Definitive equations for the fluid resistance of spheres

For calculation of terminal velocities it is convenient to express the Reynolds' number, Re, of a moving sphere as a function of the dimensionless group ψRe², where ψ is the drag coefficient. The following equations have been fitted by the method of least squares to critically selected data from a number of experimenters:

1. Re = ψRe²/24 -0.00023363(ψRe²)² + 0.0000020154(ψRe²)³ - 0.0000000069105(ψRe²)⁴ for Re<4 or ψRe²<140. This tends to Stokes' law for low values ofRe. It is specially suited to calculation of the sedimentation of air-borne particles. The upper limit corresponds to a sphere weighing 1.5 μg. falling in the normal atmosphere, that is, one having a diameter of 142 μ for unit density.

2. logRe=-1.29536+0.986 (logψRe²)-0.046677 (logψRe²)²+0.0011235 (logψRe²)³ for 3<Re<10,000 or 100<ψRe²<4.5.10⁷.

Correction for slip in gases should be applied to Stokes' law by the following expression, based on the best results available:

1 + l/a[1.257 + 0.400exp(-1.10a/l)],

where the mean free path l is given by η/0.499σc.

This conveniently transforms to the following for the sedimentation of particles in air at pressure p cm. mercury

1 + l/pa[6.32.10^-4 + 2.01.10^-4exp(-2190ap)]