singular Perturbation Problems in Ship Hydrodynamios 



[L] <!>rr +7«^r +-^<l»ea= - 4>xx; (2-57') 



(H] It — t -*'""/-°'-°> ♦•'O. 



a IN 



on r = ro{x,0). (2-58') 



The definition of N is analogous to that in (2-47'). It is a unit 

 vector lying in the cross section plane at some x, perpendicular to 

 the contour of the body in that cross section. It has the three com- 

 ponents: 



(0,-l,ro^ro) 



V[l +(ro/ro)2] ' 



measured in the x, r, and 9 directions, respectively. Equation 

 (2-58'), like (2-58), expresses the fact that 9<^/9n = 0, where n 

 is the unit vector normal to the body surface. 



Let the inner expansion be expressed as follows: 



N 



<^(x,y,z)~ y #n(x,y»z) as e -^ for fixed (x.y/c.z/c). 

 Substitute this expansion into the [ L] and [ H] conditions above: 

 [L] Vy,,(^o + ^l +*2 +*3 + ...) = - (*0,/^|,/ ..•); 



The operator Vy ^ is the 2-D Laplaclan In the y-z plane, that Is, 



2 2 2 2 



2 ^ j^. +_L__iI_+i_L+J_-§l 



y»* ~q2 j.2"j,2 r8r 2aQ2* 



8y 8z 8r r 89 



It can be proven that the first term In the expansion, 4»q, 

 represents just the uniform stream: 



$Q(x,y,z) = Ux. 



717 



