Editing Vortex ring (section)

===Historical studies===
The formation of vortex rings has fascinated the scientific community for more than a century, starting with [[William Barton Rogers]]<ref>{{cite journal |last1= Rogers|first1= W. B. |date=1858 |title=On the formation of rotating rings by air and liquids under certain conditions of discharge |url=https://www.biodiversitylibrary.org/item/113539#page/255/mode/1up |journal=Am. J. Sci. Arts |volume=26 |pages=246–258 }}</ref> who made sounding observations of the formation process of air vortex rings in air, air rings in liquids, and liquid rings in liquids. In particular, [[William Barton Rogers]] made use of the simple experimental method of letting a drop of liquid fall on a free liquid surface; a falling colored drop of liquid, such as milk or dyed water, will inevitably form a vortex ring at the interface due to the surface tension.{{fact|date=April 2024}}

Vortex rings were first mathematically analyzed by the German physicist [[Hermann von Helmholtz]], in his 1858 paper ''On Integrals of the Hydrodynamical Equations which Express Vortex-motion''.<ref name="helmholtz1858">{{cite journal |last1=Helmholtz |first1=H. |date=1858 |title=3. Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen |url=https://ia800708.us.archive.org/view_archive.php?archive=/22/items/crossref-pre-1909-scholarly-works/10.1515%252Fcrll.1857.53.1.zip&file=10.1515%252Fcrll.1858.55.25.pdf |journal=Journal für die reine und angewandte Mathematik |volume=55 |pages=25–55 |doi=10.1515/9783112336489-003|isbn=9783112336472 }}</ref><ref name="helmholtz1867">{{cite journal |last1=Helmholtz |first1=H. |date=1867 |title=LXIII. On integrals of the hydrodynamical equations, which express vortex-motion |url=https://www.tandfonline.com/doi/pdf/10.1080/14786446708639824 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=33 |issue=226 |pages=485–512 |doi=10.1080/14786446708639824|url-access=subscription }}</ref><ref>{{cite book |last1=Moffatt |first1=K. |chapter=Vortex Dynamics: The Legacy of Helmholtz and Kelvin |date=2008 |editor1-last=Borisov |editor1-first=A. V. |editor2-last=Kozlov |editor2-first=V. V. |editor3-last=Mamaev |editor3-first=I. S. |editor4-last=Sokolovskiy |editor4-first=M. A. |title=IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence |chapter-url=https://link.springer.com/chapter/10.1007%2F978-1-4020-6744-0_1 |series=IUTAM Bookseries |publisher=Springer Netherlands |volume=6 |pages=1–10 |doi=10.1007/978-1-4020-6744-0_1|isbn=978-1-4020-6743-3 }}</ref>