1 , 



-7— curl 



47T 



A Vortex Theory for the Maneuvering Ship 



V (M) when M is in Q , 



r-^'^^'^hi 11$ '"■<-•' 



(2) 



VgCM) when M is in fi^ , 



since Vp.n = V„n on S. 



Let us consider now the surface of the hull as covered by a very thin bound- 

 ary layer (of thickness S); we see that a vortex equal to (1/S) n AV^ on the mean 

 surface S(ni) between the internal face S- and the external face s^ of the bound- 

 ary layer, and to 20 in n^ , generates an absolute fluid motion which has the fol- 

 lowing properties: outside the body, the motion is identical to that of the fluid; 

 inside the body, the fluid is at rest with respect to the body. 



3.2. A Distribution of Doublets is Equivalent to a Distribution 



of Bound Vortices When the Angular Velocity is Equal to Zero 



When Q = 0, we have a velocity potential in n^ and in ii. , which may be re- 

 garded as due to doublets normal to the hull: 



477 JJ 



V^ in Qg , 

 , V-) ^l^m^ ^S(m), V.-gradO^ = (3) 



s(m) "> U in n. , 



n being the unit vector normal to the hull and positive outwards. 



Therefore, m. and m^ being on the normal n(m) to S(m), m. on s., m^ on s^ , 

 we have 



with 



f(m) = U[i^x(m.) + iyy(m.) + i^zCm-)] + constant.' (5) 



Equation (5) is Fredholm's equation of the 2nd kind relative to an interior 

 Dirichlet's Problem. For any value C of the constant in the right member of (5), 

 the solution of (4) is unique. One has 



Therefore when another constant C is substituted for c, ^Jm^) is changed in 

 $^(mg) + c'- c. Hence the motion of the fluid outside the body does not depend 

 upon the value of the constant c. 



Let us assume this constant chosen in such a way that yjm) = o on the 

 forebody. Let C^(m) be the rings normal to Vr„(m), a^ the arc of this ring, a' 

 the arc of their orthogonal trajectories C^ (<^o°= ^ ^^ ^^® forebody, >0 behind), 

 i^, i; the unit vectors tangent to c^ and to (E„ respectively, the directions of 

 these vectors being those of da^ >0, da^ >o, and these directions themselves 

 being chosen in such a way that - n = io ^^o* 



823 



