J, = -jrA/l 



Figure 65 — Flow net due to a line source between walls, or forming one member of a 



row of sources; the source mentioned is at the origin. 



See Section 44. (Copied from Reference 253.) 



sin nB/a for sinh nz/a rotates everything through 90 deg, with the sources or vortices on the 

 af-axis. 



A number of interesting physical cases can be constructed by the suitable insertion of 

 walls. 



(A) Line Source Between Parallel Walls 



According to Equation [44c], xb is constant along each of the lines y = - a/2. (Note 

 that tan 90 deg = - <x, tan 270 deg = - ex,.) Rigid walls may be inserted along these lines, and 

 the formulas then represent flow due to a line source midway between two walls separated by 

 a distance a. The flow net is shown in Figure 65. As indicated on the figure, a consistent 

 set of values of i/f, which is many-valued, is as follows: lA = on the positive a;-axis; 

 r{f = - n A/2 along the positive y-axis to y = a/2 and then in both directions along the wall 

 that lies at y = a/2; \]r = - tt A on the negative a;-axis; ifj = - i rr A/2 on the negative y-axis 

 and along the other wall at y = - a/2; lA -» - 27tA just below the ai-axis. 



As a; -♦ i oo, v -» 0, u -* (n A/a) tanh (2Trx/a) -* -n A/a. Thus the flow becomes uniform, 

 and the total rate of outflow from the source is 2 (a |u|) = 2ttA, as found before. In general, on 

 the ic-axis v = and 



102 



