378 Mr.W. Sutherland on Thermal 



as the ends of the tube, the gas enters at one end with 

 velocity and leaves at the other with velocity u ; in unit 

 time the mass nmku passes out with momentum nmAu 2 , and 

 this therefore is the force exerted by the tube on the gas in 

 it ; this force acts only near the entrance in the part where 

 the velocity is rising from to u, so that in this part the total 

 traction exceeds the total friction by nmAu 2 . In the velocity 

 u we have the cause of thermal transpiration, while that of 

 radiometer motion is implied in the equation 



total unequilibrated traction = nmAu' i . . . (5) 



If the spaces at the ends of the tube instead of being 

 infinite are finite, the gas will flow till a fall of pressure is 

 established to arrest it, but we cannot secure that u = Q all 

 over any section of the tube by an application of pressure, 

 because the flow established by excess of pressure at one end 

 of a capillary tube is not of uniform velocity throughout each 

 section, but has a maximum velocity at the axis and a 

 minimum at the surface ; hence to secure that there shall be 

 no total flow in such a tube we have to establish a difference 

 of pressure which acting alone would discharge a volume Au 

 per unit time in the opposite direction to that of u. Thus, 

 then, our solution for the motion in a conducting tube when 

 there is no total flow of gas along it consists of the super- 

 position of a uniform velocity u and opposite velocities varying 

 in conformity with the laws of flow in a capillary tube of 

 uniform temperature, the result being to give a surface of 

 zero velocity somewhere between the axis and the wall, with 

 a circulation going up between this surface and the wall, and 

 backward between this surface and the axis. 



According to the theory of the flow of gas in a capillary 

 tube, if dp/cLv or p' is the rate of fall of pressure along the 

 tube, where the pressure is p and w is the viscosity, then B, 

 the volume measured at p delivered in unit time from a 

 circular tube of radius R (0. E. Meyer, Pogg. Ann. cxxvii.), is 



B = 7r/RV8^ (6) 



when the slipping of the gas on the walls can be neglected ; 

 but if slipping is to be taken account of let its coefficient be 

 £ ; then 



B=7r/R 4 (l+4f/R)/8i? (7) 



As the importance of £ depends entirely on its ratio to R, and 

 as we wish to discuss tubes of auy minuteness whatever, a 

 discussion of slipping becomes of first-rate importance to the 

 subject in hand. 



