fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...

, 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 In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by s ...

, which is used to calculate the drag force on small particles, assumes a

no-slip condition In fluid dynamics, the no-slip condition for viscous fluids assumes that at a solid boundary, the fluid will have zero velocity relative to the boundary. The fluid velocity at all fluid–solid boundaries is equal to that of the solid boundary. C ...

which is no longer correct at high

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 ...

. The Cunningham slip correction factor allows predicting the

drag force In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...

on a particle moving a fluid with

between the continuum regime and

free molecular flow Free molecular flow describes the fluid dynamics of gas where the mean free path of the molecules is larger than the size of the chamber or of the object under test. For tubes/objects of the size of several cm, this means pressures well below 10� ...

. The

drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equ ...

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

Ebenezer Cunningham Ebenezer Cunningham (7 May 1881 in Hackney, London – 12 February 1977) was a British mathematician who is remembered for his research and exposition at the dawn of special relativity. Biography Cunningham went up to St John's College, Camb ...

Cunningham, E., "On the velocity of steady fall of spherical particles through fluid medium," ''Proc. Roy. Soc. A'' 83(1910)357. derived the correction factor in 1910 and with

Robert Andrews Millikan Robert Andrews Millikan (March 22, 1868 – December 19, 1953) was an American experimental physicist honored with the Nobel Prize for Physics in 1923 for the measurement of the elementary electric charge and for his work on the photoelectric e ...

, verified the correction in the same year. :

C = 1+ \frac\cdot (A_1+A_2\cdot e^)

where :''C'' is the correction factor :λ is the

mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...

:''d'' is the particle diameter :''A_n'' are experimentally determined coefficients. :For air (Davies, 1945): ::''A''₁ = 1.257 ::''A''₂ = 0.400 ::''A''₃ = 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 Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

must be taken into account.

References

Fluid dynamics Dimensionless numbers Aerosols {{fluiddynamics-stub