On the classification of dense fluids

It is commonly accepted that the existence of entropy imposes restrictions on the constutive functions in the Navier-Stokes-Fourier equations. In the paper: S. Piekarski, "On the Navier-Stokes equation for water" (Archives of Acoustics, 31, 2, 265-271, 2006) it has been shown that if the energy per unit mass is a function of the temperature T only, then the pressure p is an arbitrary function of the density ρ multiplied by the temperature T.
Now the general form of the relations between the energy density and the pressure is given (both quantities are understood as functions of the mass density and the temperature).
These relations can be approximated in different ways and different approximations suggest different classifications of dense fluids (some of them are similar to the virial expansions).