Figure 160 — Two slender circular cylinders in motion. See Section 95. 



The first approximation to the velocity potential represents a dipole on the axis of 

 each cylinder and is, from Equation [37s] in Section 37, 



^ ' = a^ F 



cos (^i -a) cos (02 - iS) 



2 T/ + ^,2 ,^ _ 



[95a] 



where r., r^, f? , 9 are adequately defined in Figure 160. The flow due to each of these 

 dipoles then violates the boundary condition at the surface of the other cylinder, hence their 

 respective images in the other cylinder must be added, then, for a similar reason, the images 

 of these images, and so on. Only the first pair of images will be included here. 



Furthermore, to the same degree of approximation the displacement of the first images 

 from the axis of the cylinder containing them may be neglected. Their contribution to the 

 potential is then, from Equation [52c], 



a-^ b' 



(^2+") a^b^ cos (9^+13) 



[95b] 



where r is the distance between the axes of the cylinders. 



The contribution of cylinder B to the integral in Equation [17d], which expresses the 

 kinetic energy of the fluid, is then 



27r 2n 



-P j ^%(^s-JP \ {^^nW cos {9^- li)bd9^. 



236 



