PX285 - L5a - vortex lines

the flow around a vortex:

\vec{u} = \frac{K}{2 π r} \hat{θ}

where, $K = \oint \vec{y} \cdot d \vec{l}$ is the circulation around any closed loop enclosing the axis, $r = 0$

if the rotation axis is not attached to a material object, the vortex is called a free vortex or vortex line

the closed path above does not enclose the vortex line
the total circulation is:

\begin{aligned} \oint \vec{u} \cdot d \vec{l} & = \int_{Γ_{A}} \vec{u} \cdot d \vec{l} + \int_{L_{A B}} \vec{u} \cdot d \vec{l} + \int_{Γ_{B}} \vec{u} \cdot d \vec{l} + \int_{L_{B A}} \vec{u} \cdot d \vec{l} \\ = \int_{Γ_{A}} \vec{u} \cdot d \vec{l} + \int_{Γ_{B}} \vec{u} \cdot d \vec{l} \end{aligned}

this is because the paths $L_{A B}$ and $L_{B A}$ cancel

⟹ \int_{Γ_{A}} \vec{u} \cdot d \vec{l} = - \int_{Γ_{B}} \vec{u} \cdot d \vec{l}

$Γ_{A}$ and $Γ_{B}$ are in opposite directions around the vortex line
if $Γ_{B}$ is changed such that both are clockwise, $\oint \vec{u} \cdot d \vec{l}$ around the vortex line in a fixed direction is constant along the line