The Physics of Thin Air
The Knudsen number ($Kn$) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. It is named after Danish physicist Martin Knudsen.
In standard fluid dynamics (like water in a pipe or air over a car), we assume the fluid is a continuous, unbroken 'continuum.' However, if you have an extremely small pipe (nanotechnology), or if the air is extremely thin (outer space), you can no longer ignore the fact that air is actually made of discrete, bouncing molecules. The Knudsen number tells you exactly when the continuum assumption violently breaks down.
Mean Free Path
The "mean free path" is the average distance a gas molecule travels before colliding with another gas molecule.
- At sea level: The mean free path is microscopic (about 68 nanometers). Molecules collide constantly, acting like a fluid.
- In Low Earth Orbit: The air is so thin that a molecule might travel kilometers before hitting another one.
If the mean free path is larger than your physical object (like a tiny satellite), the air doesn't flow around it; individual molecules simply smash into it like microscopic bullets. This is called 'free molecular flow.'
The Formula
Example Calculation
A tiny MEMS (Micro-Electromechanical System) device has a functional channel width of $1 \cdot 10^{-6} , \text{meters}$ (1 micrometer). The mean free path of the air inside it is $6.8 \cdot 10^{-8} , \text{meters}$ (68 nanometers).
- Divide Mean Free Path by Length: $(6.8 \cdot 10^{-8}) / (1 \cdot 10^{-6}) = 0.068$.
The Knudsen number is $0.068$. Because it is between $0.01$ and $0.1$, the flow is in the 'slip flow' regime, meaning standard fluid dynamics equations will be slightly inaccurate and must be modified to account for molecules 'slipping' along the walls.