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. This length scale could be, for example, the radius of a body in a fluid. The number is named after Danish physicist Martin Knudsen (1871–1949).
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The Knudsen number is a dimensionless number defined as:
where
For an ideal gas, the mean free path may be readily calculated so that:
where
For particle dynamics in the atmosphere, and assuming standard temperature and pressure, i.e. 25 °C and 1 atm, we have λ ≈ 8 × 10^{−8} m, or approximately 2.6 × 10^{−9} ft.
The Knudsen number can be related to the Mach number and the Reynolds number:
Noting the following:
Average molecule speed (from MaxwellBoltzmann distribution),
thus the mean free path,
dividing through by L (some characteristic length) the Knudsen number is obtained:
where
The dimensionless Mach number can be written:
where the speed of sound is given by
where
The dimensionless Reynolds number can be written:
Dividing the Mach number by the Reynolds number,
and by multiplying by ,
the Knudsen number is obtained.
The Mach, Reynolds and Knudsen numbers are therefore related by:
The Knudsen number is useful for determining whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used: If the Knudsen number is near or greater than one, the mean free path of a molecule is comparable to a length scale of the problem, and the continuum assumption of fluid mechanics is no longer a good approximation. In this case statistical methods must be used.
Problems with high Knudsen numbers include the calculation of the motion of a dust particle through the lower atmosphere, or the motion of a satellite through the exosphere. The solution of the flow around an aircraft has a low Knudsen number. Using the Knudsen number an adjustment for Stokes' Law can be used in the Cunningham correction factor, this is a drag force correction due to slip in small particles (i.e. d_{p} < 5 µm).

