f >0 



Figure 99 — Diagram for flow past 

 a circular cylinder. 



= C I r + — ) cos i9, tp = U ir j sin 



[67e,fl 



in terms of polar coordinates r, 0, such that x = r cos 6 and y = r sin 6 as illustrated in 

 Figure 99. The components of velocity are 



f ^^ ~ y^\ 2a^xy 

 u^Vi-l + a^ — 1, v = , 



[67g,h] 



qr= ui-1 + —] cos 9, gQ=v(l+ — jsin 0, 



/ 2a^ a'* \ 



o2 = [/2 1 cos 2(9 + . 



\ ,2 ,4 / 



[67i,j] 



[67k] 



There is a singularity at the origin, where q -* <x>. Stagnation points occur at (a,0) and (-a,0). 



The a;-axis represents a plane of symmetry for the flow, and the flow net has also a 

 plane of geometrical symmetry along the y-axis. 



At large distances <(> -* V x and the fluid is flowing toward negative x (if U > 0), or to- 

 ward d = 7T, with uniform velocity V. The formulas represent, in fact, such a uniform stream 

 superposed upon the flow due to a line dipole at the origin of dipole moment ^ = a'^ V, as is 

 evident from formulas in Sections 35 and 37. The axis of the dipole is directed oppositely to 

 the stream. 



Along the a;-axis ip = 0, and i/» = also on the circle r = a. This circle may be taken 

 to represent a circular cylinder, and the formulas then represent a stream, uniform at infinity, 

 flowing past such a cylinder. On the cylinder itself <^ - 2 U x, q^ = 0, and q = |ga| where 



149 



