potential flow
. Note that the velocity potential is undefined to
an arbitrary additive constant.
We have demonstrated that a velocity potential necessarily exists in a fluid whose velocity field is irrotational.
Conversely, when a velocity potential exists the flow is necessarily irrotational.
This follows because [see Equa
tion (A.176)]
ω
=
∇ ×
v
=
−∇ × ∇
φ
=
0
.
(5.30)
Incidentally, the fluid velocity at any given point in an irrotational fluid is normal to the constant
φ
surface that passes
through that point.
If a flow pattern is both irrotational and incompressible then we have
v
=
−∇
φ
(5.31)
and
∇ ·
v
=
0
.
(5.32)
These two expressions can be combined to give (see Section A.21)
∇
2
φ
=
0
.
(5.33)
In other words, the velocity potential in an incompressible irrotational fluid satisfies Laplace’s equation.
According to Equation (5.24), if the flow pattern in an incompressible inviscid fluid is also irrotational, so that
ω
=
0
and
v
=
−∇
φ
, then we can write
∇
parenleftBigg
p
ρ
+
1
2
v
2
+
Ψ
−
∂φ
∂
t
parenrightBigg
=
0
,
(5.34)
which implies that
p
ρ
+
1
2
v
2
+
Ψ
−
∂φ
∂
t
=
C
(
t
)
,
(5.35)
Incompressible Inviscid Fluid Dynamics
83
P
C
B
A
Figure 5.4:
Twodimensional flow.
where
C
(
t
) is uniform in space, but can vary in time. In fact, the time variation of
C
(
t
) can be eliminated by adding the
appropriate function of time (but not of space) to the velocity potential,
φ
. Note that such a procedure does not modify
the instantaneous velocity field
v
derived from
φ
. Thus, the above equation can be rewritten
p
ρ
+
1
2
v
2
+
Ψ
−
∂φ
∂
t
=
C
,
(5.36)
where
C
is constant in both space and time. Expression (5.36) is a generalization of Bernoulli’s theorem (see Sec
tion 5.3) that takes nonsteady flow into account. However, this generalization is only valid for irrotational flow. For
the special case of steady flow, we get
p
ρ
+
1
2
v
2
+
Ψ
=
C
,
(5.37)
which demonstrates that for steady irrotational flow the constant in Bernoulli’s theorem is the same on all streamlines.
(See Section 5.3.)
5.8
TwoDimensional Flow
Fluid motion is said to be
twodimensional
when the velocity at every point is parallel to a fixed plane, and is the same
everywhere on a given normal to that plane. Thus, in Cartesian coordinates, if the fixed plane is the
x

y
plane then we
can express a general twodimensional flow pattern in the form
v
=
v
x
(
x
,y,
t
)
e
x
+
v
y
(
x
,y,
t
)
e
y
.
(5.38)
Let
A
be a fixed point in the
x

y
plane, and let
ABP
and
ACP
be two curves, also in the
x

y
plane, that join
A
to an
arbitrary point
P
. See Figure 5.4. Suppose that fluid is neither created nor destroyed in the region,
R
(say), bounded by
these curves. Since the fluid is incompressible, which essentially means that its density is uniform and constant, fluid
continuity requires that the rate at which the fluid flows into the region
R
, from right to left across the curve
ABP
, is
equal to the rate at which it flows out the of the region, from right to left across the curve
ACP
. Now, the rate of fluid
flow across a surface is generally termed the
flux
. Thus, the flux (per unit length parallel to the
z
axis) from right to
left across
ABP
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 Fall '12
 Fluid Dynamics, Fluid Mechanics, stress tensor, Fluid Motion