Flow of Viscous Liquids. 357 



readily be given by means of Green's theorem. For if yjr 

 and ^ are any two functions of x and ?/, 



J 



{x d £-+in} d HHx^-w°x} d * d y> • ( 6 ) 



the integrations being taken round and over the area in 

 question. If we suppose that yjr and dty/dn are zero over 

 the boundary, the left-hand member vanishes. If, further, 

 ^=y 2 -v|r ; we have 



jycv«w , *»4r-JF+v*+ , «fe<fri • • • (7) 



of which the right-hand member vanishes by (5). Hence 

 V 2 ^ vanishes all over the area, and by a known theorem, as 

 yfr vanishes on the contour, this requires that ^ vanish 

 throughout. 



We will now investigate in detail the slow motion of 

 viscous fluid within a circular boundary. In virtue of (5) 

 V 2 ^, which represents the vorticity, satisfies Laplace's equa- 

 tion, and may therefore be expanded in positive and nega- 

 tive integral powers of r, each term such as r n , or r~ n , being 

 accompanied by the factor cos (n6 + a). But if, as we shall 

 suppose, the vorticity be finite at the centre of the circle, 

 where r=0, the negative powers are excluded, and we have 

 to consider only such terms as 



*H£+;i+ M^> =, *" cos (n0+a) - ■ (8) 



The solution of this is readily obtained. If we assume 



yfr = r m cos (n6 + a), ..... (9) 



we find m = n + 2, To this may be added, as satisfying 

 V 2 <\Jr = 0, a term corresponding to m = n; so that the type of 

 solution for n6 is 



<t/r=A n r n+2 cos {n6 + a) +B n r n cos (nd + P). . (10) 



By differentiation, 



^ = {n + 2)A n r n+1 cos (n6 + *) + nB,/ 1 - * cos (n<9 + /3). (11) 



The first problem to which we will apply these equations is 

 that of motion within the circle r=l under the condition 

 that the tangential motion vanishes at every part of the 



