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National Advisory Committee for Aeronautics, Technical Notes - Notes on Aerodynamic Forces - II - Curvilinear Motion

This note deals with the steady motion of a rigid body on a
curvilinear path through a perfect fluid otherwise at rest, so I
that the position of the body relative to the path remains con—
stant. The body turns with constant angular velocity around a
geometrical axis fixed with respect to space and with respect to
the body. Hence the flow of the fluid is stationary relative to
the body. The problem stands in close relation to that of the
body moving straight forward, which indeed is a special case of
the present problem. The methods employed with the investigation
or the straight motion can partly be used for the more general
problem too, but care must be taken that they are applied properly
and only as far as they are still valid.

It is to be understood at the outset that the forces between
the body and the fluid are now by no means the same whether the
real motion is considered, or whether the body is supposed to be
at rest and the entire fluid rotating about the center of rotation.
This latter case would imply the presence of the centrifugal
forces of the rotating fluid, giving rise to changes in the pres-
sure distribution and of the forces between the body and the fluid.
Nor is the pressure in each point now determined by the sQuare of
the velocity according to the law of Berncuilli, for the motion is
not really stationary since the body is moving. In the case of
non-stationary motion the pressure is the sum of the pressure due
to the Bernoulli (Vega/2) and the product of the change of the
velocity potential per unit of time and the density (Lamb, p.19).
The additional pressure can be transformed if, as in the present
problem, the flow is quasi-stationary and changes only its posi—
tion, each point of the configuration of the flow moving With a
velocity U. Let m be the potential, then'the change duvdt I
of the potential per unit of time is U dayds where -dB is a
linear element in the direction of U. Therefore the pressure add
-ditional to .V2 0/2 eQuals the product of the density, the veloc—
ity of the configuration in the point considered and the component
of the flow in the same direction.