singular Perturbation Problems in Ship Hydrodynamiaa 



dltlons on the near -field expansion as follows: 

 [ L] $, + $2 + $3 + . . . 



b J ' YY YY YY 



--e^L^i +^, +$2 +$2 +...]; (2-17) 



"• 'xx 'zz ^xx 'zz •' 



on Y = ± H(x,z) . (2-18) 



Solution of the 4»| problem . From the [ L] condition above , 

 it is clear that: 



^lyY=0 (2-19) 



in the fluid domain. Therefore $| must be a linear function of Y. 

 In view of the synnmetry of the problem, we can set: 



$,(x,Y,z;€) = A,(x,z;c) +B, (x,z;€)|y|, for |y|>H(x,z). 



(2-20) 

 The body condition reduces to: 



$,^(x,±H(x,z),z;€) = ± €^UH^(x,z) = ± B,(x,z;€) = 0(€^). (2-21) 



It appears that we have determined the value of B|(x,z;€) -- but this 

 is wrong, as we shall see in a moment. The two-term inner expan- 

 sion appears to be: 



<Kx,y,z;€) ~ Ux + A,(x,z;€) + B,(x,z;€) |y|. 



Its outer expansion is obtained by setting Y = y/e: 



<^(x,y,z;c) ~ Ux + — B|(x,z;€) |y | + A|(x,z;€) . 

 0(1) 0(e) 0(6^) 



The order-of-magnitude estimates were obtained as follows: B| is 

 O(c^) , from (2-21). If our expansion is consistent (as we insist), 

 then A| is also O(c^) , by (2-20). Now, in the outer expansion of 

 the inner expansion, the B| term is lower order than the A, termi. 



685 



