Definitive equations for the fluid resistance of spheres

TLDR

Reynolds number of a moving sphere is expressed as a function of the dimensionless group ψ Re², which reduces to Stokes’ law at low Re and is particularly useful for calculating sedimentation of air‑borne particles, with an upper limit corresponding to a 1.5 µg sphere (≈142 µm diameter) in normal atmosphere. The authors provide least‑squares fitted equations for Re as a function of ψ Re² over different ranges, a logarithmic fit for intermediate Re, and a slip‑correction factor for gases based on mean free path, which can be transformed for particle sedimentation in air at various pressures.

Abstract

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 ψ Re 2 , 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: Re = ψ Re 2 /24 -0.00023363(ψ Re 2 ) 2 + 0.0000020154(ψ Re 2 ) 3 - 0.0000000069105(ψ Re 2 ) 4 for Re <4 or ψ Re 2 <140. This tends to Stokes' law for low values of Re. 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. log Re =-1.29536+0.986 (logψ Re 2 )-0.046677 (logψ Re 2 ) 2 +0.0011235 (logψ Re 2 ) 3 for 3< Re <10,000 or 100<ψ Re 2 <4.5.10 7 . 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(-1.10 a / 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 -4 exp(-2190 ap )]