In terms of cylindrical coordinates a;, 9 such that a; = &j cos 6, y = o) sin d, 



<i = w/rS^sin2 0, li = — 6j ^ t:^'^ cos 2 ^; [lOSe.h] 



2 2 



hence 



5'~ = &j A; S sin 2 6, Qq = cu k a cos 2 d, [105i, j] 



q =\co\k w. [105k] 



At the end of the minor axis, 6 - 77/2, q— =0, qn = - co k co, so that the fluid is circulating 

 backwards. At the end of the major axis, = 0, qn = a kTlj; but the velocity of the shell is 

 oi CO and so exceeds qa, since ^ < 1. Thus relatively to the shell the fluid circulates in the 

 opposite direction, in order to keep its motion irrotational in space. 



The pressure at any point, if the rotation is steady, is, from Equation [lie]. 



p = po) kco I cos 2 6- — k\ + constant. 



The kinetic energy of the fluid, per unit length of the cylinder, is 



[1051] 



where the boundary of the region of integration is the ellipse defined by Equation [105a]. 

 Substituting x = ax', y - by', and the value of q. 



1 ' ' 2 



Tj = — pabcj^k^ll (a} x''^ + h^ y''^) dx' dy', 



and the region of integration for a;' and y'is now a circle of unit radius. Changing to polar 

 coordinates so that x' = r cos 6, and replacing dx' dy'hy r dr dd, 



1 2n 



I I y' dx' dy' =1 j x' dx' dy' = j r dr j c 



27r 



os^ 6dd = — . 

 4 





 Hence 



pk^ ab {a^ + 6^) w^. [105m] 



257 



