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Knudsen number

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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Definition

The Knudsen number is defined as:

$\mathit{Kn} = \frac {\lambda}{L} = \frac {k_B T}{\sqrt{2}\pi\sigma^2 P L}$

where

$λ$ = mean free path (m)^*
L = representative physical length scale (m)
k_B = Boltzmann's constant (1.38×10^-23 J/K)
T = temperature (K)
$σ$ = particle diameter (m)
P = total pressure (Pa)

(* For particle dynamics in the atmosphere, and assuming standard temperature and pressure, i.e. 25°C, 1 atm, we have $λ$ = 8×10^-8 m. )

Application

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).