Circulatory Flow, li w = AX, cp = A\^, ip = AX^, where A is real, the X^ parabolas 

 become streamlines in a type of flow in which there is a singularity at the origin and the 

 velocity vanishes at infinity. For, then 



? = 



dw 

 dz 



dw /dX 

 dx/ dz 



Ml 



f^r 



[87h] 



This might represent a sort of circulatory flow past a parabolic cylinder whose cross-sectional 

 profile is represented by one of the A2 parabolas, or between two such cylinders corresponding 

 to two values of X„. 



Streaming Flow. Let 



(i-x^-^, 



w = - V (— X^ - i/3A ) , £/ a»d /3 real and jS > 0, 



<^ = - (; [y (X2 - xl) + /3xJ =U{x-^ VTT^) 



<A = - ^Xj (X2 - i3) = [/ (y + j3 s/T^r^) = Uy 



\ \Jr+ xj 



[871] 



[87j] 



[87k] 



Here, in accord with the labeling in Figure 139, continuity has been secured, except on the 

 negative a;-axis, by assuming that X, ^ 0; then the°sign of X, and the sign before \Jt - x 

 must be taken opposite to the sign of y, but \Jr + x is positive. For the velocity 



u = U I - 1 + yjr+x I , V = -=:z^ , 



\ 2r J 2rJr+ x 



[871, m] 



..(..£^.£!) 



[87n] 



Thus toward infinity v -* Q, u-* - U, and the flow becomes a uniform stream at velocity U 

 toward negative x. On the positive ar-axis r = x and 



\ y/2^ J 



[87o] 



The value = occurs on the positive a;-axis, where X, = 0, on the parabola at X2 = ^• 

 On this parabola \/r + a; = j8 so that its apex, which represents the stagnation line, is at 



X = P^/2; its semi-latus-rectum is the value of |y| when a; = and r = y, or ^■^. A solid 



cylinder may be inserted along this parabola. On the cylinder 



209 



