# What is a “perfect fluid”, really?

In Statistical Mechanics, unless I'm making a confusion mistake, a perfect fluid is defined as a large collection of particles without any "internal" interactions (except maybe from point collisions). Long and short range forces are all neglected. This gives the usual perfect gas law: $$\begin{equation}\tag{1} p V = N k T. \end{equation}$$ Assuming adiabatic irreversibility, this implies the polytrope relation, which is a special case of a barotrope state relation : $$\begin{equation}\tag{2} p = \kappa \, \rho_{\text{mass}}^{\gamma}, \end{equation}$$ where $$\kappa$$ is a constant and $$\gamma$$ is the adiabatic index of the fluid. Of course, $$\rho_{\text{mass}}$$ is the proper mass density of the fluid. We could also find $$\begin{equation}\tag{3} p = (\gamma - 1) \, \rho_{\text{int}}, \end{equation}$$ where $$\rho_{\text{int}}$$ is the internal energy density, defined as $$\rho_{\text{int}} = \rho - \rho_{\text{mass}}$$ if $$\rho$$ is the total energy density (I'm using natural units so $$c \equiv 1$$).

Now, in Special (and General) Relativity, a perfect fluid is defined as any substance that doesn't show any macroscopic viscosity and shear (this $$\underline{\text{suggest}}$$ no internal microscopic interactions, but this isn't obvious), and such that the energy-momentum of the fluid is diagonal and isotropic in the proper reference frame: $$\begin{equation}\tag{4} T_{ab} = \begin{bmatrix} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{bmatrix}. \end{equation}$$ Here, a barotropic relation may admit any function $$p(\rho)$$, and not just (2) or (3) above. For example, we may admit the Van der Waals law which isn't a perfect fluid in statistical mechanics (there are some short range forces in action) : $$\begin{equation}\tag{5} p = \frac{c \, \rho_{\text{mass}} \, T}{1 - a \, \rho_{\text{mass}}} - b \, \rho_{\text{mass}}^2. \end{equation}$$ This special relativistic definition isn't the same as the statistical mechanics definition, since it may admit fluids with some internal interactions (yet without showing any macroscopic shear and viscosity).

Now, I'm finding myself irritated by these two definitions, which aren't exactly equivalent. So what really is a "perfect fluid"? The $$\underline{\text{statistical}}$$ one (without any microscopic internal interactions), or the $$\underline{\text{relativistic}}$$ one (which may admit internal interactions)?

Or is there two different "names" that may differentiate the two definitions, something like "ideal fluid" and "perfect fluid" or something else?

I don't like the two inequivalent definitions having the same name, since it opens the door to confusion. I don't want a sloppy use of the same name, just because we are working in different fields (classical statistical mechanicians, or general relativists, or fluid dynamicians, ...).

I have the impression that the statistical definition is the right one, from an historical perspective, and that the relativists should call their perfect fluid as "ideal fluid", instead. Is that right? Or maybe it should be the reverse??

The sloppy Wikipedia appears to reverse the names: https://en.wikipedia.org/wiki/Perfect_fluid, which apparently was written by a relativist! And https://en.wikipedia.org/wiki/Ideal_gas which calls "ideal gas" the perfect fluid of statistical mechanics. Now I'm all confused! Wikipedia isn't a good reference for physics definitions, since there are frequently many inconsistencies.

I think you are not distinguishing properly between ideal/perfect fluids and ideal/perfect gases as they are different concepts:

In traditional fluid mechanics a perfect or ideal fluid is only characterised by the absence of dissipation, the viscosity and the coefficient of heat conduction are zero (Landau & Lafshitz) and the process as a consequence reversible. This means that its equations of motion are given by the Euler equations (including constant entropy) and not the full Navier-Stokes equations that contain a viscous term as well and were written down almost 100 years later in the 1840s. A perfect 'inviscid' fluid can be any fluid, a gas or a liquid, and poses no restrictions regarding the material law. It is particularly useful in aerodynamics where viscous contributions generally only dominate near the walls. Assuming an ideal fluid you may be able to obtain analytical solutions without the need for numerical simulations that take viscous effects directly into account.

An ideal gas on the other hand is a model for the material law: In this case the members of a rarefied gas are assumed as point-particles only interacting in elastic collisions that are assumed to obey Newtonian physics. This means the particles have only translational degrees of freedom and no complex far-field interactions between particles are admitted. If furthermore the heat capacity can be assumed constant, it is considered a perfect gas. In kinetic theory of gases this allows for a simple estimation of transport coefficients and most compressible mechanics are based on this simplification.

• So you're saying that what I called a "perfect fluid" in statistical mechanics is actually a "perfect gas", not a fluid? This would make sense. – Cham Jan 30 '20 at 21:18
• @Cham To be precise, what you describe as perfect fluid in statistical mechanics in the first section of your question would be an ideal gas for most people, including myself. I am not aware of statistical mechanics literature that actually defines it the way you described it. E.g. Landau & Lafshitz, as already mentioned, adhere to the nomenclature I used. – 2b-t Jan 30 '20 at 21:33
• I'm not sure to understand your previous comment. The statistical definition of an ideal gas (what I wrongly called "perfect fluid") is a collection of particles without any internal interaction. It's a bunch of free particles. This is how we derive the formula $p V = N k T$, and so the polytrope formula (2). – Cham Jan 30 '20 at 21:42
• @Cham Yeah, that is correct. I only intended to say that I think you mixed up the perfect fluid and the perfect gas in the first section of your question and the literature (also in statistical mechanics) is to my knowledge consistent in distinguishing the two as different concepts. I have noticed that in scientific papers people do not distinguish cleanly between inviscid flow, perfect/ideal fluid and perfect/ideal gas but books are generally consistent and precise in this regard. – 2b-t Jan 30 '20 at 21:50
• Ok, I think things are getting clearer now, thanks. Yes, many authors are very sloppy in the terms and nomenclature used (like you said), and this is why I got confused in the long run (after reading a lot of papers). It is worst in French (my native language), since we tend to say "gaz parfait" (perfect gas, instead of ideal gas): fr.wikipedia.org/wiki/Gaz_parfait, and "fluide parfait" (perfect fluid): fr.wikipedia.org/wiki/Fluide_parfait, and often mixing the words "gas" and "fluid"! – Cham Jan 30 '20 at 22:08

That Friedman used a perfect fluid is crazy.
Space is not a perfect fluid as it seems infinitely compressible. But it Could also be a fluid with varying viscosity. More viscous at low density, less viscous at high density. This Could account for dark components. High viscosity in low density regions gives illusion of mass/inertia. Low viscosity in dense regions would cause increased micro turbulence which could absorb smooth rotational flow.