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Dynamics of Inviscid Fluids 103

Let us consider the simple case when (C) is a circular disk centered
at A. First we remark that, in this case, the function f")(z) is constant
and consequently we could eliminate the free of z term Q (t) f@) (z).
This result is obvious because the rotation of the disk with respect to its
center does not influence the ideal fluid flow. The case when a = 1 and
3 = 0 corresponds to the situation when (C) is performing a uniform

translation along the Oz axis; with respect to (C) (the system Axy),
the flow is steady with a velocity at infinity parallel to the Oz axis
and whose algebraic magnitude, versus the same axis, is —1; then the

complex potential associated to this relative flow is — (« + w), R being
the radius of (C) and consequently the absolute flow watched from the
fixed system OXY, has as complex potential

R? R?
)--4 ,
f(z) =2- (2+

which corresponds to a doublet located at the origin A of the plane
z, and whose axis is collinear with the velocity. From here, we could
deduce, at once, that in the case when the circular cylinder translates
with arbitrary components (a, 8), the corresponding complex potential

is

2
6p)
f(2)=—(a+ —.

An important generalization of the above situation is the situation
when the displacement of the obstacle in the fluid mass takes place in
the presence of an unlimited wall (as it is the case of a profile moving
in the proximity of the ground, that is the “ground effect” problem).

At the same time a great interest arises from the fluid flow induced
by a general rototranslation of a system of n arbitrary obstacles in the
mass of the fluid. We will come again to this problem after the next
section, by pointing out a new general method for approaching the plane
hydrodynamics problem [111].

9.3 The 3-Dimensional Case
Consider now the three-dimensional flow induced by the motion of
a rigid spatial body (obstacle) in the mass of fluid at rest at far field,
1.e., it is about a generalization of the previous study made in the plane

case. Let then O.X;.X»X3 be the three-rectangular fixed system and the
velocity potential of the absolute fluid flow ® (X1, X2, X3,t) be, at any
moment, a harmonic function of X; whose gradient (velocity) is zero at
infinity. Introducing also the mobile system Az,29273 — rigidly linked to