singular Perturbation Problems in Ship Hydrodynamios 



h„(e)-€"2, a„Jlog€r 

 m=0 



as a single term (albeit inconsistent) in the series 2;^^^)* ^^ 

 alternative way of describing this practice is to say that I consider 

 log € = 0(1) as 6—^01 I have encountered some practical prob- 

 lems whicn could apparently not be solved by the Van Dyke matching 

 principle unless treated in this way, and I have never seen or heard 

 of a problem in which this practice led to difficulties. There are 

 some good arguments for proceeding in this way, but I know of no 

 proof that either way is the correct way. (Some of my colleagues 

 will call this a cheap trick, rather than a higher expression of an 

 art form. ) 



The classical example in physics of this kind of mathematical 

 problem is the boundary layer first described by Prandtl in 1904. The 

 thickness of the boundary layer becomes smaller and smaller as the 

 small parameter, i/vR approaches zero (R is the Reynolds num- 

 ber), but the presence of the boundary layer cannot be neglected, 

 because then the governing differential equation becomes lower order, 

 and the body boundary conditions cannot all be satisfied. Unfor- 

 tunately, Prandtl did not realize the generality of the analysis which 

 he introduced into the viscous -fluid problem, and, lacking the 

 modern formalism for treating such problems, he could not obtain 

 higher-order approximations. 



Perhaps I should include a discussion of Prandtl's problem 

 in this paper, since it might be considered as a "singular perturba- 

 tion problem in ship hydrodynamics." However, I shall not do this, 

 for several reasons. Van Dyke's coverage of the problem is excel- 

 lent, I think. Also, the analysis concerns only laminar boundary 

 layers, and they are really of quite limited interest in ship hydro- 

 dynamics. Finally, the formal procedure breaks down completely 

 at the leading edge of a body, and the singularities that occur there 

 cause major difficulties in all attempts to use the formalism to 

 obtain higher-order approximations. 



One final point should be emphasized, even at the risk of 

 insulting the intelligence of readers who have read this far. When- 

 ever we write, "e -*• , " we are implying the existence of a se- 

 quence of physical problems in which the geometry of some funda- 

 mental parameter varies. For example, in Prandtl's boundary- 

 layer problem, we may consider that viscosity changes as 

 C = 1/"/r -^ 0. In the simple ordinary-differential-equation example 

 presented above, we may think of a spring-mass system in which the 

 mass is changed systematically from one experiment to the next. 

 Later, when we treat slender-body theory, we consider a sequence 

 of problems in which the body changes each time. The theory always 

 implies the possible existence of such a series of problems , and the 

 quality of the predictions Improves as the problem inore nearly fits 



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