84 for a variety of nonspherical particles. The ratio of the terminal velocity of fall for ellipsoids to the terminal velocity of spheres of the same volume is presented in one of the graphs to follow. These curves") have been computed from the theoretical formulas of Oberbeck,‘?) which agree with the limited number of experi- mental measurements available. In general, particles of irregular shape have smaller terminal velocities than spheres of the same volume. Equations (2) and (3) are not applicable to the motion of spheres in a gas if the diameter of the spherical particle is not considerably greater than the mean free path, A, of the air molecules. For ¢/A = 10, the terminal velocity as predicted by Stokes’ law is low by about 15 per cent, with the error increasing with decreasing d/X. A rough theoretical analysis of the drag for the range 2< d/A < 100 was made by Cunningham;‘*) a more complete theory was developed by Epstein.“ This range of flow is frequently referred to as the Stokes-Cunningham range. A study 50 drag formulas ore accurate to +10%) and Weber") have proposed a general empirical formula that fits the data not only in this range, but also in the ranges d < and d > A, In the latter two ranges, the empirical formula agrees with the theoretical expressions. The empirical formula, as used here, has constants adjusted to fit Millikan’s oil-drop data, and gives the following expression for the terminal velocity in air: o * uu 2 aq 10 °O Equation (4) also reduces to Eq. (3) for d/A >> 1. The computations made here are based on Eq. (4). Figures 7 through 10 give the results of the computations that have been prepated for this report. These figures are based on spherical particles of density, p’ = 2 gm/cm', and on the NACA standard atmosphere.” Figure 7 depicts the boundaries of the various flow regimes for different: altitudes and different sphere diameters. These curves are determined by the value of ellieelNoeletDolitoathe Stokes’ range, which is determined by the Reynolds number (R = 1). Figure 8 gives the terminal velocity as a function of particle diameter at various altitudes. Since the terminal velocity is proportional to the particle density p’, the values on the graphs may be readily adjusted to other values of p’. Figure 9 shows the times required for spherical particles to fall from a given initial altitude to sea level as a function of particle diameter. | O5 02 2 5 2 | 5 10 20 50 100 200 500 1000 Porticle diameter, ¢, in microns, p (104m) Fig. 7—Various flow regimesfor the free fall of spherical particles as a function of diameter and altitude Upper limit of Stokes’ flow 200 —~ 100 ‘ 5 50 % ~ 20 a Z 10 8 5 E « wv (4) & = 20 due again to Epstein.) Epstein’s formulas for the two ranges, d/A << 1 and 2 <d/A S100, depend on the ratio of the specular to the diffuse reflection in For the intermediate range, d ~ A, no theoretical formula is available. Knudsen z v £ 30 of the flow has also been made for d/A << 1, first by Cunningham and subse- the collisions between the particles and the air molecules. This ratio is not measured directly, but is chosen to fit observed drag values on small spheressettling in gases. (boundaries drawn at points where 40 quently by several other investigators. The definitive analysis for this range 1s “= re [1 + (1.644 + 0.55269 8584/4) /dJ. AY APPENDIX Ii WORLDWIDE EFFECTS OF ATOMIC WEAPONS # Z =! ' 5 z s 3 1?) € 05 02 ws r Initial SO fogitituse (km) Seo level o 3 & SE 5 VA nS Ol a2 3 1s 10 io* 2 5 2d p> 2 5 4? 2 5 Terminal velocity, # i9' 2 5, 2 5 19 2 5 107 2 (in cm/sec) Fig. 8—Terminalvelocity of spherical particles of density p’ = 2 gm/cm’, as a function of diameter, at various altitudes igures The effect of this neglect is unimportant for the long settling times and large distances of present interest, since the root-mean-square Brownian displacement varies as the square root of time and the gravity drift displacement varies linearly with time. Figure 10 indicates the effect of particle shape on settling velocity; it gives the