  We continue the investigation of thermodynamical properties of the BPS Skyrme
model. In particular, we analytically compute the baryon chemical potential
both in the full field theory and in a mean-field approximation. In the full
field theory case, we find that the baryon chemical potential is always exactly
proportional to the baryon density, for arbitrary solutions. We further find
that, in the mean-field approximation, the BPS Skyrme model approaches the
Walecka model in the limit of high density - their thermodynamical functions as
well as the equation of state agree in this limit. This fact allows to read off
some properties of the $\omega$-meson from the BPS Skyrme action, even though
the latter model is entirely based on the (pionic) $SU(2)$ Skyrme field. On the
other hand, at low densities, at the order of the usual nuclear matter density,
the equations of state of the two models are no longer universal, such that a
comparison depends on some model details. Still, also the BPS Skyrme model
gives rise to nuclear saturation in this regime, leading, in fact, to an exact
balance between repulsive and attractive forces. The perfect fluid aspects of
the BPS Skyrme model, which, together with its BPS properties, form the base of
our results, are shown to be in close formal analogy with the Eulerian
formulation of relativistic fluid dynamics. Within this analogy, the BPS Skyrme
model, in general, corresponds to a non-barotropic perfect fluid.
