Forschungsinteressen
One guiding theme in the research of Günther Grün’s group are mathematical problems related to fluid dynamics, often two-phase flow (i.e. fluid flow with two immiscible constituents) or fluids with a complex rheology (like, e.g., polymeric liquids or fluids carrying nanoparticles).
From the MODELING perspective, this includes the derivation of thermodynamically consistent PDE-models, e.g.
- for diffuse and sharp interface models with special emphasis on species transport across fluidic interfaces or on electrokinetic phenomena with or without wall effects („electrowetting“)
- for the transport of magnetic nanoparticles in fluid flow subject to external magnetic fields
- for multi-scale effects in two-phase polymeric flow
as well as
- the derivation of models on the effect of thermal noise in wetting („stochastic thin-film equation“).
From the perspective of MATHEMATICAL ANALYSIS, the group works on
- existence and regularity results for PDE-systems arising in hydrodynamics ,
- free boundary problems for (stochastic) degenerate parabolic equations of second and higher order — including (stochastic) porous-medium equations, (stochastic) thin-film equations or (stochastic) parabolic p-Laplace equations. Results include existence of solutions, optimal estimates for propagation rates of free boundaries or for the size of waiting times.
In the field of NUMERICS,
convergent schemes of the above mentioned problems are implemented and analyzed. The inhouse finite-element/finite-volume package EconDrop3D provides space-time adaptive schemes for coupled momentum-phase-field systems with species transport and has recently been extended to include multi-scale effects or geometric flows . Monte-Carlo simulations to validate models based on the stochastic thin-film equation have been performed as well. The group pioneered convergence analysis for numerical schemes related to degenerate fourth-order parabolic equations like the thin-film equation or equivalently like degenerate Cahn-Hilliard equations.
Selected publications:
(Stochastic) degenerate parabolic equations
- Dareiotis K., Gess B., Gnann M., Grün G.:
Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise
In: Archive for Rational Mechanics and Analysis (2021)
ISSN: 0003-9527
DOI: 10.1007/s00205-021-01682-z - Fischer J., Grün G.:
Existence of positive solutions to stochastic thin-film equations
In: SIAM Journal on Mathematical Analysis 50 (2018), S. 411-455
ISSN: 0036-1410 - Fischer J., Grün G.:
Finite speed of propagation and waiting times for the stochastic porous medium equation — a unifying approach
In: SIAM Journal on Mathematical Analysis 47 (2015), S. 825-854
ISSN: 0036-1410 - Grün G., Rauscher M., Mecke K.:
Thin-film flow influenced by thermal noise
In: Journal of Statistical Physics 122 (2006), S. 1261-1291
ISSN: 0022-4715 - Giacomelli L., Grün G.:
Lower bounds on waiting time for degenerate parabolic equations and systems
In: Interfaces and Free Boundaries 8 (2006), S. 111-129
ISSN: 1463-9971 - Grün G.:
Droplet spreading under weak slippage: existence for the Cauchy problem
In: Communications in Partial Differential Equations (2004), S. 1697-1744
ISSN: 0360-5302
Modeling and analysis in hydrodynamics
- Grün G., Weiß P.:
On the Field-Induced Transport of Magnetic Nanoparticles in Incompressible Flow: Existence of Global Solutions
In: Journal of Mathematical Fluid Mechanics 23 (2021), Art.Nr.: 10
ISSN: 1422-6928
DOI: 10.1007/s00021-020-00523-5 - Grün G., Metzger S.:
On micro-macro-models for two-phase flow with dilute polymeric solutions — modeling and analysis
In: Mathematical Models & Methods in Applied Sciences 26 (2016), S. 823-866
ISSN: 0218-2025
DOI: 10.1142/S0218202516500196 - Abels H., Garcke H., Grün G.:
Thermodynamically consistent diffuse interface models for incompressible two-phase flows with different densities
In: Mathematical Models & Methods in Applied Sciences 22 (2012), S. 1150013-1150052
ISSN: 0218-2025 - Fontelos MA., Grün G., Jörres S.:
On a phase-field model for electrowetting and other electrokinetic phenomena
In: SIAM Journal on Mathematical Analysis 43 (2011), S. 527-563
ISSN: 0036-1410
Numerics
- Sieber O.:
Analysis and Numerics of Two-Phase Flows of Active Liquid Crystals with Willmore-type Interfacial Energy: A Micro-Macro Approach.
PhD-thesis, Erlangen (December 2021) - Grün G.:
On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities
In: SIAM Journal on Numerical Analysis 51 (2013), S. 3036-3061
ISSN: 0036-1429
DOI: 10.1137/130908208 - Grün G., Rumpf M.:
Nonnegativity preserving convergent schemes for the thin film equation
In: Numerische Mathematik 87 (2000), S. 113-152
ISSN: 0029-599X