Plane of Symmetry 



Figure 63 — Streamlines due to a line source 

 S near a wall. 



\ 



'^u 



/ 



\ 3o/2 



\ 



X a 



/ 



\ an 



\ 



/ 



/ 

 \ 



/ 

 \ 



/ 



\ 



/ 

 \ 

 / 

 \ 



(a) 



T 



(b) 



Figure 64 — A row of equal line sources 

 or vortices. 



if ny/a and tan~^ are both negative and in tlie fourth quadrant, tan~^ decreases with decreas- 

 ing a; to - TT + tan~^ [tan {-n y/a)/tanh (77 a; /a)] at a? = - a; . 

 Hence, 



tA . 2n-y 



nA 27tx 



u = — sinh , 



aH a 



aH 



n^A^ I 



C( 



, tt'^A'^ I ,2ttX 27Ty\ 2nx 2ny 



'2 (cosh + cos — - ] , H = cosh cos- 



[44d,e] 



[44f,g] 



The expressions for 0, u,v, and g are periodic in the y direction with a period equal to 



a. The y-axis and the lines y = 0, ~ a/2, - a, - 3/2 a, - 2a all represent planes of 



flow symmetry. 



The flow is that due to equal line sources spaced a distance a apart along the y-axis; 

 at each of the points (0,0), (0, - a), (0, - 2a), etc,, there occurs a line source emitting a volume 

 2tt A per second per unit length. For, as the origin, for example, is encircled positively, tan~^ 

 increases by 277 and decreases by 27t A. If ^ < 0, line sinks occur at these points. See 

 Section 40 and Figure 64a. 



In the corresponding conjugate flow, with potential <^'= ip and stream function ip '=-(/), 

 the sources are replaced by line vortices with circulation F = 2 n A about each. The velocity 

 components are u'=-v, v'=u. At large distances from the row of vortices m'=0, v'^ n A/a = 

 r/2a, since sinh (277 a;/a)/cosh (277 a;/ a) -> i 1. See Figure 64b. Thus, if a large width A of the 

 plane containing the vortices is encircled by a path, the circulation about this path is 

 2Ar/2a = hT/a, in agreement with the fact that h/a vortices are encircled. The substitution 



101 



