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In fluid dynamics (Navier-Stokes, incompressible case), a vortex filament, as I understand it, is a curve of points $x\in\mathbb R^3$ such that the velocity field $v\colon \mathbb R^3\times\mathbb R\to\mathbb R^3$ is parallel to its curl, $\nabla \times v$, also known as the vorticity. In other words, for each time $t$ we have a curve $\gamma_t\colon (a,b)\to\mathbb R^3$ that satisfies $$v(\gamma_t(s),t)=\lambda_{s,t}\nabla\times v(\gamma_t(s),t),$$ for some $\lambda_{s,t}\in\mathbb R$.

What is the equation governing the movement $d\gamma_t/dt$ of the vortex filament $\gamma_t$?

Thanks a lot in advance!

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  • $\begingroup$ You need to specify which media the vortex filaments is in otherwise there's no answer. i.e. quantum and classical description were quite different. $\endgroup$ – ShoutOutAndCalculate May 24 '19 at 10:45
  • $\begingroup$ Thanks. Does it work with that edit I did? $\endgroup$ – Zatrapilla May 24 '19 at 10:50

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