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Potential Flow

When the flow is irrotational,
\nabla \times {\bf v} = 0,
\end{displaymath} (2.21)

there exists a velocity potential, $F$, such that
{\bf v} = - \nabla F,
\end{displaymath} (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.
\end{displaymath} (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 $p$.

Prof. Alan Hood