models - Why are continuum fluid mechanics accurate when constituents are discrete objects of finite size?

Suppose we view fluids classically, i.e., as a collection of molecules (with some finite size) interacting via e&m and gravitational forces. Presumably we model fluids as continuous objects that satisfy some differential equation. What mathematical result says that modeling fluids as continuous objects can accurately predict the discrete behavior of the particles? I don't know anything about fluid mechanics, so my initial assumption may in itself be wrong.

Answer

There are many physical intuitions often presented in various texts on fluid dynamics. I won't mention those here. I will, however, mention that mathematically the passage from a particle point of view to a continuum point of view is still a largely un-resolved problem. (With suitable interpretation, this problem was already posed by Hilbert as his 6th of 23 problems.)

We can interpret the problem as one of starting from "a Newtonian description of particles interacting through collisions" and try to end up with "an approximation of the physical system by a continuum obeying certain laws of fluid dynamics (Euler, Navier-Stokes, etc.)"

Most work up through now takes an intermediate step through the Boltzmann equation: in this kinetic theory model, instead of individual particles we consider distributions of particles, where the "density" of particles is given based on both position and velocity. So it makes one level of continuum approximation. But it still keeps the facet of Newtonian theory where particles interact through direct collisions. Under an assumption known as molecular chaos (more on this later), that Boltzmann's equation follows from Newtonian laws of motion have been demonstrated, to various degrees of rigour, by Boltzmann himself, as well as Grad, Cercignani, and Lanford, building on the work of Bogoliubov, Born, Green, Kirkwood, and Yvon. For a mathematically sophisticated, but more or less self-contained description one can refer to Uchiyama's write-up. There are a few issues with this derivation.

1. The problem of potentials. The derivations listed above made assumption that the particles are hard spheres: that the only interaction between two particles is when they actually collide (so no inter-molecular forces mediated by electromagnetism, like hydrogen bonds and such), and that the particles are spherical. This is satisfactory for monatomic gases, but less so for diatomic molecules or ones with even stranger shapes. Most people don't think of this as a big problem though.


2. The derivation is only valid under the so-called Grad limit assumption. To take the continuum limit, generally the assumption is made that the particle diameter decreases to zero, while the number of particles (per unit volume) increases to infinity. Exactly how these two limits balance out affects what the physical laws look like in the continuum limit. The Grad limit assumes that the square of the particle diameter scales like the inverse of the number density. This means that the actual volume occupied by the particles themselves (as opposed to the free space between the particles) decreases to zero in this limit. So in the Grad limit one actually obtains an infinitely dilute gas. This is somewhat of a problem.


3. The derivation also makes use of what is called molecular chaos: it assumes that, basically speaking, the only type of collision that matters is that between two particles, and that the particle, after its collision, "forgets" about its previous zig-zag among its cousins in the dilute gas. In particular, we completely ignore the case of three or more particles colliding simultaneously, and sort of ignore the billiards-trick-shot like multiple bounces. While both of this can be somewhat justified based on physical intuition (the first by the fact that if you have a lot of small particles spaced far apart, the chances that three of them hit at the same time is much much much smaller than two of them colliding; the second by the fact that you assume some sort of local thermodynamic equilibrium [hence the name molecular chaos]), one should be aware that they are taken as assumptions in the Boltzmann picture.

Starting from the Boltzmann equation, one can arrive at the Euler and Navier-Stokes equations with quite a lot of work. There has been a lot of recent mathematical literature devoted to this problem, and under different assumptions (basically how the Reynolds and Knudsen numbers behave in the limit) one gets different versions of the fluid equations. A decent survey of the literature was written by F. Golse, while a heavily mathematical discussion of the state-of-the-art can be found in Laure Saint-Raymond's Hydrodynamic limits of the Boltzmann equation.

It is perhaps important to note that there are still regimes in which the connection between Boltzmann equation and the fluid limits are not completely understood. And more important is to note that even were the connections between the kinetic (Boltzmann) picture and the fluid limits, there is still the various assumptions made during the derivation of the Boltzmann equation. Thus we are still quite far from being able to rigorously justify the continuum picture of fluids from the particle picture of Newtonian dynamics.