530 



VISCOSITY. 



[CHAP, xi 



295. The results of Arts. 293, 294 can be applied to the 

 solution of a number of problems where the boundary conditions 

 have relation to spherical surfaces. The most interesting cases 

 fall under one or other of two classes, viz. we either have 



xu -f yv + zw = (1) 



everywhere, and therefore p n = 0, (f&amp;gt; n = ; or 



af+yi/ + *f=0 (2), 



and therefore % n = 0. 



1. Let us investigate the steady motion of a liquid past a fixed spherical 

 obstacle. If we take the origin at the centre, and the axis of x parallel to the 

 flow, the boundary conditions are that w = 0, v = 0, w = for r=a (the radius), 

 and w = u, v = 0, w = for r=o&amp;gt;. It is obvious that the vortex-lines will be 

 circles about the axis of x, so that the relation (2) will be fulfilled. Again, the 

 equation (9) of Art. 294, taken in conjunction with the condition to be 

 satisfied at infinity, shews that as regards the functions p n and n we are 

 limited to surface-harmonics of the first order, and therefore to the cases 

 = 1, n= -2. Also, we must evidently have ^ = 0. Assuming, then, 



we find 



3B 



The condition of no slipping at the surface r=a gives 



whence 

 Hence 





These make 



:0 &amp;gt; * *2 



(V), 

 (vi). 



