Brard 
It, 
can be less than ee 
At time t, the surface Sy generates a volume W, when 
X increases from -=>- to its maximum value. We therefore deal with 
a three-dimensional wake W,, which was not the case in the preceding 
sections A,Bor C. Within W, a vorticity OF is continuously distri- 
buted. The vector w(M) is tangent to the arc Jy ¢: Ix ,! which pas- 
ses through M. 
oma 
The total velocity Vs induced by the vortex Slama cabana te) 
at a point M located inside or outside W;, is given by Poincaré's or 
Biot-Savart's formula: 
> 1 a fs does 
V (M) = curl) — era cat dX, (P) 
2] 2 
Bite t=) hyphae or 
+H, sy 2%, ()+ Te ito aw, ay (5. 35) 
pa; t 
= 
The velocity V; is irrotational outside (W, + its boundaries). It is, in 
particular, irrotational within D; . The other vortex distributions to 
be considered are the distributions 
—> 
QG, = (Since 4 (D; » 28), 
Lote Ora ' (5. 36) 
The velocity V' induced ryY is null outside the hull, and 
curl V' = 22, within D . (5. 37) 
The vortex =e See! consists of ring vortices on at Each 
ring is made of two arcs Bolas on 235 and Bi, t Bo on Disk The 
intensity of the ring is a constant_dy Y5(X, t) along the ring. Onan 
arc B,B, 4 located onthe part2,, of 2, , there exists the vortex 
filament of intensity dy T; (X, t) already considered in formula (5. 35) 
- see integral extended to ‘Be . The vortex distribution 2 is equiva- 
lent to a normal dipole distribution (2, Mon). It generates a velocity 
V5 » and one has 
Mics ee (V,, iN 4) -V, a“ vo within D, (5. 38) 
where Bia is due to some incident flow on the body. Outside the body, 
the total velocity is 
— —> = = 
V(M) = V,(M) + V,(M) + V (M) . (5. 39) 
Let us put Vv, =2 P. (outside W, + its boundaries) (5. 40) 
£228 
