Editing Knudsen number (section)

==Relationship to Mach and Reynolds numbers in gases==
The Knudsen number can be related to the [[Mach number]] and the [[Reynolds number]].

Using the [[dynamic viscosity]]
:<math>\mu = \frac{1}{2}\rho \bar{c} \lambda,</math>

with the average molecule speed (from [[Maxwell–Boltzmann distribution]])
:<math>\bar{c} = \sqrt{\frac{8 k_\text{B} T}{\pi m}},</math>

the [[mean free path]] is determined as follows:<ref name = "thermal">{{cite journal | last1= Dai | first1= W. |display-authors=etal | title= Influence of gas pressure on the effective thermal conductivity of ceramic breeder pebble beds | journal = [[Fusion Engineering and Design]] | year=2017 | volume=118| pages= 45–51|doi= 10.1016/j.fusengdes.2017.03.073 | bibcode= 2017FusED.118...45D }}</ref>
:<math>\lambda = \frac{\mu}{\rho} \sqrt{\frac{\pi m}{2 k_\text{B} T}}.</math>

Dividing through by ''L'' (some characteristic length), the Knudsen number is obtained:
:<math> \mathrm{Kn}\ = \frac{\lambda}{L} = \frac{\mu}{\rho L} \sqrt{\frac{\pi m}{2 k_\text{B} T}},</math>

where
: <math>\bar{c}</math> is the average molecular speed from the [[Maxwell–Boltzmann distribution]] [L<sup>1</sup> T<sup>−1</sup>],
: ''T'' is the [[thermodynamic temperature]] [θ<sup>1</sup>],
: ''μ'' is the [[dynamic viscosity]] [M<sup>1</sup> L<sup>−1</sup> T<sup>−1</sup>],
: ''m'' is the [[molecular mass]] [M<sup>1</sup>],
: ''k<sub>B</sub>'' is the [[Boltzmann constant]] [M<sup>1</sup> L<sup>2</sup> T<sup>−2</sup> θ<sup>−1</sup>],
: <math>\rho</math> is the density [M<sup>1</sup> L<sup>−3</sup>].

The dimensionless Mach number can be written as
:<math>\mathrm{Ma} = \frac {U_\infty}{c_\text{s}},</math>

where the speed of sound is given by
:<math>c_\text{s} = \sqrt{\frac{\gamma R T}{M}} = \sqrt{\frac{\gamma k_\text{B}T}{m}},</math>

where
: ''U<sub>∞</sub>'' is the freestream speed [L<sup>1</sup> T<sup>−1</sup>],
: ''R'' is the Universal [[gas constant]] (in [[SI]], 8.314 47215 J K<sup>−1</sup> mol<sup>−1</sup>) [M<sup>1</sup> L<sup>2</sup> T<sup>−2</sup> θ<sup>−1</sup> mol<sup>−1</sup>],
: ''M'' is the [[molar mass]] [M<sup>1</sup> mol<sup>−1</sup>],
: <math>\gamma</math> is the [[ratio of specific heats]] [1].

The dimensionless [[Reynolds number]] can be written as
:<math>\mathrm{Re} = \frac {\rho U_\infty L}{\mu}.</math>

Dividing the Mach number by the Reynolds number:

:<math>\frac{\mathrm{Ma}}{\mathrm{Re}} = \frac{U_\infty / c_\text{s}}{\rho U_\infty L / \mu} = \frac{\mu}{\rho L c_\text{s}} = \frac{\mu}{\rho L \sqrt{\frac{\gamma k_\text{B} T}{m}}} = \frac{\mu}{\rho L} \sqrt{\frac{m}{\gamma k_\text{B} T}}</math>

and by multiplying by <math>\sqrt{\frac{\gamma \pi}{2}}</math> yields the Knudsen number:

:<math>\frac{\mu}{\rho L} \sqrt{\frac{m}{\gamma k_\text{B}T}} \sqrt{\frac{\gamma \pi}{2}} = \frac{\mu}{\rho L} \sqrt{\frac{\pi m}{2k_\text{B} T}} = \mathrm{Kn}.</math>

The Mach, Reynolds and Knudsen numbers are therefore related by

:<math>\mathrm{Kn}\ = \frac{\mathrm{Ma}}{\mathrm{Re}} \sqrt{\frac{\gamma \pi}{2}}.</math>