Ogilvie 



cylindrical coordinates (x,r,0) In a very conventional manner. 

 As c "*■ , the slender body becomes more and more slender, 

 shrinking down to a line which coincides with part of the x axis. 

 (This is the line segment that I defined as A previously.) In the 

 limit, there is no body at all and thus no disturbance of the incident 

 uniform flow. In the far field, the disturbance Is always o(l). 

 Therefore the far field consists of the entire space except the x 

 axis, and the potential function must satisfy the Laplace equation 

 everywhere except possibly on the x axis. 



At Infinity, it is reasonable to require that the perturbation 

 of the incident flow should vanish, which implies that the pertur- 

 bation potential must be regular even at infinity. A velocity potential 

 cannot be regular throughout space, including infinity, unless it is 

 trivial. Therefore the velocity potential must be singular some- 

 where, and the only place in the far field where such behavior Is 

 permitted is on the x axis. Our far-field slender-body problems 

 all reduce to finding appropriate singularity distributions on the x 

 axis. 



The Far- Field Singularity Distributions. In the far field, 

 the first term in the asymptotic expansion for the potential function 

 will be Ux. All of the following terms must represent flow fields 

 for which the velocity approaches zero at infinity; they represent 

 distributions of singularities on the x axis. The nature of the 

 singularities can only be determined in the matching process, and 

 so we must generally be prepared to handle all kinds of singularities. 



One of the easier ways of doing this is to apply a Fourier 

 transform to the Laplace equation, replacing the x dependence by 

 a wave-number dependence. The resulting partial differential 

 equation in two dimensions can be solved by separation of variables 

 in cylindrical coordinates. When we require that the potential 

 functions be single valued, we find that the solutions must all be 

 products of: 



K„(|k|r) or Ij^(|k|r) and sin n0 or cos n9, 



where Kn and In denote modified Bessel functions. Since I^ is 

 poorly behaved when its argument is large, we reject it, so that the 

 solution consists of terms: 



Kn(|k|r)[Q' cos nG + p sin n0] . 



The quajitltles a and P are constants with respect to r and 0, 

 but they are both functions of k. They also depend on the index n, 

 of course. The general solution is obtained by combining all such 

 possible solutions. Any term in the far-field expansion of the 

 potential function might be of the form: 



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