524 VISCOSITY. [CHAP, xi 



nor losing angular momentum, so that the above expression must 

 be independent of r. This gives 



(2). 



If the fluid extend to infinity, while the internal boundary is that 

 of a solid cylinder of radius a, whose angular velocity is o&amp;gt; , we 

 have 



a&amp;gt; = co a 2 /r* .......................... (3). 



The frictional couple on the cylinder is therefore 



47TyU.a 4 ft) ........................... (4). 



If the fluid were bounded externally by a fixed coaxial cylin 

 drical surface of radius b we should find 



which gives a frictional couple 



*-,. -(6)*- 



292. A similar solution, restricted however to the case of 

 infinitely small motions, can be obtained for the steady motion of 

 a fluid surrounding a solid sphere which is made to rotate 

 uniformly about a diameter. Taking the centre as origin, and the 

 axis of rotation as axis of a?, we assume 



u = coy, v = wx ) w = (1), 



where o&amp;gt; is a function of the radius vector r, only. If we put 



P=fa&amp;gt;rdr (2), 



these equations may be written 



u = -dP/dy, v = dP/d.x, w = Q (3); 



and it appears on substitution in Art. 286 (4) that, provided we 

 neglect the terms of the second order in the velocities, the 

 equations are satisfied by 



p = const., V 2 P = const (4). 



* This problem was first treated, not quite accurately, by Newton, Principia, 

 Lib. ii., Prop. 51. The above results were given substantially by Stokes, I. c. ante, 

 p. 515. 



