. Hence, the above equations
imply that if a cylindrical obstacle is placed in a uniformly flowing inviscid fluid then there is
zero drag
. On the other
hand, as long as there is net circulation of the flow around the cylinder, the lift is nonzero. Now, lift is generated
because (negative) circulation tends to increase the fluid speed directly above, and to decrease it directly below, the
cylinder. Thus, from Bernoulli’s theorem, the fluid pressure is decreased above, and increased below, the cylinder,
giving rise to a net upward force (
i.e.
, a force in the
+
y
direction).
Suppose that the cylinder is placed in a fluid which is initially at rest, and that the fluid’s uniform flow velocity,
V
0
,
is then very slowly ramped up (in such a manner that no vorticity is induced in the upstream flow at infinity). Since
the flow pattern is initially irrotational, and since the flow pattern well upstream of the cylinder is assumed to remain
irrotational, the Kelvin circulation theorem indicates that the flow pattern around the cylinder also remains irrotational.
Consider the time evolution of the circulation,
Γ
=
contintegraltext
C
v
·
d
r
, around some
fixed
curve
C
that lies entirely within the
fluid, and encloses the cylinder. We have
d
Γ
C
dt
=
contintegraldisplay
C
∂
v
∂
t
·
d
r
=
contintegraldisplay
C
bracketleftBigg
−∇
parenleftBigg
p
ρ
+
1
2
v
2
parenrightBigg
+
v
×
ω
bracketrightBigg
·
d
r
=
contintegraldisplay
C
v
×
ω
·
d
r
,
(5.108)
where use has been made of (5.24) (with
Ψ
assumed constant). However,
ω
=
ω
z
e
z
in twodimensional flow, and
d
r
×
e
z
=
d
S
, where
d
S
is an outward surface element of a unit depth (in the
z
direction) surface whose normal lies in
the
x

y
plane, and that cuts the
x

y
plane at
C
. In other words,
d
Γ
C
dt
=
−
contintegraldisplay
S
ω
z
v
·
d
S
.
(5.109)
We, thus, conclude that the rate of change of the circulation around
C
is equal to minus the flux of the vorticity across
S
[assuming that vorticity is convected by the flow, which follows from (5.25), the fact that
ω
=
ω
z
e
z
, and the fact
94
FLUID MECHANICS
−
5
−
4
−
3
−
2
−
1
0
1
2
3
4
5
y/a
−
5
−
4
−
3
−
2
−
1
0
1
2
3
4
5
x/a
Figure 5.11:
Streamlines of the flow generated by a cylindrical obstacle of radius a, whose axis runs along the zaxis,
placed in the uniform flow field
v
=
V
0
e
x
. The normalized circulation is
γ
=
2
.
5
.
that
∂/∂
z
=
0 in twodimensional flow]. However, we have already seen that the flow field surrounding the cylinder
is
irrotational
(
i.e.
, such that
ω
z
=
0). It follows that
Γ
C
is
constant
in time. Moreover, since
Γ
C
=
0 originally,
because the fluid surrounding the cylinder was initially at rest, we deduce that
Γ
C
=
0 at all subsequent times. Hence,
we conclude that, in an inviscid fluid, if the circulation of the flow around the cylinder is initially zero then it remains
zero. It follows, from the above analysis, that, in such a fluid, zero drag force and zero lift force are exerted on the
cylinder as a consequence of the fluid flow. This result is known as
d’Alembert’s paradox
, after the French scientist