With a suitable definition of tan ^, ^A = occurs on the a;-axis and also on the dividing 

 surface <S defined by 



1.2a):!=tan-l 



ny nx 



tan — / tanh — 



[44r] 



This surface passes through the point {x , 0), and toward a; = - ~ its sides become parallel and 

 a distance L apart where L represents twice the limiting numerical value of y or 



L = 



ga 

 a+g 



[Us] 





y 



a/1 









_______^ 







^SIZIZZ~^^~ 



^r~^~-- 



s ~— -^ 



s:^-- 



Figure 66 — Flow past a semi-infinite cylinder 



between walls a apart, or along a channel 



narrowed in a certain manner. Constructed 



with use of a line source at 0. 



See Section 44. 



Here, for continuity, tan"^- tan (7TL/2a)] is interpreted as tv - nL/2a. On the walls, i/f = aU/2 

 at ?/ = a/2, and ip = - aU/2 at y = - a/2. 



A semi-infinite cylinder can be inserted along S, and the flow is then represented 

 between the walls and past this cylinder. The fluid approaches fromi a; = + =» at velocity U 

 and leaves, at a; = - «, at velocity U' = U (1 + g/a) = aU/ (a- L). 



Streamlines for g/a = 1 are shown in Figure 66, for y > only, since the a;-axis repre- 

 sents a plane of symmetry. On the cylinder, q = U Sit P, where x = 0.087a. Other possible 

 forms of the cylinder, for g/a = 1/2 and g/a = 3, are shown by broken curves. 



A wall could also be inserted along the positive a;-axis up to Xq and then along S, 

 forming, with a wall at y = a/2, a channel narrowed in a certain manner. 



If the sign of U is changed, all velocities are reversed without change in the geometri- 

 cal flow net or the pressure. The diagram can be most easily reversed from left to right by 

 reversing the positive direction for x. 



(C) Line Source in One Wall or Corner of a Channel 



In Case A, an additional wall can be inserted along the y-axis. It is then convenient 

 to take ijj = - n 4/2 for y > and i/r = rr A/2 for y < 0, so that i/i is continuous on the positive 

 a;-axis. The formulas of Case A, and the right-hand half of Figure 65, will then represent the 

 flow in a semi-infinite channel with plane walls a apart, due to a line source along the middle 

 of its base. See Figure 67a. The source may represent approximately fluid entering or leav- 

 ing through a narrow slit; a volume n A passes through the slit per second, between any two 

 planes of flow unit distance apart. 



104 



