Geometric aspects of the isentropic liquid dynamics and vorticity invariants

TY - JOUR

T1 - Geometric aspects of the isentropic liquid dynamics and vorticity invariants

AU - Balinsky, Alexander A.

AU - Blackmore, Denis

AU - Kycia, Radosław

AU - Prykarpatski, Anatolij K.

N1 - Publisher Copyright: © 2020 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2020/11

Y1 - 2020/11

N2 - 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.

AB - 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.

KW - Charged liquid fluid dynamics

KW - Diffeomorphism group

KW - Hydrodynamic Euler equations

KW - Isentropic hydrodynamic invariants

KW - Lie-Poisson structure

KW - Liquid flow

KW - Symmetry reduction

KW - Vortex invariants

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U2 - 10.3390/e22111241

DO - 10.3390/e22111241

M3 - Review article

SN - 1099-4300

VL - 22

SP - 1

EP - 26

JO - Entropy

JF - Entropy

IS - 11

M1 - 1241

ER -