The group Mathematical Modeling and Analysis (MMA), headed by Prof. Dr. rer. nat. Dieter Bothe, represents an interdisciplinary group. It roots in the Department of Mathematics at Technische Universität Darmstadt with its research activities being embedded within the Profile Area “Thermofluids & Interfaces” at Technische Universität Darmstadt.
The research topics of MMA are substantially motivated by open problems in process and chemical engineering. The corresponding research areas are
Mathematical Modeling and Applied Analysis
- Two-Phase Free Boundary/Interface Problems
- Reaction-Diffusion(-Electromigration) Systems
- Contact Line Dynamics
- Phase Transition Problems
- Viscoelastic Fluids
Our research is based on the continuum modeling of two-phase flows employing and further developing sharp-interface models based on Continuum Thermodynamics. These models correspond to increasing levels of physico-chemical interface properties, starting from capillary interfaces to the case when the interface is a phase on its own with surface tension, interfacial viscosities and adsorbed species. For the different levels the corresponding mathematical models are mathematically analysed regarding solvability and their qualitative properties.
For a deeper understanding of complex fluid behavior a rigorous mathematical analysis of the corresponding fluid models is indispensable, e.g. by means of analysis of nonlinear evolutions in Banach spaces. By the intricate structure of the systems of nonlinear PDEs such a treatment requires the use of deep and subtle analytical tools. On the other hand, well-posedness of the models represents the fundamental prerequisite not only for further analytical investigations but also for numerical computations and simulations.
Besides existence and uniqueness of solutions, stability issues are of preferential interest. This includes examinations on convergence to equilibria or rigorous proofs of linear and nonlinear instability. Further main objectives are asymptotics and precise dependence on related parameters of solutions.
- Two-Phase Flows and Free-Surface Flows
- Transport Processes at Fluidic Interfaces
- Interfacial Mass and Heat Transfer
- Surfactant Mixtures
- Reaction-Advection-Diffusion Systems
- Contact Line Dynamics
- Thermal and Solutal Marangoni-driven Flows
- Viscoelastic Fluids
- Hydrodynamics of Drops and Bubbles
Our research is focused on Computational Fluid Dynamics (CFD) by means of Direct Numerical Simulation (DNS) including the numerical modeling of interfacial transport processes and physico-chemical phenomena in two- and multiphase systems. For a deep understanding of elementary transport and transfer processes at fluid interfaces, we actively develop techniques for different numerical simulation methods. The central aim is to provide taylor-made high-fidelity methods which exhibit distinct advantages for the specific interfacial transport physics under consideration.
We develop and deploy Volume-Tracking, Interface-Tracking and Front-Tracking Methods and hybrid variants of those. The code bases used are Free-Surface 3D (FS3D), which is an in-house Fortran code, and Open Field Operation and Manipulation (OpenFOAM), which is an open source C++ library for computational continuum mechanics.
Free Surface 3D (FS3D)
Part of the simulations in the group of Mathematical Modeling and Analysis are performed with the inhouse Volume-Of-Fluid (VOF) code “Free Surface 3D” (FS3D), originally developed by Rieber and Frohn  and Rieber  at the Institute of Aerospace Thermodynamics (ITLR), University Stuttgart, Germany. Since then, the code has been massively expanded by the ITLR and the group of Mathematical Modeling and Analysis (MMA) in Darmstadt, Germany.
The research focus of the MMA group has been mainly on the mathematical and numerical modeling of multiphysical and multiscale two-phase flows.
A noticeable difference to original implementation of FS3D relate to substantially improved surface tension treatment within the VOF framework:
- The improved surface tension treatment of the balanced continuum surface force (bCSF) model, with the curvature being calculated from local height functions, was demonstrated for the hydrodynamics of wavy laminar falling films. Simulations using the bCSF model combined with a high-quality curvature calculation allowed to find good agreement with experiments.
- Binary collisions of shear-thinning droplets were investigated. The newly developed lamella stabilization algorithm allows to capture liquid films thinner than one computational cell with the VOF method. The same technique was also used to study the onset of instabilities in high energy head-on droplet collisions.
