  Recent work giving a classification of kinematic and vorticity conservation
laws of compressible fluid flow for barotropic equations of state (where
pressure is a function only of the fluid density) in $n>1$ spatial dimensions
is extended to general non-isentropic equations of state in which the pressure
is also a function of the dynamical entropy (per unit mass) of the fluid. Two
main results are obtained. First, we find that apart from the familiar
conserved integrals for mass, momentum, energy, angular momentum and Galilean
momentum, and volumetric entropy, additional kinematic conserved integrals
arise only for non-isentropic equations of state given by a generalized form of
the well-known polytropic equation of state with dimension-dependent exponent
$\gamma=1+2/n$, such that the proportionality coefficient is an arbitrary
function of the entropy (per unit mass). Second, we show that the only
vorticity conserved integrals consist of a circulatory entropy (which vanishes
precisely when the fluid flow is irrotational) in all even dimensions. In
particular, the vorticity integrals for helicity in odd dimensions and
enstrophy in even dimensions are found to be no longer conserved for any
non-isentropic equation of state.
