Ogitvie 



This appears so obvious that I pass on Immediately to the $| 

 problem. From the [ L] and [ H] conditions, we find: 



ft 



[L|] Vyz ^1=0 in the fluid domain; 



[H.] ^=__=^:E1— - on r=r,. (2-63) 



aN 



V[i +(V"o)'] 



Finding ^, is strictly a problem in two dimensions. In fact, it is 

 just the problem that the early aerodynamiclsts put forth intuitively 

 at the beginning of their slender-body analysis. (It was also the end 

 of their analysis f) For an arbitrary body shape, we might have to 

 solve this boundary-value problem numerically; that is not much of 

 a problem today. However, we are not yet ready to work with num- 

 bers, because the formulation of the problem is not quite complete: 

 we have not specified the behavior of $| at infinity. To do so 

 requires that we match the unknown solution of this problem to the 

 far-field expansion. 



First, note what (2-63) tells us about the order of mag- 

 nitude of ^1 . The right-hand member is 0(€) and the left-hand 

 member is 0(^| /e) (because of the differentiation in the transverse 

 direction) , which together imply: 



4»,= 0(6^). 



Actually, (2-63) says only that ^| cannot be higher order than e ; 

 it could be lower order if the matching Introduced some effect that 

 required e^ to be o(^, ), but this does not happen. 



This 4», problemi is remarkably similar to the ^g problem 

 in Section 2,2. If we can assume that V^i is bounded at infinity, 

 then we can express $| in a series just like the one in (2-51). 

 Whether V$| really is bounded at infinity can only be determined 

 from the matching, of course, but we go ahead with the assumption, 

 trusting that our method will show us If we have made unwarranted 

 assumptions . 



It should be noted too that there are important differences 

 between this problem and the problem of Section 2.2. The Neumann- 

 type of condition on the body was homogeneous there, but it is not 

 homogeneous here. Thus, one may expect that there may be a non- 

 zero net source strength inside the body in the present problem. 

 What happens at Infinity is also different. In the earlier problem, 

 the potential had to represent a uniform flow at infinity, and we 

 supposed that there might be the proper circumstances that a circu- 

 lation flow could occur. In the present problem, the uniform flow 



718 



