Project heads  Volker Mehrmann^{ ☆}, Dirk Peschka^{ ★}, Matthias Rosenau^{ ✻}, Marita Thomas^{ ★}, Barbara Wagner^{ ★} 
Staff  M. Hassan Farshbaf Shaker^{ ★} 
Institutes  ☆ Technische Universität Berlin, ★ Weierstrass Institute, ✻ GFZ (HelmholtzZentrum Potsdam) 
Collaboration 

Acknowledgement: This research is carried out in the framework of the DFG funded Cluster of Excellence EXC 2046 MATH^{+}: The Berlin Mathematics Research Center within the Application Area Materials, Light and Devices. The funding period of the project is from January 2019 until December 2020.
Background and goals
The goal of the MATH^{+} project AA24 Modeling and analysis of suspension flows is to derive a unified continuum model applicable from dilute to dense, jammed suspensions. Their mathematical description provides an ongoing challenge since suspensions show much different behavior in different concentration regimes: While dilute suspensions, i.e., mixtures with low volume fraction and large particle distances, have small or negligible interparticle interaction, concentrated or dense suspensions have strong interactions that lead to constitutive material laws not present in standard fluids: Here, interparticle interactions generate a normal pressure, which is a major driving force for shear induced particle migration. While the effective material laws for dilute suspensions are satisfactorily captured by the thesis of Albert Einstein, these laws clearly fail for the transition to the dense or jammed regime. For the prediction of suspension behavior in applications it would be a major success to establish a single model that captures all these different effects and thus allows for the transition between the different regimes. It is a joint project with the Technische Universität Berlin (V. Mehrmann) and the Weierstrass Institute (D. Peschka, M. Thomas, B. Wagner), that also involves the close collaboration with the German Research Centre for Geosciences, GFZ (M. Rosenau). 

The mathematical model, i.e., the partial differential equation (PDE) in the preliminary work [5], is constructed using a generalized gradient flow approach based on an energy generating forces and a nonsmooth convex dissipation functional which generates the PDE dynamics from these forces.
In this way, we deduce one possible model to describe flows of twophase mixtures with free, evolving boundaries and we provide the underlying construction for gravitydriven and surfacetension driven flows, see [5].
It is the goal of this project to extend this model by conservative effects thus leading to a portHamiltonian or GENERIC system in the spirit of [4] and to investigate the resulting system analytically and numerically.
Mathematical challenges of such a twophase mixture flow model are:


Project related events
 Workshop: PDE 2019: "Partial Differential Equations in Fluids and Solids", Weierstrass Institute Berlin, 913 September 2019.
 Minisymposium: ``Recent advances in understanding suspensions and granular media flow'' organized by A. Münch (Oxford) and D. Peschka at 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019) in Valencia, July 1519, 2019.
 Upcoming Thematic Einstein Semester WS2020/21: ``Energybased mathematical methods for reactive multiphase flows'' organized by M. Thomas, B. Wagner, A. Mielke, V. Mehrmann, D. Peschka
Guests
 Prof. Luca Heltai (long term), SISSA, Trieste, May 29July 31, 2019.
 Prof. Uwe Thiele (jointly with RG7, talk Material Modeling Seminar), Münster, February 2728, 2019.
Preliminary work and projectrelated publications
[6]  M.H. FarshbafShaker, M. Thomas, Analysis of a compressible Stokesflow with degenerating and singular viscosity, pdf file, 2020. 
[6]  D. Peschka, M. Rosenau, Twophase flows for sedimentation of suspensions, WIAS Preprint 2743, 2020. 
[5]  D. Peschka, M. Thomas, T. Ahnert, A. Münch, and B. Wagner, Gradient structures for flows of concentrated suspensions, SpringerCIM Series, 2019. 
[4]  A. M. Badlyan, B. Maschke, C. Beattie, and V. Mehrmann, Open physical systems: from GENERIC to portHamiltonian systems, MTNS Hong Kong, 2018. 
[3]  T. Ahnert, A. Münch, B. Niethammer, and B. Wagner, Stability of concentrated suspensions under Couette and Poiseuille flow, J. Eng. Math., 2018. 
[2]  T. Ahnert, A. Münch, and B. Wagner, Models for the twophase flow of concentrated suspensions, Eur. J. Appl. Math., 2018. 
[1]  M.C. Ritter, T. Santimano, M. Rosenau, K. Leever, and O. Oncken, Sandbox rheometry: Coevolution of stress and strain in riedeland critical wedgeexperiments, Tectonophysics, 2018. 
Talks and posters
 Poster: M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of syspension flows, Visit of the scientific advisory board of MATH+, November 11, and MATH+ Day, December 13, 2019.
 Invited Talk: M. Thomas, GENERIC structures with bulkinterface interaction, SFB 910 Symposium "Energy based modeling, simulation and control", TU Berlin, October 25, 2019.
 Invited Talk: M. Thomas, "Gradient structures for flows of concentrated suspensions", Thematic Minisymposium MS ME075 "Recent advances in understanding suspensions and granular media flow", ""9th International Congress on Industrial and Applied Mathematics" (ICIAM 2019, Valencia), July 1519, 2019.
 Invited Talk: D. Peschka, Dynamic contact angles via generalized gradient flows, Modelling of Thin Liquid Films  Asymptotic Approach vs. Gradient Dynamics, Banff International Research Station, Banff, Canada, April 30, 2019.
 Invited Talk: D. Peschka, Gradient formulations with flow maps  mathematical and numerical approaches to free boundary problems, Kolloquium des Graduiertenkollegs 2339, Universität Regensburg, May 24, 2019.
 Talk: D. Peschka, Modeling, simulation, and experiments for flows of concentrated suspensions, LangenbachSeminar, WIAS Berlin, April 24, 2019.
 Poster: D. Peschka, Dynamic Contact Angles via Gradient Flows, 694. Heraeus seminar on "Wetting on soft or microstructured surfaces", Bad Honnef, Germany, April 1113, 2019.
 Talk: D. Peschka, Gradient structures for flows of concentrated suspensions  jamming and free boundaries, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S11 ``Interfacial Flows", Technische Universität Wien, Austria, February 20, 2019.
 Invited Talk: D. Peschka, Mathematical modeling of fluid flows using gradient systems, Seminar in PDE and Applications, Delft University of Technology, Netherlands, May 28, 2019.