= Solution
For constant density $\rho$, the <Euler equations for an inviscid fluid> with a conservative force per unit mass are
$$
\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u=-\nabla(p/\rho+\Phi),\qquad \nabla\cdot\mathbf u=0.
$$
Set $\boldsymbol\omega=\nabla\times\mathbf u$. The vector identity $(\mathbf u\cdot\nabla)\mathbf u=\nabla(|\mathbf u|^2/2)-\mathbf u\times\boldsymbol\omega$ and vanishing <curl> of a gradient give
$$
\partial_t\boldsymbol\omega=\nabla\times(\mathbf u\times\boldsymbol\omega)
=(\boldsymbol\omega\cdot\nabla)\mathbf u-(\mathbf u\cdot\nabla)\boldsymbol\omega,
$$
where both divergence terms vanish. Thus the <vorticity equation> is
$$
\boxed{\frac{D\boldsymbol\omega}{Dt}=(\boldsymbol\omega\cdot\nabla)\mathbf u.}
$$
The partial time derivative measures Eulerian change; $(\mathbf u\cdot\nabla)\boldsymbol\omega$ is advection; the right-hand side describes stretching and tilting of vortex lines by <velocity> gradients. A conservative body force supplies no <curl> source, and constant density removes any baroclinic <pressure> source.
For planar <velocity> independent of $z$, <vorticity> is perpendicular to the plane, $\boldsymbol\omega=(0,0,\omega)$, and $(\boldsymbol\omega\cdot\nabla)\mathbf u=\omega\partial_z\mathbf u=0$. Hence \b[each material particle conserves its <vorticity>], $D\omega/Dt=0$. Choose the <stream function> convention $u=\psi_y$, $v=-\psi_x$. Then
$$
\boxed{\omega=v_x-u_y=-\Delta\psi.}
$$
In a steady flow, $\psi_y\omega_x-\psi_x\omega_y=0$, so <vorticity> is constant along <streamlines>. Where $\nabla\psi\ne0$, use $\psi$ as a local transverse coordinate: this equality makes $\omega$ independent of the coordinate along a streamline. Consequently
$$
\boxed{\Delta\psi=F(\psi)\quad\text{locally on a regular streamline patch}.}
$$
This is <steady planar vorticity as a local function of stream function>.
A global single-valued $F$ additionally requires compatible <vorticity> values on all components of each stream-function level set. That condition is not supplied by steady Euler flow alone. For example, $\psi=x^3-x$ gives $(u,v)=(0,1-3x^2)$, a smooth steady incompressible shear with zero advective acceleration and constant <pressure>. Yet $\psi(0,y)=\psi(1,y)=0$ while $\Delta\psi(0,y)=0$ and $\Delta\psi(1,y)=6$. Thus \b[the unrestricted global deduction is false; the local result, or a suitable connected-level-set hypothesis, is the valid conclusion].
Back to article page
