= Solution
The <vorticity> of a <fluid> <velocity> field is $\boldsymbol\omega=\nabla\times\mathbf v$. A <vortex line> is an integral curve tangent to that vector, not generally a particle path or <streamline>. Since $\nabla\cdot\boldsymbol\omega=0$, <vorticity> flux through successive cross-sections of a <vortex tube> is equal; a regular line cannot simply end inside the <fluid>. By Stokes's theorem, the tube strength is the <circulation>
$$
\Gamma=\int_S\boldsymbol\omega\cdot d\mathbf A=\oint_{\partial S}\mathbf v\cdot d\mathbf l.
$$
For an inviscid <barotropic fluid> with conservative body <force>, <Kelvin's circulation theorem> conserves <circulation> around a material circuit. Under those assumptions <vortex tubes> move with the <fluid>. In three-dimensional incompressible ideal <flow>,
$$
\frac{D\boldsymbol\omega}{Dt}=(\boldsymbol\omega\cdot\nabla)\mathbf v;
$$
stretching a material tube raises its <vorticity> as its cross-section shrinks. In two-dimensional <incompressible flow> there is no such stretching, so scalar <vorticity> is materially conserved. Constant <viscosity> adds a diffusion term $\nu\nabla^2\boldsymbol\omega$; <baroclinicity> and boundaries can also generate <vorticity>.
An ideal line vortex has $v_\phi=\Gamma/(2\pi r)$ and zero ordinary <vorticity> for $r>0$, with its <circulation> concentrated in the singular core. Circular trajectories therefore do not imply nonzero local <vorticity> everywhere. A finite physical core removes the <velocity> singularity.
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