  In this paper, we will deepen the understanding of some fluid models proposed
by other authors for the description of dark energy. Specifically, we will show
that the so-called (Modified) Berthelot fluid is the hydrodynamic realization
of the free Dirac-Born-Infeld theory and that the Dieterici fluid admits a
non-relativistic $k$-essence formulation; for the former model the evolution of
the scalar field will be written in terms of some cosmographic parameters. The
latter model will also be tested using Machine Learning algorithms with respect
to cosmic chronometers data, and results about the dynamics at a background
level will be compared with those arising when other fluids (Generalized
Chaplygin Gas and Anton-Schmidt) are considered. Due to some cosmic opacity
effects, the background cosmology of universes filled by these inequivalent
fluids, as they arise in physically different theories, may not be enough for
discriminating among them. Thus, a perturbation analysis in the long-wavelength
limit is carried out revealing a rich variety of possible behaviors. It will
also be shown that the free Dirac-Born-Infeld theory cannot account for flat
galactic rotation curves, and therefore we derive an appropriate relationship
between the scalar field potential and the brane tension for achieving this
goal; this provides an estimate for the dark matter adiabatic speed of sound
inside the halo consistent with other literature. A certain relationship
between the Newtonian gravitational potential within the galaxy and the
Lagrangian potential in the non-relativistic regime for the (Modified)
Berthelot fluid will also be enlightened.
