

Flow of Viscous Liquids. 361 



Thus if PAO be denoted by <£, the value of i/r in the 

 neighbourhood of A is given by 



7r.^ = sin2(/> + 2<£ (22') 



That the functions of <f) which occur in (22') satisfy the 

 fundamental equation may be readily seen. 



By calculation from (22') we get the following values for 

 </> expressed as fractions of degrees : — 



^ 0, -25, -50, -75, 1-00 



<f> 0, ll°-40, 23°*83, 39-40, 90°*00 



This example is of interest, from its bearing upon the laws 

 of flow at a place where a channel is enlarged. In actual 

 fluids there would be a tendency to shoot directly across from 

 A to B_, the region about C being occupied by an eddy or 

 backwater such that the motion of the fluid near the wall was 

 reversed. Nothing of the kind is indicated by the present 

 solution. In (22) dyjr/dr represents the velocity across the 

 line 6=^7r, and we see that there is no change of sign. In 

 fact the velocity decreases, as r increases, all the way from 

 r=0 to r=l. The formation of a backwater may thus be 

 connected with the terms involving the squares of the 

 velocities, which are neglected in the present solution. And 

 we may infer that if the motion were slow enough, or if the 

 fluid were viscous enough, the backwater, usually observed in 

 practice, would disappear. 



Another particular case of some interest, included in the 

 general solution already indicated, would be obtained by 

 supposing similar sources to be situated at = 0, 6 = tt, and 

 equal sinks at 6 = ^ir, = J7r. 



We will now suppose that it is the radial velocity which 

 vanishes at every point of the circumference r = l, and that 

 the tangential velocity also vanishes except in the neighbour- 

 hood of = 0. In this case, by the symmetry, i|r in (10) 

 reduces to a series of cosines. And 



- ^f = Sn sin nO(A n r n + 2 + B n r n ), 



which is to vanish when r=l for all values of 6. Hence 



A n + B w =0; (23) 



so that 



f = "(l-f 2 )JB/ cos nd, . . . (24) 



^=B% n cosn0{nr n - 1 --(n + 2)r n + 1 }. . . (25) 

 Phil. Mag. S. 5. Vol. 36. No. 221. Oct 1893. 2 B 



