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Geometric aspects of the isentropic liquid dynamics and vorticity invariants

Balinsky, Alexander A. ORCID:, Blackmore, Denis, Kycia, Rados?aw and Prykarpatski, Anatolij K. 2020. Geometric aspects of the isentropic liquid dynamics and vorticity invariants. Entropy 22 (11) , 1241. 10.3390/e22111241

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We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented. Keywords: liquid flow; hydrodynamic Euler equations; diffeomorphism group; Lie-Poisson structure; isentropic hydrodynamic invariants; vortex invariants; charged liquid fluid dynamics; symmetry reduction

Item Type: Article
Date Type: Published Online
Status: Published
Schools: Mathematics
Additional Information: Attribution 4.0 International (CC BY 4.0)
Publisher: MDPI
ISSN: 1099-4300
Date of First Compliant Deposit: 2 November 2020
Date of Acceptance: 26 October 2020
Last Modified: 07 May 2023 02:56

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