71. CYLINDER AND VORTICES IN A STREAM 



(1) A single vortex. Let a cylinder of radius a be stationary in a stream approaching at 

 velocity U toward negative x\ let there be circulation F about the cylinder, and also a line 

 vortex with circulation F. located at the external point 2 = 6 = -Ae'^, or (- A cos y, ~h sin y), 

 the origin being taken on the axis of the cylinder. Here h and y are real, and the vortex is lo- 

 cated on a radial line inclined at an angle y to a radius drawn in the direction of the stream; 

 see Figure 107. This case is of interest in the theory of wakes. 



The complex potential representing the partial flow caused by the vortex is given by 

 Equation [42a] with A replaced by F /2n, s - c by 2 - & or s + Ae'>', and 2 + c by 2 - 6 'where 

 ?)'= -(a^/h)e^y; for, as shown in Section 42(B), the image vortex lies on the inverse line with 

 respect to the cylinder and hence on the same radial line as the actual vortex but at a distance 

 a'^/h from the axis of the cylinder. In this flow the circulation around the cylinder is -F,. 

 Circulation T + F must then be added in order to make the total equal to F. This can be ac- 

 complished, in superposing the flow due to the stream, by using Equation [69a] with F re- 

 placed by F + F,. The complete complex potential thus constructed is 



^2\ ^'(r + Fi) 3 iF^ 3_j 



V { 3 + - — I + In — + In 



2n a 2n 



b' 



[71a] 



Expressions for (fi, 0, and the velocity are easily derived if needed. 



The force on the cylinder may be found from the Blasius theorem, Equation [74g], pro- 

 vided the vortex is assumed to remain stationary. Here 



Figure 107 — A line vortex at "6" near 

 a circular cylinder in a stream. 



158 



