/^ 



\ 



i:_Nk 



/^ 



(a) (b) (c) 



Figure 67 — Line source on a rectilinear boundary or in a corner. 

 See Section 44(C). 



Or, alternatively, walls may be inserted along ?/ = and y = a/2 only. Then the line 

 source, of the same strength, lies on a side extending to infinity in both directions, as in 

 Figure 67b, and the streamlines are illustrated by the upper half of Figure 65. 



Finally, walls may be inserted again along y = and y = a/2, and also along y = 0. 

 Then the formulas and one quarter of Figure 65 represent the flow due to a source or sink in 

 one corner of a rectangular channel of width a/2, as in Figure 67c. The source emits nA/2 

 units of volume into the channel per unit time, between planes of flow unit distance apart. 



(D) Channel with a Smooth Contraction 



The mathematical formulas can be extended upward to y = a without passing any singu- 

 larity. The diagram between y = and y = a is then symmetric with respect to the line y =a/2. 

 Let a uniform stream be superposed, as in Case B, and let curved infinite walls be inserted 

 along any two streamlines lying between y = and y = a. Then the diagram between these 

 lines represents flow along a two-dimensional channel whose cross-section changes smoothly 

 at a certain place along its length. See Figure 68, where any two curves may represent the 

 walls. 



Equations [44k] to [44p] are applicable. Assume i,'/ = and tan~' [(tan rry/ a) /tanb 

 (77 x/d)^ = on the a>axis. Then, for continuity, tan~^ passes from the first quadrant into the 

 second as y increases past y = a/2, provided a; > 0, but, if x < 0, these two quadrants are 

 interchanged. Thus, a.s x -* + ao and tanh (n x/a) -* 1, tan~^ [(tan (rry/a)/ tanh (ttx/o)] -» n y/a; 

 whereas, as x -* - ~, tan~^ -» 77 - (ny/a). Hence 



asa;-» + oo, t/;-> Uy; 

 as a; -• - oc, 1/; -. U 



Hyi] 



Thus, if I'l ^ and t';„ are the values of 1'/ on the curved walls, their distance apart, given by the 

 difference in y, is 



7 _ a iIj ^ - llj , llj ry -ill . 



^'tttt^-^ i at a; =-00, L = ^ - ^ * at a; = -^ 



a + ^ U ' 2 (y 



105 



