.Phys., 122(6):1261-1291, 2006.
• L. Giacomelli, G. Grün, Lower bounds on waiting times for degenerate parabolic equations and systems, Interfaces and Free Boundaries, 8:111-129, 2006 (with L. Giacomelli).

The goal of this project is to investigate scaling laws in rheological processes. More precisely, analytical tools to rigorously infer such scaling laws shall be developed. A common feature of the friction dominated processes to be investigated is their gradient flow structure. This structure translates the physical free energy and the underlying dissipation mechanism into the mathematical terms of functional and metric tensor, respectively. The idea is to use global PDE methods, rather than local methods, like matched asymptotic expansions, to analyze the scaling laws. Phenomena to be studied include spreading of viscous films, case-II diffusion of solvents in polymeric solids, and the demixing of polymer solutions and sponge-like pattern formation.

•  G. Grün, Droplet spreading under weak slippage: existence for the Cauchy problem, Comm.Part.Diff.Equations, 29:1697-1744, 2004.
• G. Grün, Droplet spreading under weak slippage: the waiting time phenomenon, Ann. I. H. Poincare, Analyse non lineaire, 21:255-269, 2004.
• G. Grün, Droplet spreading under weak slippage: the optimal asymptotic propagation rate in the multi-dimensional case, Interfaces and Free Boundaries, 4:309-323, 2002.

 This project is concerned with the analytical description of the qualitative and asymptotic behaviour of solutions to certain higher order parabolic differential equations arising in modelling phenomena of wetting and film rupture. It is planned to develop efficient numerical tools — in particular algorithms based on Finite-Volume-Methods — to enable computer simulations of film rupture in space dimension N = 3.
The outline is as follows:
— Part 1: Analysis of degenerate parabolic equations of fourth order which are obtained as lubrication limit from the Navier- Stokes equations.
— Part 2: Derivation and analysis of evolution equations to model the formation of liquid structures on inhomogeneous ma- terial interfaces.
— Part 3: Generalization to higher spatial dimensions of a new finite-volume-algorithm which recently has been developped by M. Rumpf and G. Grün. Implementation of this algorithm. Numerical simulations and evaluation of proposed models.