PX285 - L5b - interacting vortex lines

opposite directions

considering two vortex lines, $A$ and $B$ , separated by a distance, $d$ , rotating in opposite directions
the streamline of $A$ generates a flow at location of $B$
this leads to a force: ${\vec{F}}_{21} = ρ {\vec{u}}_{1} \times {\vec{K}}_{2}$ , which will be directed towards $A$
$B$ rotates in the irrotational rotation of $A$ , at a distance $d$ ,

\begin{matrix} A : u_{θ} = \frac{K}{2 π r} \\ at B : u_{θ} = \frac{K}{2 π d} \end{matrix}

u_{θ} (1) > u_{θ} (2) ⟹ P (1) < P (2)

likewise, $A$ is pushed towards vortex $B$
but this is just the initial phase of the vortex interaction
the velocity caused by the magnus force causes another magnus force
the pair of vortices drift in the $y$ -direction
the drift becomes steady

u_{d r i f t} = \frac{K}{2 π d}

this speed is exactly the speed of the flow of A at B
since there is no shear, there is no magnus effect, no acceleration, and therefore the motion is steady
thus, the vortices are not attracted towards each other

same direction

now considering that the vortices are rotating in the same direction
initially, they repel each other
the vortices rotate around the axis between them