Editing Vortex (section)

====Irrotational vortices====
[[File:IrrotationalVortexFlow-anim-frame0.png|thumb|upright|Pathlines of fluid particles around the axis (dashed line) of an ideal irrotational vortex. (See [[commons:File:IrrotationalVortexFlow-anim.gif|animation]].)]]
In the absence of external forces, a vortex usually evolves fairly quickly toward the irrotational flow pattern{{Citation needed|date=August 2015}}, where the flow velocity {{mvar|u}} is inversely proportional to the distance {{mvar|r}}. Irrotational vortices are also called ''free vortices''.

For an irrotational vortex, the [[circulation (fluid dynamics)|circulation]] is zero along any closed contour that does not enclose the vortex axis; and has a fixed value, {{math|Γ}}, for any contour that does enclose the axis once.<ref name=LJC7.5>{{harvnb|Clancy|1975|loc=sub-section 7.5}}</ref> The tangential component of the particle velocity is then <math>u_{\theta} = \tfrac{\Gamma}{2 \pi r}</math>. The angular momentum per unit mass relative to the vortex axis is therefore constant, <math> r u_{\theta} = \tfrac{\Gamma}{2 \pi}</math>.

The ideal irrotational vortex flow in free space is not physically realizable, since it would imply that the particle speed (and hence the force needed to keep particles in their circular paths) would grow without bound as one approaches the vortex axis. Indeed, in real vortices there is always a core region surrounding the axis where the particle velocity stops increasing and then decreases to zero as {{mvar|r}} goes to zero. Within that region, the flow is no longer irrotational: the vorticity <math>\vec \omega</math> becomes non-zero, with direction roughly parallel to the vortex axis. The [[Rankine vortex]] is a model that assumes a rigid-body rotational flow where {{mvar|r}} is less than a fixed distance {{mvar|r}}<sub>0</sub>, and irrotational flow outside that core regions.

In a viscous fluid, irrotational flow contains viscous dissipation everywhere, yet there are no net viscous forces, only viscous stresses.<ref name="SirakovGreitzer2005">{{cite journal|last1=Sirakov|first1=B. T.|last2=Greitzer|first2=E. M.|last3=Tan|first3=C. S.|title=A note on irrotational viscous flow|journal=Physics of Fluids|volume=17|issue=10|year=2005|pages=108102–108102–3|issn=1070-6631|doi=10.1063/1.2104550|bibcode=2005PhFl...17j8102S}}</ref> Due to the dissipation, this means that sustaining an irrotational viscous vortex requires continuous input of work at the core (for example, by steadily turning a cylinder at the core). In free space there is no energy input at the core, and thus the compact vorticity held in the core will naturally diffuse outwards, converting the core to a gradually-slowing and gradually-growing rigid-body flow, surrounded by the original irrotational flow. Such a decaying irrotational vortex has an exact solution of the viscous [[Navier–Stokes equations]], known as a [[Lamb–Oseen vortex]].