Mathematical Modeling and Analysis
The group Mathematical Modeling and Analysis (MMA), headed by Prof. Dr. rer. nat. Dieter Bothe, has a strongly interdisciplinary research portfolio. MMA is part of the Analysis group within the Department of Mathematics at the TU Darmstadt, with its research activities being embedded into the Profile Area Thermofluids & Interfaces at TU 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 Analysis
- Sharp-Interface Continuum Modeling
- Multicomponent Diffusion Modeling
- Bulk-Surface Reaction-Diffusion Systems
- Kinematics of Two-Phase Flows
- Contact Line Dynamics
- 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 (see, e.g. ) 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.
The fundamental mathematical modeling of dynamic wetting is investigated in the scope of the Collaborative Research Center 1194 The kinematics of moving interfaces and moving contact lines is investigated in detail [3,5]. We prove the well-posedness of the discontinuous ODE describing the streamlines of the flow  and derive an evolution equation for the co-moving evolution of the interface normal and the contact angle . The latter result allows to analyze qualitative properties of regular solutions of sharp-interface continuum mechanical models of dynamic wetting. In particular, we prove rigorously that physically reasonable solutions to the “standard model” of wetting based on the two-phase Navier Stokes equations and the Navier slip condition must be singular at the moving contact line . Moreover, the analytical insights lead to the development of new numerical methods with increased accuracy at the moving contact line.
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.
 Bothe, D., Dreyer, W. (2015). Continuum thermodynamics of chemically reacting fluid mixtures. Acta Mechanica, 226(6), 1757–1805. https://doi.org/10.1007/s00707-014-1275-1
 Bothe, D., Prüss, J. (2016). On the Interface Formation Model for Dynamic Triple Lines. In H. Amann, Y. Giga, H. Kozono, H. Okamoto, & M. Yamazaki (Eds.), Recent Developments of Mathematical Fluid Mechanics (pp. 25–47). Springer. https://doi.org/10.1007/978-4-431-56457-7_2
 Fricke, M., Köhne, M., Bothe, D. (2019). A kinematic evolution equation for the dynamic contact angle and some consequences. Physica D: Nonlinear Phenomena, 394, 26–43. https://doi.org/10.1016/j.physd.2019.01.008
 Fricke, M., Bothe, D. (2020). Boundary conditions for dynamic wetting – A mathematical analysis. The European Physical Journal Special Topics, 229(10), 1849–1865. https://doi.org/10.1140/epjst/e2020-900249-7
 Bothe, D. (2020). On moving hypersurfaces and the discontinuous ODE-system associated with two-phase flows. Nonlinearity, 33(10), 5425–5456. https://doi.org/10.1088/1361-6544/ab987d
[P1] Bothe, D., Druet, P.-E. (2020). On the structure of continuum thermodynamical diffusion fluxes -- A novel closure scheme and its relation to the Maxwell-Stefan and the Fick-Onsager approach, http://arxiv.org/abs/2008.05327
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 detailed understanding of elementary transport and transfer processes at fluid interfaces, we actively develop techniques for different numerical simulation methods. For this purpose, we develop taylor-made high-fidelity methods which exhibit distinct advantages for the specific interfacial transport physics under consideration [10, 11, 9, 13].
These fundamentally different numerical approaches are combined under one roof and inside a single numerical framework – OpenFOAM. This open-source C++ based library for Computational Continuum Mechanics and Multiphysics allows us to provide the cutting edge numerical methods for emerging physical models and collaborate, share, and discuss with our partners in science and industry.
OpenFOAM is a versatile and flexible simulation software to address complex interfacial transport process and phenomena involved therein. 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 thisparticularly 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 .
The group actively developes both sharp and diffuse interface models:
- Volume-Of-Fluid(VOF) Interface-Capturing (algebraic and geometric) Methods are strictly volume conservative, handle strong deformations of the interface as well as topolgical changes, with one remaining challenge: an accurate and stable curvature approximation. We are developing unstructured unsplit VOF methods [1,2] as well as algebraic VOF methods for challenging viscoelastic flows  and species transfer .
