150 Tables 136-137. 



TABLE 136. — Aerodynamics. 

 Kinetics of Bodies in Resisting Medium. 



The differential equation of a body falling in a resisting medium is dtt'dt = g — kifi. The ve- 

 locity tends asymptotically to a certain terminal velocity, V = -^'glk. Integration gives w =- 

 ¥• tanh {gt/V), x =—' log cosh (gt/V) \l u = x = t =^ o. 



When body is projected upwards, du/dt = —g _ ku'^, and if «o 'S velocity of projection, then 

 tan-i u/V = tan-i [tio/V) -gt/ F, x ={V^/2g) log (F^ + «o^) (V^ + u^). The particle comes to 

 rest when / = {V / g) tarr^ («o/F) andx = (F/sg) log (i - mVV^). 



For small velocities the resistance is more nearly proportional to the velocity. 



Stokes' Law for the rate of fall of a spherical drop of radius a under gravity g gives for the 

 velocity, v, 



. = — (cr-p), 



where a and p are the densities of the drop and the medium, 17 the viscosity of the medium. 

 This depends on five assumptions: (i) that the sphere is large compared to the inhomogeneities 

 of the medium; (2) that it falls as in a medium of unlimited extent; (3) that it is smooth and 

 rigid; (4) that there is no slipping of the medium over its surface; (5) that its velocity is so 

 small that the resistance is all due to the viscosity of the medium and not to the inertia of the 

 latter. Because of 5, the law does not hold unless the radius of the sphere is small compared 

 with yj/vp (critical radius). Arnold showed that a must be less than 0.6 this radius. 



If the medium is contained in a circular cylinder of radius R and length L, Ladenburg showed 

 that the following formula is applicable (Ann. d. Phys. 22, 287, 1907, 23, 447, 1908): 



y ^2 gaHoi^p) 



9 r]{i + 2.4a/R) (i + z.\alL) 



As the spheres diminish in size the medium behaves as if inhomogeneous because of its molec- 

 ular structure, and the velocity becomes a function of II a, where / is the mean free path of the 

 molecules. Stokes' formula should then be modified by the addition of a factor^ viz.: 



1,1 = - 1^ ((T - p) j I + (0.864 + 0.29^ '-25 WO) - I 

 (See chapter V, Millikan, The Electron, 191 7 ; also Physical Review 15, p. 545, 1920.) 



TABLE 137. —Flow of Gases through Tubes.* 



"When the dimensions of a tube are comparable with the mean free path (Z) of the molecules of 

 a gas, Knudsen (Ann. der Phys. 28, 75, 199, 1908) derives the following equation correct to 5% 

 even when DjL = 0.4: Q, the quantity of gas in terms of /"F which flows in a second through a 

 tube of diameter D, length /, connecting two vessels at low pressure, difference of pressure 

 /'2 — A. equals {P2 — Pi)/l^\/'p where pis the density of the gas at one bar (i dyne/cm^) = (mo- 

 lecular weight)/(S3.i5 X lO^T^) and /F,* which is of the nature of a resistance, = 2. 394i//Z>8 + 

 3.i84/Z)2. The following table gives the cm^ of air and Nat i bar which would flow through dif- 

 ferent sized tubes, difference of pressure i bar, room temperature. 



/ = icm. D = icm. W = 5.58 Q, cm^ of air, 5200. cm^ of I/2, 19700. 



10 I 27.1 1070. 4050. 



I 0.1 2710. 10.7 40.5 



10 0.1 24300. 1.20 3.60 



Knudsen derives the following equation, equivalent to Poiseuille's at higher, and to the above at 

 lower ])ressures : 

 Q = (/'g-y'i) [aP +3(1+ ciP)/(i + c-iP)} where a = ■irD*Ii2?,-fil (Poiseuille's constant) ; * = 



1/ fF\/p7 (coefficient of molecular flow) ; ci = ^'p D/i); and C2 = 1.24 x/p" /?/??; »j = viscosity coef- 

 ficient. The following are the volumes in cm^ at i bar, 20°C, that flow through tube, D = \ cm, 

 / = locm. Pi — P\ - I bar, average pressure of /"bars: 



P = 10.6 Q = 13,000,000. -^ = 5- Q = 1026. P = I. Q = 1044. cm^ 



100. 2,227. 4. 1024. 0.1 1065. 



10. 1,058. 3. 1025. o.oi 1070. 



When the velocity of flow is below a critical value, F (density, viscosity, diameter of tube), the 

 stream lines are parallel to the axis of the tube. Above this critical velocity, F„, the flow is tur- 

 bulent. F„ = Utj pr for small pipes up to about 5 cm diameter, where A' is a constant, and r the 

 tube radius. When these are in cgs units, /& is 10^ in round numbers. Below Fc the pressure drop 

 along the tube is proportional to the velocity of gas flow ; above it to the square of the velocity. 



* See Dushman, The Production and Measurement of High Vacua, General Elec. Rev. 23, p. 493, 19J0 

 Smithsonian Tables. 



