Figure 151 — Two equal line dipoles with longitudinal axes 

 (in the direction of x). 



q^ = U^-2^U 



cos 2 d. 



cos 2 d. 



+ (i 



2 cos (2 0J -3^2)' 



r2 ,2 

 '1 ^^2 



[92f] 



On the a;-axis u - u^ and on the y-axis u = u^ where 



x^ + b^ b^ - y^ 

 i^- - V + 2^1 , M = - 6' + 2/x 



{x^ - 6^) 



ib' + y') 



[92g,h] 



A. Two Line Dipoles Alone. If (7 = 0, the field of flow is that due to two equal line 

 dipoles whose axes are parallel and directed along the line through the locations of the dipoles, 

 which are at (- b, 0); compare Equation [37a]. Stagnation points occur on the y-axis at 



(0, - b), where 0j = 135 deg, 6^ = 45 deg, and 1// = + fjL/'b. Streamlines for |0| < |/i|/6 run 

 through both dipoles; those for |0l> |/i|/6 consist of two disconnected loops, one associated 

 with each dipole. Streamlines above the a;-axis are shown in Figure 152. 



B. Flow Past One or Two Similar Cylinders of Special Shape. Assume that U/fi > 0, 

 so that the dipole axes are oppositely directed to the stream at infinity, whose velocity is 



U toward negative x. Then the streamline for t// = consists of the a;-axis and the curve 

 S defined by 



[92g,h] 



If c is large, S is an oval curve cutting the a;-axis in two stagnation points; as c-*<x, it approxi- 

 mates a circle. If c = 6, it contracts in the middle to a point at the origin; for c<b, it consists 

 of two loops, one surrounding each dipole. Several possible forms of S are shown in Figure 153. 



The formulas may represent flow past a cylinder, or two parallel similar cylinders, in- 

 serted along S. If c<b/2, the cross-sections of the cylinders are nearly circular; even if their 



226 



