Irrotational Flow

(4.82) |

because is a constant. However, from Equation (A.171),

(4.83) |

Thus, we obtain

Taking the curl of this equation, and making use of the vector identities [see Equation (A.176)], [see Equation (A.173)], as well as the identity (A.179), and the fact that in an incompressible fluid, we obtain the vorticity evolution equation

Thus, if

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. Because the circulation around such a curve in an irrotational fluid is zero, we can write

(4.86) |

which implies that

(4.87) |

(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)

(4.88) |

The previous equation becomes exact in the limit that

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

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 (A.176)]

(4.90) |

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

(4.91) |

and

(4.92) |

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

(4.93) |

In other words, the velocity potential in an incompressible irrotational fluid satisfies Laplace's equation.

According to Equation (4.84), if the flow pattern in an incompressible inviscid fluid is also irrotational, so that
**
**
and
, then we can write

(4.94) |

which implies that

(4.95) |

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 previous equation can be rewritten

where is constant in both space and time. Expression (4.96) is a generalization of Bernoulli's theorem (see Section 4.3) that takes non-steady flow into account. However, this generalization is only valid for irrotational flow. For the special case of steady flow, we get

which demonstrates that for steady irrotational flow the constant in Bernoulli's theorem is the same on all streamlines. (See Section 4.3.)