Abstract
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.
Highlights
Our review is devoted to compressible liquid or gas motions in which entropy remains locally constant throughout the flowfield, i.e., the flow for which the entropy of a moving element along a streamline remains constant, is called isentropic
We presented a detailed enough differential geometric description of the isentropic fluid motion phase space and featuring it in the diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints
We presented a modification of the Hamiltonian analysis in the case of isothermal liquid dynamics
Summary
Our review is devoted to compressible liquid or gas motions in which entropy remains locally constant throughout the flowfield, i.e., the flow for which the entropy of a moving element along a streamline remains constant, is called isentropic. The equality of partial derivatives above means, owing to the well known Montel-Menchoff-Young theorem [1,2,3], the existence of such a differentiable thermodynamic state function R2+ 3 (ρ, σ) → e ∈ R, that its differential satisfies the following equality: δe(ρ, σ ) = Tδσ + pδρ/ρ2 The latter expression presents exactly the written down second thermodynamic law with respect to the locally defined variables, if the smooth function R2+ 3 (ρ, σ ) → e ∈ R is interpreted as the specific medium energy of the system at the internal absolute temperature T = T (ρ, σ ) and pressure p(ρ, σ ). Taking in addition that our medium is imbedded into some domain M ⊂ R3 , moving in space-time, our task is to describe adequately the related motion spatial phase space variables, compatible with the corresponding Euler evolution equations
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