Current projects

 
Free boundary propagation and noise: analysis and numerics of stochastic degenerate parabolic equations

DFG Research Grant

Dates: 2018-2020
Participants: Prof. Dr. Günther Grün, Hubertus Grillmeier, M.Sc.


The porous-medium equation and the thin-film equation are prominent examples of nonnegativity preserving degenerate parabolic equations which give rise to free boundary problems with the free boundary at time t > 0 defined as the boundary of the solution’s support at that time.
As they are supposed to describe the spreading of gas in a porous-medium or the spreading of a viscous droplet on a horizontal surface, respectively, mathematical results on the propagation of free boundaries become relevant in applications. In contrast to, e.g., the heat equation, where solutions to initial value problems with compactly supported nonnegative initial data
instantaneously become globally positive, finite propagation and waiting time phenomena are characteristic features of degenerate parabolic equations.
In this project, stochastic partial differential equations shall be studied which arise from the aforementioned degenerate parabolic equations by adding multiplicative noise in form of source terms or of convective terms. The scope is to investigate the impact of noise on the propagation of free boundaries, including in particular necessary and sufficient conditions for the occurrence
of waiting time phenomena and results on the size of waiting times. Technically, the project relies both on rigorous mathematical analysis and on numerical simulation.

Interfaces, ComplexStructures, and Singular Limits in Continuum Mechanics
DFG Research Training Group GRK 2339

Dates: 2018-2022 (first funding period),
Participants at FAU: Proff. Bänsch, Burger, Grün (Co-Spokesperson), Knabner, Pratelli, Neuss-
Radu, and Dr. Ray.

                                                                                    
The DFG Research Training Group IntComSin is a joint doctorate program of FAU and University of Regensburg in Applied Mathematics. The research focuses on many aspects of advanced mathematical modeling, analysis and numerics with the perspective to understand complex phenomena observed in fluid mechanics, material engineering, biological systems and other applied sciences. Typically interfaces, multiple scales/fields and small parameters (singular limits) are the core features of the problems to be studied. The doctoral program offers a structured course program in partial differential equations, calculus of variations, numerical analysis, scientific computing, mathematical modeling and professional skills. The topics explored in the doctoral projects address:
• interfaces (two-phase flows, transport processes at interfaces, fluidic and elastic effects in membranes, shape optimisation, fluid-structure interactions),
• complex structures ( multiple scales, homogenisation of porous media, micro-macro models for complex fluids, microstructures generated by non-convex variational problems),
• singular limits and dimension reduction (thin film limits, plates/shells and beams, asymptotic limits in phase field models).
For further information, see www.uni-regensburg.de/IntComSin

Completed projects

Diffusive interface models for transport processes at fluidic interfaces (Part 2)

DFG Priority Programme SPP 1056 "Transport processes at fluidic interfaces"

Dates 2013 — 2016
Participants Professor Dr. Günther Grün, Stefan Metzger, M.Sc.

In recent years, diffuse interface models turned out to be a promising approach to describe fundamental features of two-phase flow like droplet break-up or coalescence. In the second funding period, novel thermodynamical consistent phase-field models for species transport in two-phase flow shall be derived with an emphasis on soluble surfactants. Additional phenomena -- ranging from microscale effects like molecule orientation over thermal effects to electrostatic interactions -- shall be included as well. On this basis, new sharp-interface models shall be derived by formal asymptotic analysis.
For selected diffuse-interface models, existence of solutions and stability of fluidic interfaces will be investigated by rigorous mathematical analysis. Stable numerical schemes shall be formulated and implemented in two and three space dimensions. By numerical simulations, partially guided by the "Leitmassnahme" Taylor-flow, the models shall be validated and further improved. By numerical analysis, convergence shall be established for the prototypical problem of species transport in two-phase flow with general mass densities.

Fronts and Interfaces in Science and Technology (FIRST) / Marie Curie Initial Training Networks
Funding Agency 7. EU-Forschungsrahmenprogramm
Funding period Jan. 2010 — Dec. 2013
Principal Investigator Professor Dr. Günther Grün
With this network, the universities of Bath, Eindhoven, Erlangen, Haifa (Technion), Madrid (Complutense), Paris (Orsay), Rome (La Sapienza), Zürich and the industrial partners EGIS and SIEMENS AG foster a joint training platform for PhD-students working on analysis and control of interfacial phenomena. Applications range from image processing over reaction-diffusion systems to complex multi-phase flow.
FAU is involved in three projects, guided by Proff. Grün, Knabner, and Leugering. The first one is concerned with the effects electric fields have on two-phase flow with electrolyte solutions. The goal is to derive thermodynamically consistent diffuse-interface models for general mass densities and ion distributions and to prove existence and regularity of solutions.
The second one is a tandem project with Prof. Peletier (TU Eindhoven) devoted to contaminant flow in porous media. There is experimental evidence that attachment to colloids strongly enhances contaminant transport. Derivation and analysis of appropriate multi-scale models are in the focus of this project.
Prof. Leugering's project -- jointly with Prof. Coron (University Pierre et Marie Curie, Paris) -- is devoted to optimal control and stabilization of flow of gas, water, and traffic in networked pipe- and road-systems. It focusses on reachability and stabilizability properties under constraints both in states and controls and on the derivation of appropriate sensitivities for a numerical treatment of optimal controls for systems of realistic size.

 

Diffusive interface models for transport processes at fluidic interfaces (Part 1)

DFG Priority Programme SPP 1056 "Transport processes at fluid interfaces"

Dates 2010 — 2013
Participants Professor Dr. Günther Grün, Dipl.-Math. Fabian Klingbeil


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.

Mathematical Analysis of Models Describing the Evolution of Liquid Patterns on Material Interfaces (Part 2 and Part 3)
DFG Priority Programme SPP 1052 "Benetzung und Strukturbildung an Grenzflächen""
Funding period1999 - 2007
ParticipantsProf. Dr. Günther Grün, Dr. Jürgen Becker

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. Opens external link in new windowWetting Kaleidoscope>>

 

DAAD project "Mathematical Analysis and Numerical Simulation of Dynamic Electrowetting" ("projektbezogener Personenaustausch Spanien")
PartnerUniversidad Autónoma Madrid (Prof. Marco Fontelos)
Funding period 2007 - 2008
Principal InvestigatorsProf. Dr. Günther Grün, Prof. Dr. Marco Fontelos (CSIC Madrid)

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>>

 

 

Complex rheologies
Project in SFB 611 "Singular phenomena and scaling in mathematical models", University of Bonn (together with Christiane Helzel and Felix Otto)
Funding period 2004 - 2006

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).

 

Scaling laws and their cross-overs: global analysis of rheological processes
Project in SFB 611 "Singular phenomena and scaling in mathematical models", University of Bonn (together with Felix Otto)
Funding period 2002 - 2004

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.
DFG project "Mathematical Analysis of Models Describing the Evolution of Liquid Patterns on Material Interfaces ")
Project in SFB 611 DFG Priority Programme SPP 1052 "Benetzung und Strukturbildung an Grenzflächen"
Funding period1997 - 1999
ParticipantsProf. Dr. Günther Grün, Dr. Jürgen Becker

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.