Editing Vortex (section)

===Vortex types===
In theory, the speed {{mvar|u}} of the particles (and, therefore, the vorticity) in a vortex may vary with the distance {{mvar|r}} from the axis in many ways.  There are two important special cases, however:
[[File:Rotational vortex.gif|thumb|A rigid-body vortex]]
* If the fluid rotates like a rigid body – that is, if the angular rotational velocity {{math|Ω}} is uniform, so that {{mvar|u}} increases proportionally to the distance {{mvar|r}} from the axis – a tiny ball carried by the flow would also rotate about its center as if it were part of that rigid body.  In such a flow, the vorticity is the same everywhere: its direction is parallel to the rotation axis, and its magnitude is equal to twice the uniform angular velocity {{math|Ω}} of the fluid around the center of rotation.
*:<math>\begin{align}
  \vec\Omega &= (0, 0, \Omega) , \quad \vec{r} = (x, y, 0) , \\ 
     \vec{u} &= \vec{\Omega} \times \vec{r} = (-\Omega y, \Omega x, 0) , \\
  \vec\omega &= \nabla \times \vec{u} = (0, 0, 2\Omega) = 2\vec{\Omega} .
\end{align}</math>
{{clear}}
[[File:Irrotational vortex.gif|thumb|An irrotational vortex]]
* If the particle speed {{mvar|u}} is inversely proportional to the distance {{mvar|r}} from the axis, then the imaginary test ball would not rotate over itself; it would maintain the same orientation while moving in a circle around the vortex axis. In this case the vorticity <math>\vec \omega</math> is zero at any point not on that axis, and the flow is said to be ''irrotational''.
*:<math>\begin{align}
  \vec{\Omega} &= \left(0, 0, \alpha r^{-2}\right) , \quad \vec{r} = (x, y, 0) , \\
       \vec{u} &= \vec{\Omega} \times \vec{r} = \left(-\alpha y r^{-2}, \alpha x r^{-2}, 0\right) , \\
  \vec{\omega} &= \nabla \times \vec{u} = 0 .
\end{align}</math>
{{clear}}

====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]].

====Rotational vortices====
[[File:Saturn north polar vortex 2012-11-27.jpg|thumb|[[Saturn|Saturn's]] north polar vortex]]
A rotational vortex – a vortex that rotates in the same way as a rigid body – cannot exist indefinitely in that state except through the application of some extra force, that is not generated by the fluid motion itself. It has non-zero vorticity everywhere outside the core. Rotational vortices are also called rigid-body vortices or forced vortices.

For example, if a water bucket is spun at constant angular speed {{mvar|w}} about its vertical axis, the water will eventually rotate in rigid-body fashion. The particles will then move along circles, with velocity {{mvar|u}} equal to {{mvar|wr}}.<ref name=LJC7.5/> In that case, the free surface of the water will assume a [[paraboloid|parabolic]] shape.

In this situation, the rigid rotating enclosure provides an extra force, namely an extra pressure [[gradient]] in the water, directed inwards, that prevents transition of the rigid-body flow to the irrotational state.