As an aside, as the question of the article is more about the mathematics as opposed to the physics: Navier-Stokes assuming a continuum is one of the fundamenta

As an aside, as the question of the article is more about the mathematics as opposed to the physics: Navier-Stokes assuming a continuum is one of the fundamental assumptions made in basic fluid dynamics, and to an extend applies to the governing equations for solid mechanics, in a whole field known as Continuum Mechanics. The continuum assumption, as it turns out, is actually incredibly accurate for most gasses that are comprised of discrete molecules. Specifically, the region in which N-S is valid for is when Knudsen Number[1] is less than about 0.01. Knudsen Number essentially characterizes how densely packed the particles are. This turns out to be most of the flows we observe on Earth. N-S becomes less accurate at around Kn = 0.1 and completely useless at Kn >= 1, as in this region the differential arguments no longer hold and NS predicts something non-physical. Some examples of flows in this regime: mass spectrometer, reentry, and inside shockwaves. In that view, you're not wrong that NS will eventually fail, but it is just applicable to most of what we do. For those working with fluids, it is important to recognize for when NS fails so they can switch to a different model for solutions. For higher Kn flows, such as Kn > 10, you can feasibly track individual particles and their trajectories. People do this for satellites and what not. However, between the range of 0.1 < Kn < 10, we do not have a good enough set of models that is able to compute accurate answers to reality with reasonable time frames and is an active field of research. If you're interested in learning more about this topic, look up the Kinetic Theory of Gas, where particles are accounted with the evolution probability distributions in temporal, spatial, and velocity space. [1]: https://en.wikipedia.org/wiki/Knudsen_number https://en.wikipedia.org/wiki/Knudsen_number