536 VISCOSITY. [CHAP, xi 



through a channel bounded by a hyperboloid of revolution (of one 

 sheet)*. 



Some examples of a different kind, relating to two-dimensional 

 steady motions in a circular cylinder, due to sources and sinks in 

 various positions on the boundary, have been recently discussed 

 by Lord Rayleigh-f. 



297. Some general theorems relating to the dissipation of 

 energy in the steady motion of a liquid under constant extra- 

 neous forces have been given by von Helmholtz and Korteweg. 

 They involve the assumption that the terms of the second order in 

 the velocities may be neglected. 



1. Considering the motion in a region bounded by any closed surface 2, 

 let u, #, w be the component velocities in the steady motion, and u + u\ v+v' t 

 w + v/ the values of the same components in any other motion subject only to 

 the condition that u', v f t w' vanish at all points of the boundary 2. By 

 Art. 287 (3), the dissipation in the altered motion is equal to 



JJJ|^+yj(+^ + .^ 



where the accent attached to any symbol indicates the value which the 

 function in question assumes when u, v, w are replaced by u', v', w'. Now the 

 formulse (2), (3) of Art. 284 shew that, in the case of an incompressible fluid, 



/jgr+2X^ (ii), 



each side being a symmetric function of a, 6, c, /, <?, h and a', 6', c', /', g', h'. 

 Hence, and by Art. 287, the expression (i) reduces to 



JJJ$ dxdydz + JJJV dxdy dz + 2 jjj(p xx a' +p vy b' +p zz c' 



+ Zpyzf 4- Zpzxd' + 2>Pxyti} dxdydz (iii). 



The last integral may be written 



du' 



and by a partial integration, remembering that u\ v', w' vanish at the 

 boundary, this becomes 



or JS jp(Xu'+ TV' + Zw') dxdydz ....................... (vi), 



* Sampson, 1. c. ante p. 134. 



t "On the Flow of Viscous Liquids, especially in Two Dimensions," PhiL Mag., 

 Oct. 1893. 



