On the Resistance Experienced by Spheres in their Motion through Gases

Abstract

Kinetic theory of the resistance to a sphere moving through a gas.— (1) Droplets small in comparison with the mean free path. The high degree of accuracy achieved in the experimental determination of the law of motions of droplets through gases, makes a careful theoretical examination of the problem desirable. Assuming the usual Maxwellian distribution of velocities in the gas, the force exerted by the impinging molecules is found to be $M$ where $M=\left(\frac{4\pi }{3}\right) \mathrm{Nm}{a}^2{c}_{m}V$, $N$, $m$, $a$, and $c_m$ being the number per unit volume, mass, radius, and mean speed of the molecules and $V$ the speed of the droplet. The force exerted by the molecules leaving the surface depends on how they leave. (1) For uniform evaporation from the whole surface, the force is $-M$; (2) for specular reflection of all the impinging molecules, $-M$; (3) for diffuse reflection with unchanged distribution of velocities, $-\left(\frac{13}{9}\right)M$; (4) for diffuse reflection with the Maxwell distribution corresponding to the effective temperature of the part of the surface they come from, $-\left(\frac{1+9\pi }{64}\right)M$, for a non-conducting droplet (4a), and $-\left(\frac{1+\pi }{8}\right)M$, for a perfectly conducting droplet (4b). Cases (1) and (2) can not be distinguished experimentally, but (2) is more probable physically. The experimental values agree with 1/10 specular reflection, case (2), and 9/10 diffuse reflection, case (4a) or (4b). For large values of $\frac{l}{a}$, the droplet behaves like a perfect conductor, case (4b). (2) Comparatively large spheres. The distribution of velocities is no longer Maxwellian because of the hydrodynamic stresses which can not now be neglected. The new law is derived (Eq. 47). The conditions at the surface of the sphere are discussed and it is shown that the diffusely reflected molecules have a Maxwellian distribution corresponding to the temperature and density of the gas, just as though they were reflected with conservation of velocity (specularly). The assumptions of Bassett are theoretically justified and a complete confirmation is obtained for the correction factor for Stokes' law [$1+0.7004 \left(\frac{2}{s-1\mathrm{}}\right) \left(\frac{l}{a}\right)$] on which Millikan's conclusions are based, especially as to the percentage of specular reflection. (3) Rotating spheres are also considered in an appendix, and the values of the resistance are derived for various cases.