A Vortex Theory for the Maneuvering Ship 



(iii) (J ) due to viscosity (friction or, more exactly, viscous drag), 



(iv) (3^) due to the velocity potential of the absolute motion, 



(v) (Jj) due to the propeller, and 



(vi) forces due to the system — if any — which reduces the freedom of the 

 body or generates its forced motion. 



The set of hydrodynamic forces is J^. We assume here that the body is not 

 fitted with planes and fins (see par. 6). 



For what follows, it is helpful to separate the set of forces (J^) into two 

 additive parts, J. and 3^', (J. ) existing alone when there is no wake, while (J') 

 is the contribution of the wake. 



It is possible to obtain rigorously this result by starting from the contribu- 

 tion in the absolute momentum of the fluid of each part of 4> (see par. 5.3). 

 However, we will firstly proceed using an approximate expression of the hydro- 

 dynamic pressure p(m^, t ') on Sg. 



The velocity potential 0(M, t ' ) is, at time t ' + dt ' , when M is at rest with 

 respect to the fixed axis: 



t/)(M,t'+dt') = 0'(x',y',z',t'+dt') ^^(x-Ugdt'.y-Vgdt', z- Wgdt', t ' + dt ') , (1) 



where u^., vg, wg are the components of the absolute velocity Vj.(M, t ') of the 

 point attached to the body which, at t ' , coincides with M. Consequently: 



^ (M,t') = ^(M,t') - VE(M,t') grad0(M,t') . (2) 



The hydrodynamic pressure is given by: 



^[p(M,t')-p^] = ^ (M,t') - yV2(M,t') , (3) 



where V(M, t ') is the absolute velocity of the fluid at M. V^. being its relative 

 velocity at the same point, one has 



^[p(M,t')-pJ = I^CM.t') + CVeV)^,,, -iv2(M,t') (4) 



i [p(M,t')-p^] = l^(M,t') + 1 Ve (M,t') - iv/(M,t'). (4-) 



Since Iv/Vgl is generally small, we could neglect the last term in the right 

 member of (4) and write 



1 / 'X 1 



— p(m^, t ) = — p + 



1^- Vggrad^]^ .t' 



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