522 VISCOSITY. [CHAP, xi 



if X = /i//3. This determines B, in (2), so that 



(ii). 



If X/a be small, this gives sensibly the same law of velocity as in a tube of 

 radius a + \, on the hypothesis of no slipping. The corresponding value 

 of the flux is 



If X were more than a very minute fraction of a in the narrowest tubes 

 employed by Poiseuille [a = 001 5 cm.J a deviation from the law of the fourth 

 power of the diameter, which was found to hold very exactly, would become 

 apparent. This is sufficient to exclude the possibility of values of X such as 

 235 cm., which were inferred by Helmholtz and Piotrowski from their 

 experiments on the torsional oscillations of a metal globe filled with water, 

 described in the paper already cited*. 



The assumption of no slipping being thus justified, the comparison of the 

 formula (4) with experiment gives a very direct means of determining the 

 value of the coefficient p, for various fluids. 



It is easily found from (3) and (4) that the rate of shear close 

 to the wall of the tube is equal to 4w /a, where w is the mean 

 velocity over the cross-section. As a numerical example, we may 

 take a case given by Poiseuille, where a mean velocity of 126 6 c. s. 

 was obtained in a tube of 01134 cm. diameter. This makes 

 4&amp;gt;w /a 89300 radians per second of time. 



290. Some theoretical results for sections other than circular 

 may be briefly noticed. 



1. The solution for a channel of annular section is readily deduced from 

 equation (2) of the preceding Art., with A retained. Thus if the boundary- 

 conditions be that w = Q for r = a and r=b, we find 



giving a flux 



2. It has been pointed out by Greenhillt that the analytical conditions 

 of the present problem are similar to those which determine the motion of a 

 Motionless liquid in a rotating prismatic vessel of the same form of section 



* For a fuller discussion of this point see Whetham, &quot; On the Alleged Slipping 

 at the Boundary of a Liquid in Motion,&quot; Phil. Trans., 1890, A. 



f &quot; On the Flow of a Viscous Liquid in a Pipe or Channel,&quot; Proc. Lond. Math. 

 Soc., t. xiii. p. 43 (1881). 



