= Solution
Take the curl. A <gradient> has zero curl, and the supplied identity, with $\nabla\cdot u=0$ and $\nabla\cdot\omega=\nabla\cdot(\nabla\times u)=0$, gives
$$
\partial_t\omega+(u\cdot\nabla)\omega-(\omega\cdot\nabla)u=0.
$$
Therefore $D\omega/Dt=(\omega\cdot\nabla)u$. In planar flow $\omega=(0,0,\partial_xv-\partial_yu)$ and $\partial_z u=0$, so the stretching term vanishes. Consequently
$$
\boxed{D\omega/Dt=0}.
$$
<Vorticity> is carried unchanged with each fluid particle. It need not be spatially uniform or independent of time at a fixed spatial point.
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