Editing Vortex ring (section)

===Thin-core vortex rings===
The discontinuity introduced by the [[Dirac delta function]] prevents the computation of the speed and the [[kinetic energy]] of a circular vortex line. It is however possible to estimate these quantities for a vortex ring having a finite small thickness. For a thin vortex ring, the core can be approximated by a disk of radius <math>a</math> which is assumed to be infinitesimal compared to the radius of the ring <math>R</math>, i.e. <math>a/R \ll 1 </math>. As a consequence, inside and in the vicinity of the core ring, one may write: <math> r_1/r_2 \ll 1 </math>, <math>r_2 \approx 2R</math> and <math> 1- \lambda^2 \approx 4 r_1/R </math>, and, in the limit of <math>\lambda \approx 1 </math>, the elliptic integrals can be approximated by <math> K(\lambda) = 1/2 \ln\left({16}/{(1-\lambda^2)}\right) </math> and <math> E(\lambda) = 1 </math>.<ref name="lamb1932"/>

For a uniform [[vorticity]] distribution <math>\omega(r,x)=\omega_0</math> in the disk, the [[Stokes stream function]] can therefore be approximated by
<!-- minus sign missing and wrong bracket, see Lamb p. 241 --></ref>
<math display="block"> \psi(r,x)=-\frac{\omega_0}{2\pi}R\iint{\left(\ln\frac{8R}{r_1}-2\right)\,dr'dx'} </math>

The resulting [[circulation (physics)|circulation]] <math>\Gamma</math>, hydrodynamic impulse <math>I</math> and [[kinetic energy]] <math>E</math> are
<math display="block">\begin{align}
\Gamma &= \pi\omega_0 a^2\\
I &= \rho\pi\Gamma R^2 \\
E &= \frac{1}{2}\rho\Gamma^2R\left(\ln\frac{8R}{a}-\frac{7}{4}\right)
\end{align}</math>

It is also possible to find the translational ring speed (which is finite) of such isolated thin-core vortex ring:
<math display="block"> U=\frac{E}{2I}+\frac{3}{8\pi}\frac{\Gamma}{R} </math>
which finally results in the well-known expression found by [[Lord Kelvin|Kelvin]] and published in the English translation by [[Peter Tait (physicist)|Tait]] of [[von Helmholtz]]'s paper:<ref name="helmholtz1858"/><ref name="helmholtz1867"/><ref name="lamb1932"/>
<math display="block"> U=\frac{\Gamma}{4\pi R}\left(\ln\frac{8R}{a}-\frac{1}{4}\right) </math>