= Solution
The surface must be a \b[material surface $S(t)$], moving with the fluid; its boundary moves with it as well. The stated <barotropic vorticity transport> law, with $\boldsymbol\omega=\boldsymbol\nabla\times\mathbf v$, gives $\boldsymbol\nabla\cdot\boldsymbol\omega=0$. Expanding its curl therefore yields
$$
\frac{D\boldsymbol\omega}{Dt}
=(\boldsymbol\omega\cdot\boldsymbol\nabla)\mathbf v
-\boldsymbol\omega(\boldsymbol\nabla\cdot\mathbf v).
$$
A material tangent vector $\mathbf t$ obeys $D\mathbf t/Dt=(\boldsymbol\nabla\mathbf v)\mathbf t$. Apply the <product rule> to the <cross product> of two such tangents to obtain <material surface element evolution>:
$$
\frac{D(d\mathbf S)}{Dt}
=\left[(\boldsymbol\nabla\cdot\mathbf v)I-(\boldsymbol\nabla\mathbf v)^{\mathsf T}\right]d\mathbf S.
$$
Here $(\boldsymbol\nabla\mathbf v)_{ij}=\partial_jv_i$. The stretching and area-change terms cancel exactly in the transported flux element:
$$
\begin{aligned}
\frac D{Dt}(\boldsymbol\omega\cdot d\mathbf S)
&=\big[(\boldsymbol\omega\cdot\boldsymbol\nabla)\mathbf v
-\boldsymbol\omega\boldsymbol\nabla\cdot\mathbf v\big]\cdot d\mathbf S\\
&\quad+\boldsymbol\omega\cdot
\big[(\boldsymbol\nabla\cdot\mathbf v)I-(\boldsymbol\nabla\mathbf v)^{\mathsf T}\big]d\mathbf S
=0.
\end{aligned}
$$
Integrate over the fixed particle labels parameterizing the material surface. This proves the <conservation of vorticity flux through a material surface>:
$$
\boxed{\frac d{dt}\int_{S(t)}\boldsymbol\omega\cdot d\mathbf S=0.}
$$
The conserved flux is also the <circulation> $\oint_{\partial S(t)}\mathbf v\cdot d\boldsymbol\ell$ by the <Stokes theorem>, so this agrees with <Kelvin's circulation theorem>. The assumptions are smooth inviscid barotropic flow and conservative body forces. A fixed open surface generally exchanges vorticity flux with its surroundings and does not satisfy this conservation statement; compressibility by itself does not invalidate the material-surface result.
Back to article page
