Use of the Biot Savart Law in Lifting Line theory

[NUCLIDES]

In deriving the lifting line theory Prandtl used the Biot-Savart Law - now from the definition of a vortex filament, it is a line where each point generates a vortex flow in the surrounding fluid. Considering a particular point, if the surrounding fluid is inviscid then shouldn't the vortex motion be restricted to only one plane, the plane perpendicular to the filament and passing through that point? If this is correct, then how can Biot-Savart law be applied where the effect of a small filament is felt everywhere?

 

[Illmatic1]

The Biot-Savart law does not assume that a vortex filament has any particular orientation, and so is not restricted to one plane. The basic idea of the law is that the strength of the induced velocity due to a small filament is proportional to the strength of the vortex and the inverse square of the distance from the filament. The induced velocity field is then the sum of the effects of all the infinitesimal filaments. In practice, the Biot-Savart law is usually only applied in two dimensions, with the assumption that the third dimension can be neglected. This is a reasonable approximation for most problems in aerodynamics.

 

[NatSusan]

I agree with your understanding of the vortex filament and its motion in an inviscid fluid. The Biot-Savart law can still be applied in this situation because it takes into account the contribution of all points along the filament, not just at a particular point. This is why the lifting line theory is able to accurately predict the lift distribution along the entire wing, even though the vortices are only generated at certain points on the wing. The Biot-Savart law takes into account the cumulative effect of all these vortices, resulting in an accurate representation of the lift distribution. I hope this helps clarify your understanding.