Figure 81 — Line source at in a 

 uniform stream; see Section 53. 



/ cos 6\ 



u^(Jl-l+g J, V = gU 



sin 6 



[53d,e] 



dw 

 dz 



V 



H} 



U 



3! + iy - g 



X + iy 



\dw\ 111 



h =(x-g^ + y^, [53f,g] 



where r^ is the distance from the stagnation point at {g,0). Note that dr/dx = x,/r = cos 0. 

 The only singularity occurs at the origin; and the a;-axis is an axis of flow symmetry. 

 The value ijj = occurs on the positive i-axis and also on a curve S defined by 



y = gO, or r = gd/s'in 6, 



[53h,i] 



\s 9 ■* 0, y -* 0; also 0/sin -► 1, so that r -* g. Hence S passes through {g,0). Toward 

 ar = - oo, 1^1 -. TT, and \y\ increases to a maximum of ng. 



Thus the streamline for ^ = follows the a;-axis to the nose of S, where it divides and 

 continues along both sides of S. Every other streamline undergoes a lateral displacement of 

 i ng from + ^ to - oo, or from d=0to6=-7T. All fluid coming from infinity remains outside 

 of S, and all fluid emitted from the source remains inside S. 



An infinite solid cylinder may be inserted along iS, extending to infinity also toward 

 negative x, where it has a maximum thickness of 2tTg. The formulas then represent flow past 

 this cylinder. They may also be used to represent the flow due to a line source inside a cy- 

 lindrical shell having the form of S. 



If the motion is steady, the values q = \V\ and p = p^ occur on S where r. = r, x = g/2, 

 from [53g]. On the ar-axis ahead of S, r = x, r^ = x ~ g, q = \U\ {1 -g/x), and 



P-P^ = V.,V^I-^-^) 

 \ X x^l 



[53j] 



122 



