On the Navier-Stokes equations for water

In general, the existence of entropy imposes restrictions on the constitutive functions in the Navier-Stokes equations. In this paper, it is 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.
However, for many fluids with the properties radically different than ideal gases (the best example here is water) the pressure as a function of ρ and T is not of the form p0(ρ)T. Therefore the energy density per unit mass in the Navier-Stokes equations for water should depend also on the mass density and the explicit form of this dependence requires further discussion.