Brard 
IV. FREE AND BOUND PARTS OF A VORTEX SHEET 
EULER'S EQUATION IN A MOVING SYSTEM OF AXES 
The fluid is assumed to be inviscid, incompressible and 
homogeneous. Consequently, its mass density p is independent of 
position and time. 
Let De be the hull surface of a body moving in the fluid. 
Let S', S denote two right-handed systems of axes. S' 
is at rest and the fluid motion with respect to S' is said the "'abso- 
lute'' motion of the fluid. S moves with the body and the motion 
with respect to S is said the ''relative'' motion. The subscript R 
refers to the relative snot F is the absolute exterior force : 
Va Ss -eurk V and ¥ =-G_ are the absolute velocity, vorticity 
and acceleration. The instantaneous motion of S with respect to S' 
is termed the''entrainment' motion, It consists of the addition of a 
rotation about a certain axis A and ofa translation parallel to. a 
If O and M are two points of S, the entrainment velocity Ve(M) 
of M can be expressed as 
V(M) ="V_{0) + QessAsgoOMas 
where Q E is the angular entrainment velocity. 
The relative velocity of a fluid point P located at M at the 
instant.t.. 1s 
R E 
One has 
Y(P aa a Pe Ae 
OP). saa Vena Vigne t = RP 
OR =i foybla lly Whe Td Pas © a ahs 
and the exterior force Eg per unit mass in the relative motion is 
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