Symmetries and Casimirs of radial compressible fluid flow and gas dynamics in n>1 dimensions

TL;DR

This paper classifies all Lie point symmetries and conserved quantities for radial compressible fluid flow in multiple dimensions, revealing new symmetries and hierarchies of Casimirs and generalized symmetries, applicable to gas dynamics.

Contribution

It provides a complete classification of symmetries and Casimirs for radial fluid flow equations, including special cases for different equations of state, extending previous results.

Findings

All Lie point symmetries are explicitly determined.

Hierarchy of conserved integrals are proven to be Casimirs for general equations.

New kinematic conserved integrals and symmetries are identified.

Abstract

Symmetries and Casimirs are studied for the Hamiltonian equations of radial compressible fluid flow in n>1 dimensions. An explicit determination of all Lie point symmetries is carried out, from which a complete classification of all maximal Lie symmetry algebras is obtained. The classification includes all Lie point symmetries that exist only for special equations of state. For a general equation of state, the hierarchy of advected conserved integrals found in recent work is proved to consist of Hamiltonian Casimirs. A second hierarchy that holds only for an entropic equation of state is explicitly shown to comprise non-Casimirs which yield a corresponding hierarchy of generalized symmetries through the Hamiltonian structure of the equations of radial fluid flow. The first-order symmetries are shown to generate a non-abelian Lie algebra. Two new kinematic conserved integrals found in recent work are likewise shown to yield additional first-order generalized symmetries holding for a barotropic equation of state and an entropic equation of state. These symmetries produce an explicit transformation group acting on solutions of the fluid equations. Since these equations are well known to be equivalent to the equations of gas dynamics, all of the results obtained for n-dimensional radial fluid flow carry over to radial gas dynamics.