291-292] ROTATING SPHERE. 525 



The latter equation may be written 



d?P 2 dP 



j-r + - j = const., 



dr* r dr 



da) ,. 



or r -7- + 3o) = const ...................... (o), 



OjT 



whence a) = A/r s + B ....................... (6). 



If the fluid extend to infinity and is at rest there, whilst o> is 

 the angular velocity of the rotating sphere (r = a], we have 



a? 



(7). 



If the external boundary be a fixed concentric sphere of radius 

 b the solution is 



"^tt-"' ( 8 >- 



The retarding couple on the sphere may be calculated directly 

 by means of the formulae of Art. 284, or, perhaps more simply, 

 by means of the Dissipation Function of Art. 287. We find 

 without difficulty that the rate of dissipation of energy 



= //, 1 1 l(x 2 + 2/ 2 ) ( -y- ) dxdydz 



JJJ \(IT / 



\dr 



r 



(9). 



If N denote the couple which must be applied to the sphere to 

 maintain the rotation, this expression must be equivalent to Na> , 

 whence 



or, in the case corresponding to (7), where 6= oo , 



N = Sirfia 9 to ............. , .......... (11).* 



The neglect of the terms of the second order in this problem involves a 

 more serious limitation of its practical value than might be expected. It is 

 not difficult to ascertain that the assumption virtually made is that the ratio 



* Kirchhofif, Mechanic, c. xxvi. 



