258 



+ V 



sp 



3w 



sp 



3x 



3u 

 sp 



3y 



(81) 



Further, u represents the sectional area in which 

 Hq-H is not equal to zero at S . 

 And, ^ 



EHP 



dV $ (e) + -^ 

 s U 



dS(Ho - H 



This equation reveals that the work transmitted to 



the fluid by the ship moving in still water with a 



constant speed, U (sum of the delivered horsepower 



and the work UAR caused by skin friction correction) 



changes in the fluid and is dissipated as heat, 



kinetic energy, potential energy, and work through 



the surface S . 

 A 



Oseen's Approximation and Problem of Variations 



p^gu 



!!lV f!!£V f^^^^ 

 3y / ■*" \3z / ~ \3x 



dz C , 

 s 



(87) 



We assxome that the hull is thin and S is placed 

 sufficiently far behind the hull. Then, the inte- 

 grations on S which appear in the right side of 

 the Eqs. (74), (77), (78), and (81) can be approx- 



imated as indicated in the Appendix, 

 following equations can be obtained: 



Hence, the 



DHP + UAR 



P,g' 



dV $ (e) + — — 

 sp 



dS(Ho - H )■ 

 sp 



Rt = P^g dS(Ho - H3 



Sa 



'^s\ ' ^ (!!s 



3y 



3x 



Pf" 



dS 



3<t> \ 2 

 sp > 



3y 



3(t) \ 2 

 sp > 



3z 



3<t) \ 2 

 sp ^ 



3x 



P^g 



dzc| , 



(82) 



p^gu 



dz S 



sp 



(88) 



AR = p^g dS(Ho - H ) 



+ ^ J dS 

 ^A 



P,g 



341 \ 2 / 3()> \ 2 

 sp 1 ^ ( sp ^ ^ 



3y 



dz C 



sp 



3(t> \ 2 

 sp ^ " 



(83) 



where Hn, H , and H represent the total head as 

 c 11 s sp 



follows : 



PO U2 

 Hn = + Y + ^- f 



H = + y + — (U+u + V- + w ) , 



s p g 2g s s s 



(84) 



(85) 



H = -§£• + y + ^ (Utu + v^ + w^ ) . (86) 

 sp p^g 2g sp sp sp 



In the Eqs. (82) and (83) for the balance of force, 

 the forces R^ and AR, given to the fluid from the 

 outside are divided into the force related to the 

 viscosity expressed by the first term and the force 

 related to the wave making expressed by the second 

 and third terms. In Eqs. (87) and (88) for the 

 balance of energy, the energies EHP and DHP + uAr 

 given to the fluid from the outside are independently 

 divided into the first and second terms which repre- 

 sent the energy related to viscosity and into the 

 third term and the fourth term which represent the 

 energy related to wave making. 



Now, using (87) and (88) which show that the 

 viscous energy and the potential energy are indepen- 

 dent of each other, it is obvious that the condition 

 for minimizing the viscous energy in (88) is a 

 necessary condition for minimizing the DHP. We 

 proceed, therefore, to obtain the minimum condition 

 of the viscous energy which corresponds to the 

 optimum condition for the energy recovery by the 

 propeller. For this discussion, we assume that in 

 the right side of Eq. (88), the first and second 

 terms change independently or that the increase 

 and decrease of the second term have, at least, a 

 positive correlation with the increase and decrease 

 of the first term. Based on this assumption, let 

 us consider the conditions required in minimizing 

 the following function: 



