Lecture 03 - Dynamics

Let us consider unbounded, inviscid, incompressible fluid flows. In the absence of external force, the motion of such fluid with unit density is described by the Euler equations:

where $u(x,t)$ is the velocity and $p$ is the pressure.

The velocity $u(x)$ can be determined from the vorticity $\omega (x)$ through the Biot-Savart Law:

One can avoid the singularity in equation $(1)$ by simply ignoring the nonlocal contribution of the filament and replace the Biot-Savart’s law by a velocity expression that depends only
on the local curvature of the vortex filament. If a vortex filament described by $\gamma (s)$, where $s$ is an arc length parameter measured along the filament. The equation of a filament is:

Recall that ${\gamma }_s=T$ and ${\gamma }_{ss}=T_s=\kappa N$
Thus: ${\gamma }_t=T\times \kappa N=\kappa B$, which is known as a binormal flow.

We now contemplate on what we call 'moving curves', $\gamma (s,t)$, which the curve is not only parameterized by arc-length by also by time. That is the the curves evolve over time.
Consider arc length $|{\partial }_s\gamma |$:

As a consequence we get the following conserved quantities
Energy: $\int |{\partial }_{ss}\gamma {|}^{2}ds=\int {\kappa }^{2}ds$
Linear Momentum: $\int \gamma \times {\partial }_s\gamma ds$
Angular Momentum: $\int \gamma \times (\gamma \times {\partial }_s\gamma )ds$
Helicity: $\int {\kappa }^2\tau ds$
Total Torsion: $\int \tau ds$

Let us define a complex curvature quantity, we associate to a curve $\gamma \to \psi =\kappa exp\left(i{\int }_0^s\tau d{s}^{\prime }\right)$.
Then: ${\psi }^{\ast }=\kappa exp(-i{\int }_0^s\tau d{s}^{\prime })$, thus: $\psi {\psi }^{\ast }={\kappa }^2$, thus $|\psi |=\kappa ⟹\frac{\psi }{|\psi |}=exp(i{\int }_0^s\tau d{s}^{\prime })$.

We now explicitly derive Da Rios system:

Since $\psi =|\psi |exp(i{\int }_0^s\tau d{s}^{\prime })$ we get that $i{\psi }_t=i{\kappa }_{t}exp(i{\int }_0^s\tau d{s}^{\prime })$

2D-Case

$\gamma$ is a solution for the planar filament equation.

iff $\kappa$ is a solution to the modified KDV equation:

Consider $\theta ={\int }_0^s\tau d{s}^{\prime }$
Bishop frame:
$\left(\begin{array}{c}T\\ N\\ B\end{array}\right)\to \left(\begin{array}{c}T\\ U\\ V\end{array}\right)$
Where:

Let us define ${\kappa }_1=\kappa cos(\theta )$ and ${\kappa }_2=\kappa sin(\theta )$ then our Frenet Serret equations change to:

Notice that $\kappa B=-{\kappa }_{2}U+{\kappa }_{1}V$, if we are considering a vortex filament then: ${\gamma }_t=\kappa B$ as noted here thus for a filament in the Bishop frame:

and since $\kappa N={\kappa }_{1}U+{\kappa }_{2}V$ we get: ${\gamma }_{ss}={\kappa }_{1}U+{\kappa }_{2}V$.

End of class after-notes