Hence (62) becomes 



n* (v -v) 



^^ P J ^PO 



'VP 4. -P J rpQ '' (66) 



a velocity field due to a source distribution of strength 



a = n • (v-v^)/47r (67) 



The form of a shows that n • v, is the normal component of the velocity on 

 the interior side of the source distribution on S. Uniqueness of solutions 

 of Neumann problems on S then shows that v is the irrotational velocity 

 field in D associated with the source distribution a on S. 



An alternative source distribution on S can also be found directly as 

 the solution of the exterior Neumann problem for the given values of v • n 

 on S, as was done previously in considering the displacement effects of the 

 boundary layer. Applied to the vorticity field BLW in the flow about a 

 body, the present approach requires that a composite bounding surface, 



S = S„ + T + A 



be used, where S^ is the surface of the given body, and A is the transverse 

 surface of the truncated wake. If the transverse section is taken suffi- 

 ciently far downstream, the effects of the source distribution on A may be 

 neglected, and the Neumann problem could be formulated as a pair of simul- 

 taneous Fredholm integral equations of the second kind. The resulting 

 source distribution, however, would not coincide with that given in (67). 

 In the present treatment, the value v • n = on S is preserved, whereas, 

 in deriving (67), the tangential component v x n = on S , in accordance 



D 



with the nonslip condition, was preserved. In the former case, S remains 

 a stream surface, in the latter, an equipotential. 



26 



