singular Perturbation Problems in Ship Hydrodynamics 



[ 1955] . For the first time, it was possible to predict with some 

 confidence how the flow around the various cross sections interacted. 

 There were some difficulties in principle, even with the new ap- 

 proach; what we now call the "outer expansion" of the problem was in 

 effect forced to satisfy body boundary conditions. The difficulty is 

 somewhat comparable to trying to force a Laurent-series solution 

 to satisfy prescribed conditions which are stated on a contour inside 

 the minimum circle of convergence. A readable, refreshing account 

 of slender-body theory in the i950's has been provided by Lighthill 

 [1960] . 



During the early 1960's, slender-body theory was applied to 

 ship hydrodynamics problems by several investigators. Probably 

 the earliest to try this on a major scale was Vossers [ 1962] ; he 

 attacked a variety of steady- and unsteady-motion problems by 

 slender-body theory. He used a Green's function approach, which 

 apparently avoids the fundamental difficulty in principle of the 

 previous method. However, it is really too much to hope to obtain 

 asymptotic estimates of five -fold integrals -- without making mis- 

 takes. Apparently Vossers did hope for too much, but Joosen [ 1963] 

 and [ 1964] corrected many of his mistakes, Newman [ 1964] also 

 advocated the Green's-function approach and produced some inter- 

 esting results. 



The modern (i.e. , fashionable) alternative is to use the 

 method of matched asymptotic expansions. In ship hydrodynamics, 

 Tuck [ 1963a] first used this method in his doctoral thesis at 

 Cambridge University. It avoids the difficulties in principle of 

 Ward's approach, and it is easier to work with than the Green's- 

 function method. Of course, the method of matched asymptotic 

 expansions has its own set of difficulties of principle. However, 

 it is the method that I shall pursue here.'' 



In any case, the analysis can be no better than the assump- 

 tions which are made at the beginning. Therefore I shall be (perhaps 

 painfully) explicit about the assumptions. 



2.31. Steady Forward Motion. Let the body surface be 

 specified by the equation: 



A very recent account of slender-body theory, particularly with 

 respect to its applications in ship hydrodynanmics , has been pub- 

 lished by Newman [ 1970] . I think that his presentation and mine 

 generally complement each other (and perhaps occasionally contra- 

 dict too). Newman has provided a survey that seems comparable 

 in intent to the one by Lighthill [ I960] , mentioned above, whereas 

 I am trying to place slender-body theory into a hierarchy of singu- 

 lar perturbation problems. My emphasis is on the development and 

 application of the method of solution. 



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