This is the complex potential for flow past the original cylinder at velocity U toward 

 negative y. Then 



cU X. fi cV X y. 



<p = cosh sin > ^ = sinh cos , 



mG 2m 2m mG 2m 2m 



dw dC, 

 dC dz 



iU 



cosh — ^ sinh ^ — sinh^ 



2m 



2m 



U 



2m^G 



(cosh A - cos \i) 



— ( cosh — + cos 



2 



[89q] 



When the fluid velocity at infinity is again suppressed, this time by adding iUZ in w 

 so as to superpose a flow toward positive y, then at large z and small (^writing 

 2 = 2c/ ^ + c^/6 .... again, 



icU 



2m 



e 



48?^' 



iUz 



icU 



12m 



2 6/ 8^ \2m 



^2 / 



[89r] 



Thus, from Equation [76d,e], with y = 7t/2, e 



-in/ 2 _ 



T, = 



277 / 1 



+ 1 c"" - S 



[89s] 



Cylindrical Boss or Groove, According to Equations [89d] and [88m], the ar-axis for 

 |a;| > c is part of a streamline; hence semi-infinite walls can be inserted there. Provided 

 m < 1, half of the field then represents flow past a plane wall interrupted by a cylindrical 

 boss of circular-arc section, which is 2c wide at the base and has an external angle y or 

 mn between its tangent and the wall, and hence a radius R - c/sin mn. Figures 143 and 

 157 and the upper part of Figure 144 may also be interpreted as showing streamlines for 

 such a flow. 



If 7K > 1, the diagram on the z-plane overlaps on itself and the whole field cannot be 

 used. Provided m ^2, however, the upper half of the w-plane taken by itself maps conformally 

 upon the part of the s-plane that lies above the part of the a;-axis on which |ar| > c and also 

 above the arc pi = mn, which now lies below the axis. For this purpose take Q ^ 6^ < 2n, 

 -n < 6 < n\ then S M < 2w and below the aj-axis fj. lies in the third or fourth quadrant. 



217 



