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In this paper, thermodynamics of a moving medium is considered. Main goal of this paper is to describe thermodynamics state equations and equations of co-existence manifolds of media on two- and three-dimensional manifolds. To this end the phase space of a medium is extended with the deformation tensor and the stress tensor.

We consider the Euler system describing a one-dimensional inviscid flows in space along curves of a certain class. Using differential invariants for the Euler system, we obtain its quotient equation. The solutions of the quotient equation that are constant along characteristic vector field provide some solutions of the Euler system. We discuss solving the quotient using asymptotic expansions of unknown functions and virial expansion of thermodynamic state equations. Thus the quotient is reduced to a series of ODE systems.

A Navier--Stokes system on a curve is discussed. The quotient equation for this system is found. The quotient is used to find some solutions of Navier--Stokes system. Using virial expansion of the Planck potential, we reduce the quotient equation to a series of systems of ordinary differential equations.

Symmetries and differential invariants of viscid flows with viscosity depending on temperature on a space curve are given. Their dependence on thermodynamic states of media is studied, and a classification of thermodynamic states is given., Comment: arXiv admin note: text overlap with arXiv:2006.14932, arXiv:2004.11906

In this paper, flows of a viscid fluids on curves are considered. Symmetry algebras and the corresponding fields of differential invariants are found. We study their dependence on thermodynamic states of media, and provide classification of thermodynamic states., Comment: arXiv admin note: text overlap with arXiv:2004.11906

We propose a geometrical approach to the mechanics of continuous media equipped with inner structures and give the basic (mass conservation, Navier-Stokes and energy conservation) equations of their motion.

Symmetries and the corresponding fields of differential invariants of the inviscid flows on a curve are given. Their dependence on thermodynamic states of media is studied, and a classification of thermodynamic states is given., Comment: 6 figures. arXiv admin note: text overlap with arXiv:2004.01567

This paper surveys results found by the authors in the previous papers (see for example, A. Duyunova, V. Lychagin, S. Tychkov, Differential invariants for spherical layer flows of a viscid fluid, Journal of Geometry and Physics, 130, 288-292 (2018)) concerning calculation and classification of symmetry Lie algebra and differential invariants for a variety of fundamental systems in fluid mechanics, including the Euler and Navier-Stokes equations on a plane, three-dimensional space, a sphere, and a spherical layer. In each case, the symmetry Lie algebra is found, followed by a list of generating differential invariants and invariant derivations for the group action. The main extension is a detailed analysis of thermodynamic states, symmetries, and differential invariants. This analysis is based on consideration of Riemannian structure naturally associated with Lagrangian manifolds that represent thermodynamic states. This approach radically changes the description of the thermodynamic part of the symmetry algebra as well as the field of differential invariants.

A non-stationary isentropic filtration of gases in porous media is considered. Thermodynamics in terms of contact and symplectic geometries is briefly discussed. Algebra of symmetries for the PDE system is found, and the classification of media with respect to admissible symmetries is given. Solution for one class of media is found and the phase transitions for this solution are studied. Keywords: gas filtration, porous media, symmetry algebra, phase

We give an overview of the experiments at the Laser Centre of the Vrije Universiteit in Amsterdam on ultracold gases of metastable helium-4 and helium-3 as well as mixtures of both isotopes. We describe our experimental setup and discuss our experiments on Bose-Einstein condensation of 4He* (more than 10 million atoms in a BEC) and sympathetic cooling of 3He* towards Fermi degeneracy (more than 1 million atoms below the Fermi temperature as well as realization of a degenerate boson-fermion mixture). We also present our results on Penning Ionization in ~1 mK clouds containing a single isotope (homonuclear losses) or a mixture of both isotopes (heteronuclear losses)., Comment: 18 pages, 11 figures; to be published in the Proceedings of the International school of physics "Enrico Ferm": Ultracold Fermi gases, Varenna, June 20-30 (2006)

We report the observation of simultaneous quantum degeneracy in a dilute gaseous Bose-Fermi mixture of metastable atoms. Sympathetic cooling of helium-3 (fermion) by helium-4 (boson), both in the lowest triplet state, allows us to produce ensembles containing more than 10^6 atoms of each isotope at temperatures below 1 micro-Kelvin, and achieve a fermionic degeneracy parameter of T/Tf=0.45. Due to their high internal energy, the detection of individual metastable atoms with sub-nanosecond time resolution is possible, permitting the study of bosonic and fermionic quantum gases with unprecedented precision. This may lead to metastable helium becoming the mainstay of quantum atom optics., Comment: 4 pages, 3 figures submitted to PRL

We have produced a Bose-Einstein condensate of metastable helium (4He*) containing over 1.5x10^7 atoms, which is a factor of 25 higher than previously achieved. The improved starting conditions for evaporative cooling are obtained by applying one-dimensional Doppler cooling inside a magnetic trap. The same technique is successfully used to cool the spin-polarized fermionic isotope (3He*), for which thermalizing collisions are highly suppressed. Our detection techniques include absorption imaging, time-of-flight measurements on a microchannel plate detector and ion counting to monitor the formation and decay of the condensate., Comment: 4 pages, 3 figures (changed content)