= Solution
{c}
Take the <curl> of the <Euler equations for an inviscid fluid>. The <gradient> on the right has zero <curl>. Use the stated vector identity with $\mathbf F=\boldsymbol\omega$, $\mathbf G=\mathbf u$:
$$
\partial_t\boldsymbol\omega
+(\mathbf u\cdot\nabla)\boldsymbol\omega
-(\boldsymbol\omega\cdot\nabla)\mathbf u
+\boldsymbol\omega(\nabla\cdot\mathbf u)
-\mathbf u(\nabla\cdot\boldsymbol\omega)=0.
$$
<incompressible flow> gives $\nabla\cdot\mathbf u=0$, while a <curl> always has zero <divergence>, so $\nabla\cdot\boldsymbol\omega=0$. The remaining terms give
$$
\boxed{\frac{D\boldsymbol\omega}{Dt}
=(\boldsymbol\omega\cdot\nabla)\mathbf u.}
$$
For a two-dimensional <velocity> independent of $z$, $\boldsymbol\omega=(0,0,v_x-u_y)$ and $(\boldsymbol\omega\cdot\nabla)\mathbf u=\omega_z\partial_z\mathbf u=0$. Thus \b[the scalar <vorticity> is conserved following each fluid particle]. It can still vary from particle to particle; this is advection, not a claim of spatially uniform <vorticity>.
Back to article page
