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Nerovnovážná termodynamika supratekutin
Thesis title in Czech: Nerovnovážná termodynamika supratekutin
Thesis title in English: Non-equilibrium thermodynamics of superfluids
Key words: Nerovnovážná termodynamika, helium, supratekutiny, směs, Poissonova závorka, GENERIC
English key words: Non-equilibrium thermodynamics, Helium, superfluids, mixture, Poisson bracket, GENERIC
Academic year of topic announcement: 2018/2019
Type of assignment: diploma thesis
Thesis language:
Department: Mathematical Institute of Charles University (32-MUUK)
Supervisor: RNDr. Michal Pavelka, Ph.D.
Author:
Advisors: RNDr. Jaroslav Hron, Ph.D.
Dr. Marco La Mantia, Ph.D.
Guidelines
We will investigate the following points:
1) GENERIC formulation of the one-component models with extra kinematics of density. Relation to the two-fluid model.
2) GENERIC formulation of the one-component models with extra kinematics of entropy and their relation to the models 1) and the two-fluid models.
3) Interpretation of quantum vortices as the irreversible evolution of the models.
4) Combined with the numerical solutions, we will compare the modeling approaches with the experimental data acquired within the proposed project, and we will make a recommendation on which class of the models compatible with both thermodynamics and mechanics gives best agreement with the experimental behavior.

References
[1] Ilya Peshkov, Michal Pavelka, Evgeniy Romenski, Miroslav Grmela, Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations, Accepted to Continuum Mechanics and Thermodynamics 2018.
[2] L. D. Landau, E. M. Lifshitz, Fluid Mechanics, vol. 6
[3] D. D. Holm, B. A. Kuperschmidt, Physical Review A, 36(8), 1987
[4] M. S. Mongoivi, D. Jou, M. Sciacca, Physics Reports, Volume 726, 6 January 2018, Pages 1-71
[5] E. Romenskii, Mathl. Comput. Modelling Vol. 28, No. 10, pp. 115-130, 1998
[6] M. Grmela and H.C Öttinger (1997). Phys. Rev. E. 56: 6620–6632 and 6633–6655. .
Preliminary scope of work in English
In general the purpose of mathematical modeling is to provide interpretation of experimental data. Although quantum turbulence has been studied for many decades, there is no consensus on how to describe it by continuous models.

The perhaps most frequently used approach is the two-fluid approach also referred to as the Landau-Tisza model [2]. Within this approach the liquid helium is seen as a mixture of two fluids - a normal component and a superfluid component. The former behaves as a Newtonian fluid, in particular it carries entropy and exhibits viscosity, while the latter moves without any friction and entropy transfer. However, there are conceptual challenges to the two-fluid model as even Landau & Lifshitz warn in the book that there are no two fluids in real liquid helium.

There are alternative approaches where the helium is regarded as a one-component fluid, but there is still no unity among them. In one class of models the helium is taken as a one-component fluid density of which exhibits an additional motion. The fluid is equipped with extra kinematics [3, 5].

Another alternative approaches prefer to interpret the superfluid motion as extra kinematics of entropy (or energy) density [4,1], leading to hyperbolic Cattaneo-like heat transfer coupled with fluid mechanics. These models have the advantage of relatively straightforward description of the ballistic regime in liquid helium.

The alternative approaches share a common feature - it seems to be possible that the reversible part of the evolution equations can be formulated as Hamiltonian mechanics. This feature has been shown important and shared among many successful theories of continuum thermodynamics. By combining the Hamiltonian evolution with irreversible evolution, a very large class of models (including turbulence modeling) compatible with both geometric mechanics and thermodynamics can be described. The combination is nowadays referred to as the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) [1,6].
 
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