Furthermore, a variety of physical models have been developed and applied to research topics of current interest:
- Two-phase flows with thermal Marangoni effect were studied, i.e. thermocapillary flows, where local changes of the interfacial temperature cause differences in the surface tension and induce a fluid flow along the interface. An additional energy transport equation is solved to obtain the temperature distribution within the two phases. For the transport of temperature and diluted species in two-phase flows, the two-scalar approach was developed. Furthermore, for the case of droplet migration on a solid wall the dynamic contact line has been modeled within the VOF framework.
- Rising fluid particles have been investigated foci on the dynamic behaviour of rising droplets and the effect of surface contamination on mass transfer at rising bubbles. Recent efforts have been on the accurate modeling of physical and reactive mass transfer. Subgrid-scale models allow to capture concentration boundary layers which can be fully embedded in only one computational cell.
The current research focuses on:
- Transfer processes at dynamic contact lines
- Stability analysis of flluid particles
- Droplet collisions and splashing
- Multiphysics of reactive mass transfer
 M. Rieber, A. Frohn: A numerical study on the mechanism of splashing. Int. J. Heat Fluid Fl.20, pp. 455-461 (1999)
 M. Rieber: Numerische Modellierung der Dynamik freier Grenzflächen in Zweiphasenströmungen. Ph.D. thesis, Universität Stuttgart (2004)
Open Field Operation and Manipulation (OpenFOAM)
OpenFOAM [1-4] is an open-source C++ class library for Computational Continuum Mechanics (CCM) and Multiphysics. OpenFOAM's Object-Oriented-Programming (OOP) paradigm enables to mimic data types and basic operations of CCM using top-level syntax as close as possible to the conventional mathematical notation for tensors and partial differential equations, the language in continuum modeling. OpenFOAM is a versatile and flexible simulation software to address complex interfacial transport process and phenomena involved therein – such as wetting and contact line dynamics, interfacial species or mass transfer (i.e., due to absorption, evaporation, condensation etc.) and interfacial species transport (i.e., due to adsorption of surfactants). Thus, OpenFOAM is a central platform for model and method development at Technische Universität Darmstadt, in particular within the group Mathematical Modeling and Analysis (MMA).
The young research group “Advanced Two-Phase and Interfacial Flow Simulations using OpenFOAM” under the lead of Dr.-Ing. Holger Marschall is focussed on the development and extension of modern methods for Direct Numerical Simulation of two-phase flows such as to allow for detailed studies of local transport processes at fluid interfaces. The group succeeded to unite the main representatives of methods ¬on a single software platform – cf. Figure 2: OpenFOAM.
The group actively develops both sharp-interface
• Arbitrary Lagrangian-Eulerian Interface-Tracking,
• Volume-Of-Fluid Interface-Capturing (algebraic and geometric), and
• Front-Tracking methods,
• Allen-Cahn as well as
• Cahn-Hilliard phase-field methods.
The modular concept in OpenFOAM (object-oriented and generic programming) is exploited in order to use synergy in method development by means of re-use of solution and discretisation techniques independent of the specific method. For the first time this particularly allows the problem-specific choice of suitable methods on one platform, thus, enabling comparisons of different approaches regarding performance and accuracy in an objective manner.
 Weller, H. G., Tabor, G., Jasak, H., and Fureby, C. (1998). A tensorial approach to computational continuum mechanics using object orientated techniques. Comput. Phys., 12(6):620–631.
 Jasak, H., Jemcov, A., and Tuković, Ž. (2007). OpenFOAM: A library for complex physics simulations. In International Workshop on Coupled Methods in Numerical Dynamics IUC, Dubrovnik, Croatia.
 H. Jasak. Error Analysis and Estimation in the Finite Volume Method with Applications to Fluid Flows. PhD thesis, Imperial College of Science, Technology & Medicine, 1996.
 Moukalled, Mangani, Darwish: The Finite Volume Method in Computational Fluid Dynamics – An Advanced Introduction with OpenFOAM and Matlab. Springer, 2015.