[vhffm]

If you are interested in some of the details:

The Navier-Stokes equations can be derived from the Boltzmann equation by applying a slight perturbation, expanding the result as a series, and taking the moments.

Taking the moments is essentially an integration, which comes with the implicit assumption that the system you're describing has sufficiently many particles. When running low on particles, this integration does not make sense. This is why the resulting equations do not apply at low densities.

The Navier-Stokes equations are the second order expansion of this procedure. The result of the first order expansion are the Euler equations.

This is called the Chapman-Enskog procedure. It's really quite illuminating when you see it for the first time. There's a great derivation in [1] if you can get your hand on it.

[1] http://www.uscibooks.com/shu3.htm

[orbifold]

When I saw this derivation during a course Theoretical Astrophysics it was indeed very enlightening, what is interesting is that it easily generalises to magneto hydrodynamics and other more complicated situations (mixture of multiple different fluids, fluids that react with each other etc.). I believe Landau Lifshitz contains some of them.