The formulas may represent the flow due to a dipole outside of such a cylinder, or, if 

 b. < a, the flow inside a cylindrical shell of radius a caused by a line dipole inside it. In 

 either case the subscript 1 refers to the given dipole, and the other dipole may be regarded 

 as the image of this one in the cylinder. 



The force on the cylinder can be found, as in Section 42, from the Blasius theorem 

 and the method of residues. Here 



dw 



(3-6j)2 62 (3_5^) 



I. \2-l 



If 6^ > a, inside the cylinder there is only a singularity at 2 = 6,, and, expanding at 3 = ft,) 

 as in Section 30, 



= (?, 6 +2_6 )-2 



1 2(3-62) 



(2-^)' ' ^'^ '' (^2-^)" (*2-^)' 



If &j < a, the singularity is located at b^ and {z-h^Y^ is expanded; the path of integration 

 is then traversed in the negative direction, so that ^ {2-b^)~^ dz = - 2n-i. In either case the 

 force per unit length on the cylinder is found to be directed toward the dipole and to be of 

 magnitude 



inpn'^a^b^ 



F. = ^ [52h] 



(bl-a^f 



The force is thus independent of the direction of the dipole axis. (See Reference 2, Section 

 8.81, 8.82.) 



53. LINE SOURCE IN UNIFORM STREAM 



Upon the flow due to a line source at the origin let there be superposed a uniform flow 

 at velocity V toward negative x. From Equations [35a] and [40a] the resultant w, 0, and ijj 

 may be written 



w = U (z - g In z) [53a] 



<f> = V(x-glnr), tp^V{y-ffe) [53b, c] 



r = {x^ + y2)W, e = tan-i (y/x), 



where y is a real positive constant, |0| ^77, and $ has the sign of y; see Figure 81. Thus 



121 



