Consider a straight vortex filament as shown below. At each point, there's a point vortex of strength $\it T$. Consider that a point $P$ is there on the outermost circle of flow induced by the vortex about the origin $O$. That is the point $P$ lies in the plane containing the flow induced by the vortex at $O$. So, the only point on the vortex filament that will induce a flow at point $P$ is $O$.

Therefore, the vortex at point $O^{'}$ will not induce any flow at point $P$. Vortex at point $O^{'}$ can only induce flow at those points that are in the plane containing the flow induced by the vortex at $O^{'}$.

Vortex Filament

So, my question is how Biot-Savart Law is applied for a straight filament vortex? How can the segment $dl$ induce a flow at a point $P$ that is not in the plane? Only the vortex centered at the point of intersection of line and perpendicular from point $P$ to the line can induce the flow at $P$. Please clear my query.

Biot-Savart Law

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    $\begingroup$ Biot-Savart law explicitly states that a vortex filament of length $dl$ will induce velocity everywhere in the fluid, not just in the plane perpendicular to it. For a straight line vortex, here's a derivation: web.mit.edu/16.unified/www/SPRING/fluids/Spring2008/… $\endgroup$ – Deep Aug 23 at 12:47
  • $\begingroup$ Thanks for the link! That cleared my doubt. I was not thinking in terms of 3D. $\endgroup$ – Pavan Aug 23 at 16:35

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