Editing Vortex ring (section)

===Vortex formation number===

Prior to Gharib ''et al.'' (1998),<ref name="gharib1998">{{Cite journal|last1=Gharib|first1=M.|last2=Rambod|first2=E.|last3=Shariff|first3=K.|date=1998|title=A universal time scale for vortex ring formation|url=http://dx.doi.org/10.1017/s0022112097008410|journal=Journal of Fluid Mechanics|volume=360|issue=1|pages=121–140|doi=10.1017/s0022112097008410|bibcode=1998JFM...360..121G|s2cid=50685764 |url-access=subscription}}</ref> few studies had focused on the formation of vortex rings generated with long stroke-to-diameter ratios <math>L/D</math>, where <math>L</math> is the length of the column of fluid discharged through the exhaust and <math>D</math> is the diameter of the exhaust. For short stroke ratios, only one isolated vortex ring is generated and no fluid is left behind in the formation process. For long stroke ratios, however, the vortex ring is followed by some energetic fluid, referred as the trailing jet. On top of showing experimental evidence of the phenomenon, an explanation of the phenomenon was provided in terms of energy maximisation invoking a variational principle first reported by [[Lord Kelvin|Kelvin]]<ref>{{Cite journal|last1=Thomson|first1=W.|date=1878|title=1. Vortex statics.|url=https://archive.org/details/proceedingsroya30edingoog/page/n82/mode/2up|journal=Proceedings of the Royal Society of Edinburgh |volume=9|pages=59–73|doi=10.1017/S0370164600031679}}</ref> and later proven by Benjamin (1976),<ref>{{cite conference |url=https://link.springer.com/chapter/10.1007/BFb0088744 |title=The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics. |first=T. B. |last=Benjamin |date=1976 |volume=503 |book-title=Applications of Methods of Functional Analysis to Problems in Mechanics |publisher=Springer Berlin Heidelberg |pages=8–29|doi=10.1007/BFb0088744 |url-access=subscription }}</ref> or Friedman & Turkington (1981).<ref>{{cite journal |last1=Friedman |first1=A. |last2=Turkington |first2=B. |date=1981 |title=Vortex rings: existence and asymptotic estimates. |url=https://www.ams.org/journals/tran/1981-268-01/S0002-9947-1981-0628444-6/S0002-9947-1981-0628444-6.pdf |journal=Transactions of the American Mathematical Society |volume=268 |issue=1 |pages=1–37|doi=10.1090/S0002-9947-1981-0628444-6 |doi-access=free }}</ref> Ultimately, Gharib ''et al.'' (1998)<ref name="gharib1998" /> observed the transition between these two states to occur at a non-dimensional time <math>t^*=Ut/D</math>, or equivalently a stroke ratio <math>L/D</math>, of about 4. The robustness of this number with respect to initial and boundary conditions suggested the quantity to be a universal constant and was thus named ''formation number''.

The phenomenon of 'pinch-off', or detachment, from the feeding starting jet is observed in a wide range of flows observed in nature.<ref>{{cite journal |last1=Dabiri |first1=J. O. |date=2009 |title=Optimal vortex formation as a unifying principle in biological propulsion |url=https://doi.org/10.1146/annurev.fluid.010908.165232 |journal=Annual Review of Fluid Mechanics |volume=41 |issue=1 |pages=17–33 |doi=10.1146/annurev.fluid.010908.165232|bibcode=2009AnRFM..41...17D }}</ref><ref name="dabiri2005">{{Cite journal|last1=Dabiri|first1=J. O.|last2=Gharib|first2=M.|date=2005|title=The role of optimal vortex formation in biological fluid transport |journal=Proceedings of the Royal Society B: Biological Sciences|volume=272|issue=1572|pages=1557–1560|doi=10.1098/rspb.2005.3109|pmid=16048770|pmc=1559837}}</ref> For instance, it was shown that biological systems such as the human heart or swimming and flying animals generate vortex rings with a stroke-to-diameter ratio close to the formation number of about 4, hence giving ground to the existence of an optimal vortex ring formation process in terms of propulsion, thrust generation and mass transport.<ref>{{cite journal |last1=Krueger |first1=P. S. |date=2003 |title=The significance of vortex ring formation to the impulse and thrust of a starting jet. |url=https://aip.scitation.org/doi/citedby/10.1063/1.1564600 |journal=Physics of Fluids |volume=15 |issue=5 |pages=1271–1281 |doi=10.1063/1.1564600|bibcode=2003PhFl...15.1271K |url-access=subscription }}</ref> In particular, the squid ''[[lolliguncula brevis]]'' was shown to propel itself by periodically emitting vortex rings at a stroke-ratio close to 4.<ref>{{Cite journal|last1=Stewart|first1=W. J.|last2=Bartol|first2=I. K.|last3=Krueger|first3=P. S.|date=2010-05-28|title=Hydrodynamic fin function of brief squid, Lolliguncula brevis|journal=Journal of Experimental Biology|volume=213|issue=12|pages=2009–2024|doi=10.1242/jeb.039057|pmid=20511514|issn=0022-0949|doi-access=free|bibcode=2010JExpB.213.2009S }}</ref><ref name="dabiri2005"/> Moreover, in another study by Gharib ''et al'' (2006),<ref name="gharib2006"/> the formation number was used as an indicator to monitor the health of the human heart and identify patients with [[dilated cardiomyopathy]].