Figure 55 - Streamlines for a vortex pair. The arrows indicate the direction of 

 motion of the vortices if they move with the fluid; see Section 41. 



If the flow is assumed to be steady, the vortices are stationary. Alternatively, each 

 vortex may be supposed to move with tlie average Velocity of the fluid in its neighborhood, as 

 a vortex does in a real fluid; in the present case, this velocity is simply that due to tne other 

 vortex, or, by [401], U = A/2c = r/4 n-c, directed toward negative y if A and F are positive. 

 The pair of vortices thus advances without change of the distance between them; see Figure 55, 

 where the direction of advance according to this assumption is shown by an arrow. The formulas 

 will continue to represent the motion at each instant provided the axes are allowed to move 

 with the vortices. 



A pair of vortices of the same sign was discussed by Greenhill^^. Trains of vortices 

 were introduced by von Karman^^; the streamlines for a typical Karman^^; the streamlines for 

 a typical Karman colum or "Karman street" are shown in Figure 56. 



The transformation w = A [ln(3 + ic) - ln(2-ic)] represents the same flow rotated 

 through 90 degree, with the source and sink or the vortices at (0,ic). 



It may be remarked also that toward infinity 



In 



^ = lnA.£Ulnfl-£U2c,i 

 !-c \ z J \ z j z 3 



.3 



so that toward infinity the flow due to either a source and sink or a vortex pair approximates 

 that of a dipole; compare Equation [37a]. 



89 



