75. THE LAGALLY THEOREMS 



The following special case of the force action on a cylinder is readily handled by 

 means of the Blasius theorem. 



Let a uniform line source be located at s = a, outside a cylinder of any shape, and let 

 the flow of the fluid be uniform at infinity, with velocity components u = - U sin y and 

 V = - U cos y. Then, if X and Y denote the x and y components of the force on unit length 

 of the cylinder, it will be shown that 



X =- pTU sin Y + 2npAu^^, Y = pY'U cos y + 27TpAv^^ 



[75a, b] 



Here F is the circulation around the cylinder; 27tA represents the volume of fluid emitted 

 per second per unit length from the line source; and u^^, v^^ are the components of the 

 partial particle velocity at the location of the source caused by the presence of the cylinder, 

 in addition to the velocity that would exist there if the cylinder and all circulation around 

 it were removed and replaced by fluid. 



To prove this theorem, the contour of integration in Equation [74g] is displaced from 

 the contour C of the cylinder and transformed into a distant contour S together with a small 

 contour a surrounding z = a; see Figure 113 and compare Section 29. The value of the 

 integral remains unaffected, since no singularities are passed in transforming the contour. 

 The theorem of residues is then used, as explained in Section 30. 



Figure 113 — A line source at "a" near 

 a cylinder C 



168 



