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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.

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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.

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Do similar problems occur when applying the lagrangian to ropes, slinkies, or other continuous objects? –  JamesMarshallX Feb 17 '12 at 2:07
    
@JamesMarshallX: for solid objects the theory is different. In fluids the particles are free to move around and bump into each other. For solids, the accepted description for the mechanical (as opposed to quantum mechanical or thermodynamic) properties for most of them assumes the atoms to be more or less fixed (relative to other atoms). This assumption vastly changes the nature of the problem: you are no longer trying to derive the continuum limit from Newton's laws since what governs bulk motion is interatomic/particle forces, and not "billiard dynamics". –  Willie Wong Feb 17 '12 at 9:10
    
Consider transverse wave propagation on a piece of string. The original derivation of the wave equation by D'Alembert assumes a chain of particles each joined to the next by springs governed by Hooke's law. The derivation from there on is plenty fine; the question one should make then is whether that particular assumption of the structure of a piece of string and the assumption on the form of the intermolecular forces are valid. In other words: in fluids we have three levels of models: molecular, kinetic, and continuum. For a rope, I don't know if there is a widely accepted molecular model. –  Willie Wong Feb 17 '12 at 9:19

The theory of fluids introduces material parameters in the stress tensor, which help model the substance. "The viscosity coefficient is the proportionality constant relating a velocity gradient in a fluid to the force required to maintain that gradient. The thermal conductivity is the proportionality constant relating the temperature gradient across a fluid to the flux of energy, that is, Fourier's law of heat conduction. Finally, the diffusion coefficient is the proportionality constant relating the gradient in species concentration of the mass flux." Of course it only works if it works.

There is the self consistent derivation of the Navier–Stokes equations, w.r.t. several conservation laws. But relevant for you are the considerations involving Boltzmann equation, a formalism for gases in the microscopic regime. Here, for many systems, you can find macroscopic expectation values, which validate fluid dynamics and give microscopic explanations for the viscosities, etc. The results are then usually said to "also hold for liquid systems".

For the limit, one might assume a pertubation of the Maxwell-Boltzmann distribution $f(t,\vec x)$, which weakly depends on space and time. This is the relaxation time approximation, or the $0.5^{th}$ order in Chapman–Enskog theory. From this one can compute average (particle) densities, mean velocities and average kinetic energies (temperatures). For example $$\vec V:=\langle \vec v\rangle$$ $$T(t,\vec x):=\frac{1}{3}\left\langle m\left(\vec v-\vec V\right)^2 \right\rangle,$$ where $\langle\dots\rangle$ is the mean w.r.t. the particle distribton given by $f(t,\vec x)$. This procedure gives a macroscopic/net velocity and a local temperature distribution. It eventually fulfulls an equation of state, like $P=\frac{N}{V}k_B T$, or rather $$P(t,\vec x)=n(t,\vec x)\ T(t,\vec x),$$ which relate macroscopic quantities like temperature to pressure, which is also given as mean and which is part of the stress tensor I mentioned above. The differential equations, which governs the dynamics of the fluids ultimately stem from momentum conservation in both cases. See Kerson Huang: Statistical Mechanics, 2nd Edition for a derivation.

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@JamesMarshallX: I see what you did there. –  NikolajK Feb 17 '12 at 10:11

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