Mathematical analysis and modeling of the evolution of magnetoelastic materials
Final Report Abstract
Magnetic materials are of great importance in technological applications. We aim to understand magnetic fluids with immersed particles of a certain intermediate size. Under applied magnetic fields, ferrofluids stay in the fluid phase, while the viscosity of the magnetorheolocial fluids increases in such a way that they become viscoelastic solids. It is interesting to mathematically describe the behavior of fluids with intermediate-sized particles, which show micromagnetic domains. Such micromagnetic fluids might have various technological applications. As a part of this project we worked on different models of magnetic and magneto-elastic fluid models. Our first aim was to find a model which can explain the partial mixing of two viscous incompressible fluids with different behaviors. In that direction we derived two mathematical models which can possibly explain the partial diffusion between two magnetic fluids. One of the models considers that the fluids involved have matched densities and the second one deals with the case where the fluid densities are different. We worked on the mathematical well-posedness theory of both the models and proved the existence of solutions with bounded energy. We further have shown some regularities of the solutions obtained, meaning that they are better behaved in a certain sense compared to the information available from the finite energy framework. We also have studied a different magneto-elastic model from the perspective of control. More elaborately the aim was to study a problem of optimizing the velocity, magnetization and the elastic behavior of a magneto-elastic fluid model by using a suitable magnetic field. Mathematically this problem is posed as a minimization of a tracking type cost functional. We proved the existence of a minimizer to the cost functional introduced. We also have shown that a magnetic field can be generated by using finitely many field coils in order to optimize velocity, magnetization and the elastic behavior of our model. In the process of proving this result on the optimal control we proved the well-posedness of our system in a strong functional framework and the stability of the same with respect to the external magnetic field.
Publications
- Existence of weak solutions to a diffuse interface model involving magnetic fluids with unmatched densities
M. Kalousek, S. Mitra, A. Schlömerkemper
(See online at https://doi.org/10.48550/arXiv.2105.04291) - Temperature dependent extensions of the Cahn-Hilliard equation
F. De Anna, C. Liu, A. Schlömerkemper, J.-E. Sulzbach
(See online at https://doi.org/10.48550/arXiv.2112.14665) - About a mathematical difficulty in magnetoviscoelasticity. Oberwolfach Reports 13, 2020
A. Schlömerkemper
(See online at https://doi.org/10.4171/owr/2020/13) - Existence of weak solutions of diffuse interface models for magnetic fluids, PAMM
M. Kalousek, S. Mitra, A. Schlömerkemper
(See online at https://doi.org/10.1002/pamm.202100205) - Global existence of weak solutions to a diffuse interface model for magnetic fluids, Nonlinear Analysis: Real World Applications, Volume 59, June 2021, 103243
M. Kalousek, S. Mitra and A. Schlömerkemper
(See online at https://doi.org/10.1016/j.nonrwa.2020.103243) - Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity, Discrete & Continuous Dynamical Systems - S, 14 (1), 17-39 (2021)
M. Kalousek, J. Kortum and A. Schlömerkemper
(See online at https://doi.org/10.3934/dcdss.2020331) - Global Existence and Uniqueness Results for Nematic Liquid Crystal and Magnetoviscoelastic Flows, PhD thesis, University of Würzburg, 2022
J. Kortum
(See online at https://dx.doi.org/10.25972/OPUS-27827) - Magnetoviscoelastic models in the context of magnetic particle imaging, International Journal on Magnetic Particle Imaging Vol 8, No 1, Suppl 1, 2022, Article ID 2203049, 3 Pages
S. Mitra, A. Schlömerkemper
(See online at https://doi.org/10.18416/IJMPI.2022.2203049) - Strong well-posedness, stability and optimal control theory for a mathematical model for magneto-viscoelastic fluids, Calculus of Variations and Partial Differential Equations, Vol. 61, Article number 179 (2022)
H. Garcke, P. Knopf, S. Mitra, A. Schlömerkemper
(See online at https://doi.org/10.1007/s00526-022-02271-y) - Struwe-like solutions for an evolutionary model of magnetoviscoelastic fluids, Journal of Differential Equations 309 (2022), 455–507
F. De Anna, J. Kortum, A. Schlömerkemper
(See online at https://doi.org/10.1016/j.jde.2021.11.034)
