Potential Flow

When the flow is irrotational,

$$\nabla \times {\bf v} = 0,$$ (2.21)

there exists a velocity potential, $F$, such that

$${\bf v} = - \nabla F,$$ (2.22)

§ so that (2.21) is satisfied identically ( $\nabla \times \nabla F = 0$). For an incompressible fluid, we have

$$\nabla \cdot {\bf v} = 0,$$

and so using (2.22) we obtain

$$\nabla \cdot \nabla F = 0, \qquad \Rightarrow \qquad \nabla^{2}F = 0.$$ (2.23)

Thus, $F$ satisfies Laplace's equation and the flow is called potential. Solving this (2.22) gives ${\bf v}$ and (2.20) determines $p$.