dz ' \ ,2/ 277 z 277 \z-b z-b'l 



[71b] 



The force is obtained from the residues at s = and s = 6'; compare Sections 30 and 42. It 

 is found that 



X = 



pr, /r + r^ hr^ > 



2n \ h 



A2- 



. cos y - -y pF^f/ sin 2y, 



[71c] 



y = prt/ + — 



pFj / r + r. Ar 



A2 -a 



1 \ 



— j sin y + — pr,fycos2y. [71d] 



«2/ a2 ^ 



A negative value of x represents a force in the direction of the stream or a drag. 



If r = -r., the formulas are simplified. Streamlines for such a case are shown in 

 Figure 108; here T is positive and the flow is from right to left. 



Figure 108 - Streamlines near a circular cylinder due to a stream with 



circulation V about the cylinder and a line vortex of circulational 



strength -P. (Copied from Reference 28.) 



(2) Several vortices. If several vortices are present, the forces due to them are simply 

 additive; each term in X and y that contains Tj is replaced by a sum of such terms, one for 

 each vortex. In Figure 109 are shown streamlines for a symmetrical case with T = and two 

 vortices. (References MilUer^s, Bickley^^^ and Morris'^^^ 



159 



