Velocity induced by a straight vortex filament

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 $$O^{'}$$.

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 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/… – Deep Aug 23 at 12:47
• Thanks for the link! That cleared my doubt. I was not thinking in terms of 3D. – Pavan Aug 23 at 16:35