Cunningham correction factor

In fluid dynamics, the Cunningham correction factor, or Cunningham slip correction factor (denoted $C$ ), is used to account for non-continuum effects when calculating the drag on small particles. The derivation of Stokes' law, which is used to calculate the drag force on small particles, assumes a no-slip condition which is no longer correct at high Knudsen numbers. The Cunningham slip correction factor allows predicting the drag force on a particle moving a fluid with Knudsen number between the continuum regime and free molecular flow.

The drag coefficient calculated with standard correlations is divided by the Cunningham correction factor, $C$ , given below.

Ebenezer Cunningham^[1] derived the correction factor in 1910 and with Robert Andrews Millikan, verified the correction in the same year.

C=1+{\frac {2\lambda }{d}}\left(A_{1}+A_{2}e^{\frac {-A_{3}d}{\lambda }}\right)

where

$C$ is the correction factor
$λ$ is the mean free path
$d$ is the particle diameter
$A n$ are experimentally determined coefficients.

For air (Davies, 1945):^[2]

A₁ = 1.257

A₂ = 0.400

A₃ = 0.55

The Cunningham correction factor becomes significant when particles become smaller than 15 micrometers, for air at ambient conditions.

For sub-micrometer particles, Brownian motion must be taken into account.

References

^ Cunningham, E., "On the velocity of steady fall of spherical particles through fluid medium," Proc. Roy. Soc. A 83(1910)357. doi:10.1098/rspa.1910.0024
^ Davies, C. (1945). "Definitive equations for the fluid resistance of spheres". Proceedings of the Physical Society. 57 (4): 259. Bibcode:1945PPS....57..259D. doi:10.1088/0959-5309/57/4/301.

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[Cunningham-1] Cunningham, E., "On the velocity of steady fall of spherical particles through fluid medium," Proc. Roy. Soc. A 83(1910)357. doi:10.1098/rspa.1910.0024

[2] Davies, C. (1945). "Definitive equations for the fluid resistance of spheres". Proceedings of the Physical Society. 57 (4): 259. Bibcode:1945PPS....57..259D. doi:10.1088/0959-5309/57/4/301.

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