2.I.8E

For a steady flow of an incompressible fluid of density $\rho$, show that

$\mathbf{u} \times \boldsymbol{\omega}=\nabla H,$

where $\boldsymbol{\omega}=\nabla \times \mathbf{u}$ is the vorticity and $H$ is to be found. Deduce that $H$ is constant along streamlines.

Now consider a flow in the $x y$-plane described by a streamfunction $\psi(x, y)$. Evaluate $\mathbf{u} \times \boldsymbol{\omega}$ and deduce from $H=H(\psi)$ that

$\frac{d H}{d \psi}+\omega=0$

*Typos? Please submit corrections to this page on GitHub.*