Biot–Savart law for vortex filaments

Remarks

The idea of the proof: since $\nabla \cdot \mathbf{u} = 0$, one can write $\mathbf{u} = \nabla \times \mathbf{A}$ with $\nabla \cdot \mathbf{A} = 0$, so $-\Delta \mathbf{A} = \boldsymbol{\omega}$. Solving with the Newtonian potential gives $\mathbf{A}(x) = \frac{1}{4\pi}\int \frac{\boldsymbol{\omega}(y)}{\abs{x - y}}\, dy$, and taking the curl gives the formula. Uniqueness holds because the difference of two solutions is divergence-free, curl-free and decays at infinity, hence it is harmonic and zero. Example: for a straight infinite filament along the $z$-axis, the formula gives the azimuthal speed $\Gamma / (2\pi r)$ at distance $r$, the velocity of a point vortex. The filament formula diverges on the curve itself. The self-induced velocity of a curved filament grows like the logarithm of one over the core radius, so a real vortex needs a core of finite size.

Going deeper

For a student

The law recovers the velocity from the vorticity, which is the key step in vortex methods and in the vorticity form of the Euler equations. In aerodynamics it gives the downwash of the trailing vortices in lifting-line theory and the influence coefficients of the vortex lattice method. For a finite straight segment it can be integrated in closed form, which makes it cheap to evaluate for networks of straight vortex pieces.