| Abstract: |
| Conserved integrals and invariants (advected scalars) are studied for the equations of radial compressible fluid flow in $n>1$ dimensions. Apart from entropy, which is a well-know invariant, three additional invariants are found from an explicit determination of invariants up to first-order. One holds for a general equation of state (EOS), and the two others hold only for entropic equations of state.
A recursion operator on invariants is presented, which produces two hierarchies of higher-order invariants. Each invariant yields a corresponding integral invariant, describing an advected conserved integral on transported radial domains. In addition, a direct determination of kinematic conserved densities uncovers two ``hidden`` non-advected conserved integrals: one describes enthalpy-flux, holding for barotropic EOS; the other describes entropy-weighted energy, holding for entropic EOS. A further explicit determination of a class of first-order conserved densities shows that the corresponding non-kinematic conserved integrals
on transported radial domains are equivalent to integral invariants, modulo trivial densities.
One of the hierarchies of invariants is proved to consist of Hamiltonian Casimirs. The second hierarchy, which holds only for an entropic EOS, is explicitly shown to comprise non-Casimirs. Through the Hamiltonian structure of the radial fluid flow equations,
these non-Casimir invariants yield a corresponding hierarchy of generalized symmetries. The first-order symmetries are shown to generate a non-abelian Lie algebra. Two new kinematic conserved integrals are likewise shown to yield additional first-order generalized symmetries holding for a barotropic EOS and an entropic EOS. These symmetries produce an explicit transformation group acting on solutions of the fluid equations. |
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