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Irrotational Flow
Flow is said to be irrotational when the vorticity
has the magnitude zero everywhere.
It immediately follows, from Equation (447), that the circulation around any arbitrary loop in an irrotational
flow pattern is zero (provided that the loop can be spanned by a surface that lies entirely within the fluid). Hence, from Kelvin's circulation theorem, if an inviscid fluid is initially irrotational
then it remains irrotational at all subsequent times. This can be seen more directly from the
equation of motion of an inviscid incompressible fluid which, according to Equations (39) and (79), takes the
form

(452) 
since is a constant. However, from Equation (1476),

(453) 
Thus, we obtain

(454) 
Taking the curl of this equation, and making use of the vector identities
[see Equation (1481)],
[see Equation (1478)], as well as the identity (1484), and the fact that
in an incompressible fluid, we obtain the vorticity evolution equation

(455) 
Thus, if
, initially, then
, and, consequently,
at all subsequent times.
Suppose that is a fixed point, and an arbitrary movable point, in an irrotational fluid. Let and be joined
by two different paths, and (say). It follows that is a closed curve. Now, since the circulation
around such a curve in an irrotational fluid is zero, we can write

(456) 
which implies that

(457) 
(say). It is clear that is a scalar function whose value depends on the
position of (and the fixed point ), but not on the path taken between and .
Thus, if is the origin of our coordinate system, and an arbitrary point whose position
vector is , then we have effectively defined a scalar field
.
Consider a point that is sufficiently close to that the velocity is constant along .
Let
be the position vector of relative to . It then follows that (see Section A.18)

(458) 
The above equation becomes exact in the limit that
. Since is arbitrary (provided that it is
sufficiently close to ), the
direction of the vector
is also arbitrary, which implies that

(459) 
We, thus, conclude that if the motion of a fluid is irrotational then the associated velocity field can always be expressed as minus the
gradient of a scalar function of position, . This scalar function is called the velocity potential, and
flow which is derived from such a potential is known as 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 Equation (1481)]

(460) 
Incidentally, the fluid velocity at any given point in an irrotational fluid is normal to the constant surface that
passes through that point.
If a flow pattern is both irrotational and incompressible then we have

(461) 
and

(462) 
These two expressions can be combined to give (see Section A.21)

(463) 
In other words, the velocity potential in an incompressible irrotational fluid satisfies Laplace's equation.
According to Equation (454), if the flow pattern in an incompressible inviscid fluid is also irrotational, so that
and
, then we can write

(464) 
which implies that

(465) 
where is uniform in space, but can vary in time. In fact, the time variation of can be eliminated
by adding the appropriate function of time (but not of space) to the velocity potential, . Note that such
a procedure does not modify the instantaneous velocity field derived from . Thus, the above equation can
be rewritten

(466) 
where is constant in both space and time. Expression (466) is a generalization of Bernoulli's theorem (see Section 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

(467) 
which demonstrates that for steady irrotational flow the constant in Bernoulli's theorem is the same on all streamlines. (See Section 5.3.)
Next: TwoDimensional Flow
Up: Incompressible Inviscid Fluid Dynamics
Previous: Kelvin Circulation Theorem
Richard Fitzpatrick
20120427