singular Perturbation Problems in Ship Hydro dynamics 



The functions w(z) and F(z) are, of course, very simply 

 related, although neither is known explicitly yet. In order to obtain 

 another relationship, one introduces the t, = ^ + it) plane, in which 

 the fluid domain is mapped into the lower half-space, as shown in 

 Fig. (5-4). We can write out the explicit expressions for mapping 

 the F and w planes into the C, plane. The first is accomplished 

 by means of the Schwarz-Christoffel transformation: 



dF _ Ua ; - c ^ . . 

 d^ " Tr(b + c) C + b ~ "^^^ ' 



which can actually be integrated, yielding: 



The second nnapping can be shown to take either of the equivalent 

 forms: 



w(z(U) = Ue'° ^-"^ 



(1 - ^c) +iV(l - c2)V(C^ - 1) 



(5-1) 



= ue'° (^ - ^c) - ivTl - cV(;g - 1) 



The solution is then completed by using the relationship between F 

 and w, along with these expressions, to obtain the relationship 

 between z and C, . Since: 



i£ = i^l£ H(4) 

 d; dF dC w(z(g) ' 



of which the right-hand side is known, we can integrate to obtain: 

 - ibV(l-c^) log [ t + Va^-D] 



- iV(r:;^)7(?:T) log i^H-y^>Va^) ) 



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