singular Perturbation Problems in Ship Hydrodynamias 



beginning of the section, note that our frame of reference Is now 

 moving with the body. Therefore, the velocity components should 

 be ordered: 



d^/dx ~ U = 0(1); B^/dy, 84»/9z, 8(j)/8r = o(l); 



8(<j> - Ux)/8x = o(8«})/8r). 



In each case, of course, the appropriate limit operation Is that 



€ -^ 0, where € Is the slenderness of the parameter. These order 



relations should be valid near the body. 



Far away, there will be the uniform stream, which Is 0(1), 

 but there Is no reason to assume that the perturbation velocity will 

 have components with differing orders of magnitude. 



These order-of-magnitude relations all come about auto- 

 matically If, In the near field, we define new variables: 



r = cR, y = cY, z = cZ, 



and assunne that differentiation with respect to x, Y, Z, R, and 9 

 all have no effect on the order of magnitude of a quantity. Thus, 

 suppose that the potential In the near field can be written: <J)(x,y,z) = 

 Ux + $(x,Y,Z). Then the derivatives have the following orders of 

 magnitude: 



||=o(*A), |4=o(*/€). 



It will turn out that ^ = 0(c ). This means that the transverse 

 velocity components, <^y, 4>2, and ^p are all 0(€) , that Is, they 

 are proportional to the slenderness parameter. Note also that a 

 circumferential velocity component would be given by (l/r)8(^/89 = 

 (l/€R)8$/8e = 0($/c), when we Interpret R = 0(1) (that Is, In the 

 near field) , and so circumferential and radlcd velocity components 

 have the same order of magnitude. The perturbation of the longi- 

 tudinal velocity component Is 0(4>) = 0(€^) , which Is, appropriately, 

 a higher order of magnitude than that of the trsmsverse velocity 

 components. 



In the far field, we assume that differentiation with respect 

 to any of the natural space variables has no order-of-magnltude 

 effect. Thus, we use the Cartesian coordinates (x,y,z) and the 



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