- ArbitraryLagrangian-Eulerian Interface-Tracking methods represent the liquid gas interface with a part of the mesh and follow its deformation with a movement of the computational mesh. hence, we apply this method, when the main objective is the resolution of physical processes on or close to the interface, such as the adsorption of surfactants and their influence on multiphase system , wetting processes , or complex mass tranfer processes at the interface .
- Front-Tracking methods, although not strictly volume conservative, are highly accurate and applicable to problems that involve strong surface tension forces and interfaces that strongly deform and may change topologically. We are developing hybrid Level Set / Front Tracking methods on unstructured meshes [5,6] with the aim of extending Front Tracking with local dynamic mesh refinement and geometrically complex problems.
- Diffuse interface models involving Allen-Cahn as well as Cahn-Hilliard phase-field methods. The smooth interface representation allows for a less demanding discretization of the interface evolution. Hence it can be applied to larger cases and complex topologies such as structured surfaces with the compromise of a somewhat less accurate localization of interface and thereby an increased complexity to include additional transport processes.
We have a close and longstanding relationship with leading members of the OpenFoam community and educate the next generation of Computational Engineers in using and developing OpenFOAM.
 Marić, T., Marschall, H., & Bothe, D. (2018). An enhanced un-split face-vertex flux-based VoF method. Journal of computational physics, 371, 967-993.
[2 Marić, T., Kothe, D. B., & Bothe, D. (2020). Unstructured un-split geometrical Volume-of-Fluid methods–A review. Journal of Computational Physics, 420, 109695.
 Niethammer, M., Brenn, G., Marschall, H., & Bothe, D. (2019). An extended volume of fluid method and its application to single bubbles rising in a viscoelastic liquid. Journal of Computational Physics, 387, 326-355.
[4 ]Deising, D., Marschall, H., & Bothe, D. (2016). A unified single-field model framework for Volume-Of-Fluid simulations of interfacial species transfer applied to bubbly flows. Chemical Engineering Science, 139, 173-195.
 Marić, T., Marschall, H., & Bothe, D. (2015). lentFoam–A hybrid Level Set/Front Tracking method on unstructured meshes. Computers & Fluids, 113, 20-31.
 Tolle, T., Bothe, D., & Marić, T. (2020). SAAMPLE: A Segregated Accuracy-driven Algorithm for Multiphase Pressure-Linked Equations. Computers & Fluids, 200, 104450.
 Gründing, D. (2020). An enhanced model for the capillary rise problem. International Journal of Multiphase Flow, 103210.
 Pesci, C., Weiner, A., Marschall, H., & Bothe, D. (2018). Computational analysis of single rising bubbles influenced by soluble surfactant. Journal of Fluid Mechanics, 856, 709-763.
 Weber, P. S., & Bothe, D. (2016). Applicability of the linearized theory of the Maxwell–Stefan equations. AIChE Journal, 62(8), 2929-2946.
 Bothe, D., & Fleckenstein, S. (2013). A volume-of-fluid-based method for mass transfer processes at fluid particles. Chemical Engineering Science, 101, 283-302.
 Gründing, D., Fleckenstein, S., & Bothe, D. (2016). A subgrid-scale model for reactive concentration boundary layers for 3D mass transfer simulations with deformable fluid interfaces. International Journal of Heat and Mass Transfer, 101, 476-487.
 Gründing, D., Smuda, M., Antritter, T., Fricke, M., Rettenmaier, D., Kummer, F., … & Bothe, D. (2020). A comparative study of transient capillary rise using direct numerical simulations. Applied Mathematical Modelling.
 Weiner, A., & Bothe, D. (2017). Advanced subgrid-scale modeling for convection-dominated species transport at fluid interfaces with application to mass transfer from rising bubbles. Journal of Computational Physics,347, 261-289.
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 improved 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 .
A novel method to initialize the volume fraction field with high accuracy (third- to fourth-order) has been developed in . Moreover, fundamental investigations on the kinematics of interfaces led to the development of novel interface reconstruction methods close to the domain boundary . The latter methods allow for a kinematically consistent transport of the dynamic contact angle and thereby increase the accuracy in the numerical simulation of wetting flows.
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 has been investigated within the VOF framework .
