Relationship To Mach and Reynolds Numbers in Gases
The Knudsen number can be related to the Mach number and the Reynolds number:
Noting the following:
Dynamic viscosity,
Average molecule speed (from Maxwell-Boltzmann distribution),
thus the mean free path,
dividing through by L (some characteristic length) the Knudsen number is obtained:
where
- is the average molecular speed from the Maxwell–Boltzmann distribution,
- T is the thermodynamic temperature,
- μ is the dynamic viscosity,
- m is the molecular mass,
- kB is the Boltzmann constant,
- ρ is the density, .
The dimensionless Mach number can be written:
where the speed of sound is given by
where
- U∞ is the freestream speed,
- R is the Universal gas constant, (in SI, 8.314 47215 J K−1 mol−1),
- M is the molar mass,
- is the ratio of specific heats, and is dimensionless.
The dimensionless Reynolds number can be written:
Dividing the Mach number by the Reynolds number,
and by multiplying by ,
yields the Knudsen number.
The Mach, Reynolds and Knudsen numbers are therefore related by:
Read more about this topic: Knudsen Number
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