361 



f(x) = B* Ixf-^/^ 



X f 0, 



(A, 26) 



1/2 



&r (x) : 6r (0) - B** x"' " , x +0, (A, 27) 

 c c 



where the constants B* and B** have the same sign. 

 The condition f> implies B* > and the condition 

 that the fluid does not penetrate the hub implies, 

 in the case of a smooth hub, that B** < 0. Hence 

 they must vanish both, which is achieved only by 

 giving C the value (A, 18). 



Finally we consider a hub of arbitrary shape. 

 By virtue of the Laplace transform we can write: 



6r (x) 



1 



2TTi 



g(A 



e dA 



X < 



(A, 28) 



C-loo 



where c is a positive number, and: 







g(X ) = 



fir (x) dx 

 c 



(A, 29) 



First we assume that or (x) is such that the integral 



in (A, 29) is absolutely convergent for A = c, but 



our results will appear to hold for a more general 



case. By virtue of the linearity of our questions, 

 the expressions (A, 22) and (A, 25) hold with B given 

 by: 



c+i" 



In this expression for L{x) we substitute iA = p, 

 take the limit c + and use some symmetry-properties 

 of H"*" (y) . Then we find: 



L{x) 



Re 



ipx H (U) 



(1-iP) 



1/2 



dP, X < 0, (A, 33) 

 1/2, 



Since the integrand in this expression is (U ' ) 

 we can derive for L(x): 



L(x) 



-1/2 



-1/2 



1- 



(A, 34) 



As stated in Section 6, the position of the 

 point of separation is determined by the condition 

 B = 0. By (A, 31) this condition becomes: 



/ 



L(x) 



k 



&r (x) + &r ( 



-] 



dx 



0. (A, 35) 



We can give an interpretation to the two terms in 

 this integrand. There are two reasons for which 

 the fluid may separate from the hub. First the 

 value of 6r^ may become negative, so that the 

 centrifugal force makes the fluid particles leave 

 the hub. This corresponds to the first term in the 

 integrand. Second, the radius of curvature of the 

 hub may be so small that the fluid particles are 

 unable to keep contact with the hub. This corres- 

 ponds to the second term in the integrand. 



1 

 2Tri 



g (A) (a2 -I- ^2) (^+1) 







""'"'^^ H'*'(iA) dA. 

 (A, 30) 



Substitution of (A, 29) into (A, 30), interchanging 

 the order of integration, and applying partial 

 integration with respect to x twice, gives: 



- ! 



where 



L(x) 



L(x) 



1 



2TTi 



{C26r^(x) 



-F 6r " (x) } dx 



c-l-ico 



-Ax 



H"^(iA) 



(X+1) 



1/2 



dA 



x < 0. 



(A, 31) 



(A, 32) 



REFERENCES 



Doetsch, G. (1943) . Laplace Transformation. Dover 



Publications. 

 Noble, B. (1958). Methods Based on the Wiener-Hopf 



Technique. Pergamon Press 

 Sparenberg, J. A. (1958). On a shrinkfit problem. 



Applied Scientific Research, Section A, Vol. 7. 

 Watson, G. N. (1922) . Theory of Bessel functions. 



Cambridge University Press. 

 Whitham, G. B. (1973) . Linear and Non Linear waves. 



J. Wiley and Sons. 

 Wu, Th. Y. (1972). Cavity and Wake flows. Annual 



Review of Fluid Mechanics, 4. 



