For a numerical application of (15) the difficulty arises that the body singu- 

 larities should include the image of the propeller singularity into the body. The abso- 

 lute flow generated by the complete system of body singularities in the propeller plane 

 is called the "effective wake" and it is this quantity which should be introduced into 

 (15) instead of the nominal wake. This complicates an application of (15) since the 

 images are difficult to determine for a general body of revolution or, from the experi- 

 mental point of view, since only the nominal displacement wake can be measured. It is 

 therefore important to know the general structure of the relation between effective and 

 nominal wake. For this purpose, it suffices to assume a simply shaped body. Con- 

 sidering the interaction of a sphere with a point sink, Martinek and Yeh have deduced 

 the following relation [17]: 



( E P (w' d , n \ ) 



W d , e = W d , n ] 1 + K, ra 2 ' 3 - I]" 2 V (7) (16) 



/ oQrVs \ w d , n ) ) 



By the subscripts e and n the effective and nominal displacement wakes are denoted, 

 respectively, and by w' d>n the derivative of the nominal wake in the axial direction. 

 From this expression, it follows that the effective wake is greater than the nominal 

 wake which is in accord with experimental experience. 



A numerical evaluation of the sphere-sink interaction [18] shows that the inter- 

 action force decreases both when the radius of the sphere decreases, the distance of 

 the sink from the sphere being constant, and when the distance increases, the radius 

 held constant. However, the decrease of the force is shown to be much more rapid 

 when the distance increases than when the radius decreases. From this it is concluded 

 that it will be more favorable for the propulsion of a deeply submerged body to move 

 the propeller away from the stern of the body than to refine the stern. This holds in 

 a non-viscous fluid. In viscous flow the opposite effects of displacement and frictional 

 wake on the propulsive power determine an optimum position of the propeller. 



An obvious extension of the theory is possible when introducing a sink disc 

 for the propeller instead of a point sink and when assuming that the input of the disc 

 depends on the radial coordinate corresponding to the thrust distribution of a propeller. 

 From Lagally's law the element of the interaction force d(\R) which is generated on 

 an annular element of the disc is represented by the relation 



d(AR) = pu d , e dEp = pu d , e e P dFp (17) 



where both the effective displacement wake and the surface sink distribution e p depend 

 on the radial coordinate x of the disc. The dependence of d(<\R) on radius necessi- 

 tates definition of a local thrust deduction coefficient, viz., 



t(x) = d(AR)/d,T (18) 



Expressing the surface density e p by the thrust distribution dC T /dx by means of the 

 energy and momentum equations the local displacement thrust deduction coefficient 

 becomes 



t d (x) = d (AR)/dT = 2w d , e / (l + <Jl + ; ) (19) 



We now face the problem of determining the image of a sink disc of variable 

 density into a body of revolution in order to ascertain the effective wake. For this 

 problem, solutions have become known recently. In a paper by Martinek and Yeh [18] 

 certain surface singularities are studied in the presence of both a sphere and a prolate 

 spheroid, among them the sink disc. For the spheroid the solution is given in terms of 

 Legendre functions using spheroidal coordinates. In a subsequent paper by Hunziker 

 [19] the restriction to bodies of revolution of special shape is abandoned and the prob- 

 lem of rotational symmetry is formulated generally. Introducing a distribution of 



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