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 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 )]