289-291] FLOW THROUGH A PIPE. 523 



(Art. 72). If the axis of z be parallel to the length of the pipe, and if we 

 assume that w is a function of x, y only, then in the case of steady motion 

 the equations reduce to 



..... 



where V 1 2 =d 2 /dx 2 +d 2 /dy 2 . Hence, denoting by P the constant pressure- 

 gradient ( - dp/dz), we have 



Vl 2 w=-P/ij. ................................. (iv), 



with the condition that w=0 at the boundary. If we write >// 4o>(# 2 

 for w, and 2o> for P/p, we reproduce the conditions of the Art. referred to. 

 This proves the analogy in question. 



In the case of an elliptic section of semi-axes a, 6, we assume 



which will satisfy (iv) provided 



n P 

 C= 



The discharge per second is therefore 



This bears to the discharge through a circular pipe of the same sectional 

 area the ratio 2a6/(a 2 + & 2 ). For small values of the eccentricity (e) this 

 fraction differs from unity by a quantity of the order e 4 . Hence considerable 

 variations may exist in the shape of the section without seriously affecting 

 the discharge, provided the sectional area be unaltered. Even when a : b - 8 : 7, 

 the discharge is diminished by less than one per cent. 



291. We consider next some simple cases of steady rotatory 

 motion. 



The first is that of two-dimensional rotation about the axis of 

 z, the angular velocity being a function of the distance (r) from 

 this axis. Writing 



u = coy, v = o)0), ........................ (1) 



we find that the rates of extension along and perpendicular to the 

 radius vector are zero, whilst the rate of shear in the plane xy is 

 rdco/dr. Hence the moment, about the origin, of the tangential 

 forces on a cylindrical surface of radius r, is per unit length of 

 the axis, = /jurdco/dr . 2?rr . r. On account of the steady motion, 

 the fluid included between two coaxial cylinders is neither gaining 



* This, with corresponding results for other forms of section, appears to have 

 been obtained by Boussinesq in 1868 ; see Hicks, Brit. Ass. Rep., 1882, p. 63. 



