The flow is thus represented past a semi-infinite cylinder whose profile in cross sec- 

 tion is S. The cylinder has a sharper edge then that obtained with a line source in Section 53. 

 Its shape is determined by g, its size by c, since increasing c and all coordinates in the same 

 ratio leaves Equation [58g] satisfied. The half width at a; = - c/2, or at the middle of the 

 source sheet, where >■ = '"^ and d = n - 6^, is 



y = h = ncg/2 = R/2 [58i] 



An example is illustrated in Figure 90, for g = 0.15; R = 0.47 c, x = c/800, approxi- 

 mately. The distribution of the excess of pressure above that at infinity, p - p^ and of the 

 velocity q, along the cylinder and along the plane of symmetry in front of it, are shown on 

 arbitrary scales. 



(For notation and method; see Section 34; Reference 3, p. 81.) 



59. THE SIMPLER SINGULARITIES AND THEIR TRANSFORMATION 



A simple type of singularity is the following. Suppose that at s = c the complex poten- 

 tial w becomes infinite in such a way that near c it approximates the function B In (z - c) 

 where 6 is a constant; let the difference w - B In {z - c) be a. finite regular function of z even 

 at 3 = c. Then, from the formulas in Section 40, it is clear that, if 6 = ^4, where /I, is real, 

 a line source exists at s = c, emitting 27tA. units of volume of fluid per second and per unit 

 length, whereas if 6 = iA^ where A^ is real, a line vortex exists there with circulation 27)- /Ij 

 around it; if 6 = A, + iA2, both source and vortex occur. Again, if w approximates similarly 

 lie /{z - c), where /x and o are real, then Equation [37r] shows that a line dipole exists at 

 z = c, with line-dipole moment fi and with axis inclined at an angle a to the positive a;-axis. 



Figure 90 — Streamlines past the cylinder 



shown as S in Figure 89. The inner 



streamlines are those due to the 



source sheet alone inside a cylindrical 



shell of contour S. The pressure and 



the velocity are shown, along the 



axes and over the cylinder. See 



Section 58. (Copied from 



Reference 7.) 



/' °--~~r 



Pressure y !'"i'^I ~~ 





— ^ 1 1 P-P^ 



1 Velocity 1 1 ]" 





133 



