There remains in addition a force of magnitude 4:TTpU a b /d inclined at an angle 

 2 a to the line of axes. This force is shown as Fjj in Figure 158. Its direction is such that 

 a stream flowing parallel to the line of axes causes the cylinders to repel each other, where- 

 as a transverse stream causes them to attract. When the stream is oblique the forces tend to 

 turn the line of axes into a direction perpendicular to the stream. 



If the stream is abolished by giving to everything an equal and opposite velocity, the 

 forces remain the same. Then the cylinders, if free from circulation, repel each other when 

 moving at equal speeds along the line of axes, but attract when moving in the same direction 

 transversely to this axis. 



It will be noted that no images at all have been included in this approximation, except 

 those associated with the stream itself. Their inclusion leads to terms of higher order in 

 1/d, for each of the three types of terms containing U'^, TV, or Fj Fj, respectively. 



If more than two slender cylinders are present, it is easily seen that the forces are 

 additive in first approximation; the presence of each cylinder modifies the lift on every other 

 and contributes interaction forces upon it according to the formulas just found. 



The complete solution for noncirculatory flow past two cylinders of equal radius was 

 written out in terms of images by Muller. ^^'^ Streamlines for F = C, and for a= and 

 a = 90 deg, respectively, are shown in Figure 159. The directions of the forces on the 

 cylinders are also shown. 



Figure 159 - Streamlines for flow in two directions past two equal circular cylinders. 



The arrows indicate the directions of the forces on the cylinders, when the 



motion is steady. (Copied from Reference 124.) 



The flow was treated with no restriction upon the sizes of the cylinders except that 

 they are external to each other, in terms of elliptic functions by Lagally,^^^ and with use 

 of bipolar coordinates and series by Endo. ^^^ 



Cylinder and a Wall. If the two cylinders just considered have equal radii and equal 

 and opposite circulations, or no circulations at all, and if the stream approaches perpendic- 

 ularly to the line of axes, then the plane that bisects the line joining the axes is a stream 



2U 



