Cunningham correction factor

In fluid dynamics, the Cunningham correction factor or Cunningham slip correction factor is used to account for noncontinuum 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 number. 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 verified with Robert Andrews Millikan the correction in the same year.

${\displaystyle C=1+{\frac {2\lambda }{d}}\cdot (A_{1}+A_{2}\cdot e^{\frac {-A_{3}\cdot d}{\lambda }})}$

where

C is the correction factor
λ is the mean free path
d is the particle diameter
An 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

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: 259. doi:10.1088/0959-5309/57/4/301.