Figure 157 — Streamlines above the plane of symmetry, outside the dividing surface S 

 obtained from two line dipoles in a stream. See Section 93(B). The boundary may 

 also represent a circular boss on a plane wall, with the a!-axis taken along the 

 wall, or half of a symmetrical circular-arc cylinder, as in Section 89. 

 (Copied from Reference 124.) 



94. TWO CIRCULAR CYLINDERS IN A STREAM; CYLINDER AND WALL 



Since a cylinder immersed in a uniform stream merely adds the flow due to a certain 

 dipole on its axis, as in Section 67, and the image of a dipole in a circular cylinder is another 

 dipole, as in Section 52, the flow around any number of cylinders in a stream can be built up 

 in terms of an infinite train of image dipoles in each cylinder. Circulation around the cyl- 

 inders may be added by assuming a suitable vortex on the axis of each, and then an infinite 

 train of image vortices inside each, in accord with results in Section 42. 



Only the first approximation to the solution will be given in detail here. 



Let two cylinders A and B have radii a and 6, and let their axes be located, respec- 

 tively, at (0, 0) and {-d, 0) so that they are d apart, and let a/d and b/d be small. Let the 

 stream approach at an angle a with a line drawn through their axes, as illustrated in Figure 

 158; and let there be circulation Fj about A and r2 about B. Then a first approximation to 

 the complex potential is 



w=^ U 



2 + d 



+ — [F, In 3 + F, In {z + d)]. [94a] 



2/7 



232 



