What makes an abstract physical system describable by a "fluid" equations of motion?

Question

We can describe (some of) the dynamics of many systems using fluid mechanics. Of course these include classical fluids like water, more exotic fluids like photon gases and the universe as a whole and even solid(ish) things over long times, like glasses and ice. Further still we can treat general classical systems in phase space using a phase fluid, quantum systems (e.g. Madelung equations) and with a little bit of hand waving anything that happens on a symplectic manifold (which gives us a Hamiltonian and hence a flow and Louiville's theorem).

So what is it about a system that makes it obey a fluid model? Are there systems that definitely do not fit a fluid model?

(I realise that I haven't said exactly what I mean by "fluid model", this is somewhat deliberate. If you like you can take it to mean "having equations of motion which are (almost) identical in some form to the Euler equations".)

edit: Since admittedly, the original question wasn't quite clear I'll try to clarify a bit. I'm not looking for a high-school answer, or one which describes only actual literal physical fluids, e.g. "a fluid doesn't support a shear stress", "a fluid is something you can wash your hair with". I've given some examples above of situations that fit this description, and without further explanation I think it's non-obvious what a "mean free path" or similar would mean in a generalised fluid. What I'm really looking for (and there may not be any) is some overarching physical or mathematical principle, or failing that an argument as to why there isn't one. I'd even be quite happy to be directed to a book or more appropriate forum. I apologise for not being clearer before and appreciate the answers already given.

Answer 1

The most basic definition of a fluid is

a fluid is a substance that continually deforms (flows) under an applied shear stress.

To model a fluid using the Euler equations, you need to satisfy the condition that the mean free path of a particle, $\ell$, is significantly smaller than the typical size of the domain, $L$ (and also that viscosity and heat conduction are negligible/zero). The ratio of $\ell/L$ gives one the Knudsen number.

You can calculate the mean free path via $$ \ell=\frac{1}{n}\frac{1}{\sqrt{2}\pi\sigma^2} $$ where $n$ is the number density and $\sigma$ the mean particle size. For a hydrogen gas cloud (astrophysical context here), we expect $n\sim10\rm cm^{-3}$ with $\sigma\sim10^{-8}\,\rm cm$ (diameter of hydrogen atom) which gives $$ \ell\sim10^{14}\,\rm cm $$ There are a few common scale lengths in astrophysics, two relevant ones are (a) the AU ($10^{13}$ cm) and (b) the parsec ($3\cdot10^{18}$ cm). Clearly (a) is less than the scale length so the fluid equations won't work for modeling it on this scale, whereas (b) the fluid approach would work (and is used).

As far as the more exotic situations, I presume that the fluid description is an approximation, but I'm not sure of the justifications for doing it (outside the Knudsen relation).

Answer 2

I think expecting "fluid" behaviour in terms of a material that does not support shear, is not useful in the context of the various systems you have listed in the question. Instead, I believe you are intuitively connecting ideas and concepts pertaining to conservation laws. So the idea that in specific systems, conserved charges (in the sense of Noether) have associated conserved currents or fluxes that cause it to "flow" within the system, respecting a continuity equation, is possibly when one uses the correspondence to fluids (in which, one typically has mass and momentum conservation, and possibly energy). This is about the extent of the connect to fluids, as the flow is in all likelihood not occurring in real space.

The other important thing to keep in mind is that continuum fluid dynamics is a phenomenological classical field theory (in the sense that it uses field variables instead of discrete objects to describe the system). This means that it applies only as a coarse-grained model at large length scales and long time scales. Hence, before using a fluid like or any field description of a system, it is imperative to understand the regime and separation of scales in the macroscopic problem.