• Rising fluid particles have been investigated with a focus 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 breakup dynamics of a wetting capillary bridge has been investigated in cooperation with the Nano- and Microfluidics group at TU Darmstadt [P1]. The interface reconstruction methods developed in  allowed for a detailed numerical simulation of the breakup process in good agreement with experimental data.
The current research focuses on:
• Transfer processes at dynamic contact lines
• Stability analysis of fluid particles
• Droplet collisions and splashing
• Multiphysics of reactive mass transfer
• Higher-order methods for geometry treatment in VOF
Selected external publications:
 Rieber, M., Frohn, A. (1999). A numerical study on the mechanism of splashing. International Journal of Heat and Fluid Flow, 20(5), 455–461. https://doi.org/10.1016/S0142-727X(99)00033-8
 M. Rieber: Numerische Modellierung der Dynamik freier Grenzflächen in Zweiphasenströmungen. Ph.D. thesis, Universität Stuttgart (2004)
Selected MMA Publications:
 Focke, C., Bothe, D. (2011). Computational analysis of binary collisions of shear-thinning droplets. Journal of Non-Newtonian Fluid Mechanics, 166(14–15), 799–810. https://doi.org/10.1016/j.jnnfm.2011.03.011
[4 ] Albert, C., Raach, H., Bothe, D. (2012). Influence of surface tension models on the hydrodynamics of wavy laminar falling films in Volume of Fluid-simulations. International Journal of Multiphase Flow, 43, 66–71. https://doi.org/10.1016/j.ijmultiphaseflow.2012.02.011
 Ma, C., Bothe, D. (2013). Numerical modeling of thermocapillary two-phase flows with evaporation using a two-scalar approach for heat transfer. Journal of Computational Physics, 233, 552–573. https://doi.org/10.1016/j.jcp.2012.09.011
 Fath, A., Bothe, D. (2015). Direct numerical simulations of thermocapillary migration of a droplet attached to a solid wall. International Journal of Multiphase Flow, 77, 209–221. https://doi.org/10.1016/j.ijmultiphaseflow.2015.08.018
 Albert, C., Kromer, J., Robertson, A. M., Bothe, D. (2015). Dynamic behaviour of buoyant high viscosity droplets rising in a quiescent liquid. Journal of Fluid Mechanics, 778, 485–533. https://doi.org/10.1017/jfm.2015.393
 Liu, M., Bothe, D. (2016). Numerical study of head-on droplet collisions at high Weber numbers. Journal of Fluid Mechanics, 789, 785–805. https://doi.org/10.1017/jfm.2015.725
 Weiner, A., Bothe, D. (2017). Advanced subgrid-scale modeling for convection-dominated species transport at fluid interfaces with application to mass transfer from rising bubbles. Journal of Computational Physics, 347, 261–289. https://doi.org/10.1016/j.jcp.2017.06.040
 Kromer, J., Bothe, D. (2019). Highly accurate computation of volume fractions using differential geometry. Journal of Computational Physics, 396, 761–784. https://doi.org/10.1016/j.jcp.2019.07.005
 Fricke, M., Marić, T., Bothe, D. (2020). Contact line advection using the geometrical Volume-of-Fluid method. Journal of Computational Physics, 407, 109221. https://doi.org/10.1016/j.jcp.2019.109221
[P1] M. Hartmann, M. Fricke, L. Hauer, D. Gründing, T. Marić, D. Bothe, S. Hardt (2020). Breakup Dynamics of Capillary Bridges on Hydrophobic Stripes, http://arxiv.org/abs/1910.01887
Young Scientists at MMA
Tomislav Marić is an Athene Young Investigator and a research group leader on Lagrangian / Eulerian Interface Approximation Methods (LEIA). He is developing the unstructured geometrical Volume-of-Fluid method and the hybrid Level Set / Front Tracking method with the goal of simulating multiphase flows driven by surface tension forces in complex geometries.
Holger Marschall leads the research group “Computational Multiphase Flow” (CMF) group, which emerged from MMA and has been established in April 2019 in the Department of Mathematics at TU Darmstadt. The group develops simulation methods for the computer-aided prediction of transport processes in multiphase flows. The scientific focus is on the development of adaptive and hybrid approaches, which are particularly suited to bridge the large spatial and temporal scales typically present in these transport processes.