# How large is the “elementary volume” in a fluid

A fluid is described as a continuum with quantities such as pressure $P$, temperature $T$, particle density $n$, and macroscopic velocity $\vec{v}$ as functions of spatial coordinates $x^i$ and time $t$.

On top of that, we assume that at every point thermodynamical relations hold. I.e., if we describe an inhomogeneous fluid which, however, is an ideal gas at every point, we have $$P(x^i,t) = n(x^i,t) k_B T(x^i,t)$$ for every $x^i,t$.

From the point of statistical physics this is a non-trivial assumption because there we typically assume that we are describing infinitely large systems of infinitely large numbers of particles. Here, on the other hand, we are describing and infinitely small fluid element which (if $n$ is finitely large) also contains an infinitely small number of particles.

Hence, the fluid description can be valid only when there is some small volume $\delta V$ such that:

1. Computing thermodynamical properties of the fluid over $\delta V$ is sufficient for usual thermodynamics to apply.
2. $\delta V$ is small enough so that it is smaller than any macroscopic scale of the fluid. (A self consistency check is that any variability length-scale in the fluid is smaller than $\delta V^{1/3}$.)

This is all nice and beautiful, but in practice we need to know how large $\delta V$ would be under the given conditions. How do we determine this $\delta V$ over which the fluid approximation can be applied?

• Closest description I've seen to yours is that the mean free path of a particle is significantly smaller than the domain. Usually one defines a fluid as something that continually deforms under an applied stress (see this related question). – Kyle Kanos Jan 30 '17 at 17:29
• As an extension of my previous comment, the fluid description works for both water flowing in a pipe just as it does for a star blowing up, despite the massive differences in scale (a few meters vs a few parsecs). So I don't think there can be a singular $\delta V$. – Kyle Kanos Jan 30 '17 at 18:32
• @KyleKanos Well, to add some context to this: I want to understand the so-called gradient expansion which iteratively takes the behaviour away from a perfect fluid. The expansion can be understood as an expansion of small $\ell_?/\ell_v$, where $\ell_v \sim f/ \nabla f$ is the variability scale and $\ell_?$ is probably $\sim \delta V^{1/3}$. I am wondering how to check the validity of that expansion etc etc. – Void Jan 31 '17 at 8:42

For a gas to be describable as a continuum, the collisions between molecules must be sufficiently frequent that the gas appears to be, on the scale in which we are interested, free from random fluctuations. The relevant concept is the mean free path (mfp), which is the typical distance over which a molecule travels between collisions. For air at standard temperature and pressure this about $10^{-7}m$.
The relevant dimensioness number is the ratio between the mfp and the linear size of an object immersed in the gas, which is called the Knudsen number (Kn). $Kn<0.1$ is about the limit at which continuum concepts still apply. $Kn>10$ is the regime in which tracking of individual molecules is required. The intermediate regime is complicated both physically and mathematically. It is encountered by reentering space vehicles. The preferred discription is a seven-dimensional probability density function $f(\bf{x},\bf{u};t)$ that expresses the probability of finding a molecule with velocity $\bf{u}$ at a location $\bf{x}$ and a time t