(5.16) |

Moreover, if the flow is irrotational then is automatically satisfied by writing , where is termed the velocity potential. (See Section 4.15.) Hence,

On the other hand, if the flow is incompressible then is automatically satisfied by writing , where is termed the stream function. (See Section 5.2.) Hence,

Finally, if the flow is both irrotational and incompressible then Equations (5.17)-(5.18) and (5.19)-(5.20) hold simultaneously, which implies that

It immediately follows, from the previous two expressions, that

(5.23) |

or

(5.24) |

Likewise, it can also be shown that

(5.25) |

We conclude that, for two-dimensional, irrotational, incompressible flow, the velocity potential and the stream function both satisfy Laplace's equation. Equations (5.21) and (5.22) also imply that

(5.26) |

In other words, the contours of the velocity potential and the stream function cross at right-angles.