= Conservation of vorticity flux through a material surface
{title2=$d\int_{S(t)}\boldsymbol\omega\cdot d\mathbf S/dt=0$}
For smooth inviscid <barotropic fluid> with potential body forces, <barotropic vorticity transport> and $\nabla\cdot\boldsymbol\omega=0$ give $D\boldsymbol\omega/Dt=(\boldsymbol\omega\cdot\nabla)\mathbf u-\boldsymbol\omega\nabla\cdot\mathbf u$. Combine this with <material surface element evolution> to obtain $D(\boldsymbol\omega\cdot d\mathbf S)/Dt=0$. Integrating over a <material surface> proves the flux law. Its boundary must move with the fluid; flux through a fixed open surface is generally not conserved. By <Stokes theorem> the same statement is <Kelvin's circulation theorem> around the material boundary.
Back to article page
