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Navier-Stokes is only an approximation
A smooth-fluid assumption
The Navier-Stokes equations presume that a fluid is a continuous substance that flows perfectly smoothly, no matter how far you zoom in. Zoom in far enough on a real one and tiny blips appear in the flow, caused by its grainy makeup of molecules and atoms. The equations lop those blips off, which makes them an approximation of the real flow.
How physics moved on
Most of physics took a different path. As the 1900s revealed the structure of the world at smaller and smaller scales, physicists revamped their theories of matter to account for atoms. Those new theories were approximations too, since tracking every last atom is impossible, but over the decades physicists rewrote and re-derived them in a more atom-friendly way. The theory of fluids kept its 19th-century form.
A rebuilt theory
A 20-year effort to rebuild the theory of fluids from the ground up has now produced a theory that accounts for the swarms of microscopic bits. It defines a fluid by its symmetries, and under it the Navier-Stokes equations are a consequence of those symmetries, which explains why the equations take the forms they do.
Why do fluids look smooth if made of molecules?
At everyday scales, a tiny volume of fluid still holds an enormous number of molecules, so their individual jostling averages out. A cubic centimeter of air at sea level contains roughly 2.5 times 10 to the 19th molecules. The average distance an air molecule travels between collisions is about 70 nanometers. Anything much larger than that sees air as a smooth, continuous substance.
Why is Navier-Stokes so hard to solve?
The equations contain terms where the fluid's velocity multiplies itself, which makes them nonlinear. Small changes can grow into large, chaotic swirls, which is what turbulence is. Exact solutions exist only for a few simplified cases. In 2000, the Clay Mathematics Institute listed the question of whether smooth solutions always exist in three dimensions as one of seven Millennium Prize Problems, with a prize of one million dollars.
Where do the equations stop working?
They fail when the distance between molecular collisions becomes comparable to the size of the flow. That happens in very thin air at high altitude, around spacecraft re-entering the atmosphere, and in channels only nanometers wide. Engineers handle those cases with a method called direct simulation Monte Carlo, which tracks sample molecules and their collisions. The Australian aerospace engineer Graeme Bird developed it in the 1960s.
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