Ogilvie 



2hfx.tt 

 Fig. (2-1). Coordinates for the High- Aspect- Ratio Wing 



N 

 <Kx,y,z) ~ Ux + ) ^j^{^,Y,7.), where <j>n^| = o(4>^) as €-^0, 



n=l 



for fixed (x,y,z). (2-39) 



(Again, the dependence on c is suppressed in the notation.) Since 

 the body shrinks to a line (x = 0, y = 0, | z | < S) in the linait as 

 € -* 0, the terms denoted by ^n 3^11 represent flow perturbations 

 which arise in the neighborhood of this singular line. They can be 

 expressed in terms of singularities on that line, and the strengths 

 of such singularities should be o(l) as € "^ 0, In an ideal fluid, 

 we could expect the occurrence of dipoles , quadripoles, etc. , on the 

 singular line. We also take the realistic point of view that viscosity 

 cannot be completely neglected and that there may be some circula- 

 tion as a result. In the usual aeronautical point of view, this implies 

 that there may be a vortex line present, complete with a set of trail- 

 ing vortices. In the point of view adopted in the previous section, 

 I assume that there may be a sheet of dipoles behind the singular line, 

 I also make the usual assumption that these wake dipoles (or vortices) 

 lie in the plane y = 0, This part of the y = plane (0 < x < oo, 

 |z| < S) will be denoted by W . (Note that H has all but disappeared 

 in the far field view. It is only a line.) 



We can now write the outer expansion in the following form: 



698 



