  We determine the k-essence Lagrangian of a relativistic barotropic fluid. The
equation of state of the fluid can be specified in different manners depending
on whether the pressure is expressed in terms of the energy density (model I),
the rest-mass density (model II), or the pseudo rest-mass density for a complex
scalar field in the Thomas-Fermi approximation (model III). In the
nonrelativistic limit, these three formulations coincide. In the relativistic
regime, they lead to different models that we study exhaustively. We provide
general results valid for an arbitrary equation of state and show how the
different models are connected to each other. For illustration, we specifically
consider polytropic and logotropic dark fluids that have been proposed as
unified dark matter and dark energy models. We recover the Born-Infeld action
of the Chaplygin gas in models I and III and obtain the explicit expression of
the reduced action of the logotropic dark fluid in models II and III. We also
derive the two-fluid representation of the Chaplygin and logotropic models. Our
general formalism can be applied to many other situations such as Bose-Einstein
condensates with a $|\varphi|^4$ (or more general) self-interaction, dark
matter superfluids, and mixed models.
