Editing Vorticity equation

{{Short description|Equation describing the evolution of the vorticity of a fluid particle as it flows}}
{{Continuum mechanics|cTopic=Fluid mechanics}}

The '''vorticity equation''' of [[fluid dynamics]] describes the evolution of the [[vorticity]] {{math|'''ω'''}} of a particle of a [[fluid dynamics|fluid]] as it moves with its [[flow (fluid)|flow]]; that is, the local rotation of the fluid (in terms of [[vector calculus]] this is the [[curl (mathematics)|curl]] of the [[flow velocity]]). The governing equation is:{{Equation box 1|cellpadding|border|indent=:|equation=<math> \begin{align}
\frac{D\boldsymbol\omega}{Dt} &= \frac{\partial \boldsymbol \omega}{\partial t} + (\mathbf u \cdot \nabla) \boldsymbol \omega \\
&= (\boldsymbol \omega \cdot \nabla) \mathbf u - \boldsymbol \omega (\nabla \cdot \mathbf u) + \frac{1}{\rho^2}\nabla \rho \times \nabla p + \nabla \times \left( \frac{\nabla \cdot \tau}{\rho} \right) + \nabla \times \left( \frac{\mathbf B}{\rho} \right) \end{align} </math>|border colour=#0073CF|background colour=#F5FFFA}}where {{math|{{sfrac|''D''|''Dt''}}}} is the [[material derivative]] operator, {{math|'''u'''}} is the [[flow velocity]], {{mvar|ρ}} is the local fluid [[density]], {{mvar|p}} is the local [[pressure]], {{mvar|τ}} is the [[viscous stress tensor]] and {{math|'''B'''}} represents the sum of the external [[body force]]s. The first source term on the right hand side represents [[vortex stretching]].

The equation is valid in the absence of any concentrated [[torque]]s and line forces for a [[Compressibility|compressible]], [[Newtonian fluid]]. In the case of [[incompressible flow]] (i.e., low [[Mach number]]) and [[isotropic]] fluids, with [[conservative force|conservative]] body forces, the equation simplifies to the '''vorticity transport equation''':
:<math>\frac{D\boldsymbol\omega}{Dt} = \left(\boldsymbol{\omega} \cdot \nabla\right) \mathbf{u} + \nu \nabla^2 \boldsymbol{\omega}</math>
where {{mvar|ν}} is the [[viscosity|kinematic viscosity]] and <math>\nabla^{2}</math> is the [[Laplace operator]]. Under the further assumption of two-dimensional flow, the equation simplifies to:
:<math>\frac{D\boldsymbol\omega}{Dt} =  \nu \nabla^2 \boldsymbol{\omega}</math>

==Physical interpretation==

* The term {{math|{{sfrac|''D'''''ω'''|''Dt''}}}} on the left-hand side is the [[substantive derivative|material derivative]] of the vorticity vector {{math|'''ω'''}}. It describes the rate of change of vorticity of the moving fluid particle. This change can be attributed to [[steady state flow|unsteadiness]] in the flow ({{math|{{sfrac|∂'''ω'''|∂''t''}}}}, the ''unsteady term'') or due to the motion of the fluid particle as it moves from one point to another ({{math|('''u''' ∙ ∇)'''ω'''}}, the ''[[convection]] term'').
* The term {{math|('''ω''' ∙ ∇) '''u'''}} on the right-hand side describes the stretching or tilting of vorticity due to the flow velocity gradients. Note that {{math|('''ω''' ∙ ∇) '''u'''}} is a vector quantity, as {{math|'''ω''' ∙ ∇}} is a scalar differential operator, while {{math|∇'''u'''}} is a nine-element tensor quantity.
* The term {{math|'''ω'''(∇ ∙ '''u''')}} describes [[vortex stretching|stretching of vorticity]] due to flow compressibility. It follows from the Navier-Stokes equation for [[continuity equation|continuity]], namely <math display="block">\begin{align}
  \frac{\partial\rho}{\partial t} + \nabla \cdot\left(\rho \mathbf u\right) &= 0 \\
  \Longleftrightarrow \nabla \cdot \mathbf{u} &= -\frac{1}{\rho}\frac{d\rho}{dt} = \frac{1}{v}\frac{dv}{dt}
\end{align}</math> where {{math|1=''v'' = {{sfrac|1|''ρ''}}}} is the [[specific volume]] of the fluid element. One can think of {{math|∇ ∙ '''u'''}} as a measure of flow compressibility. Sometimes the negative sign is included in the term.
* The term {{math|{{sfrac|1|''ρ''<sup>2</sup>}}∇''ρ'' × ∇''p''}} is the [[baroclinity|baroclinic term]]. It accounts for the changes in the vorticity due to the intersection of density and pressure surfaces.
* The term {{math|∇ × ({{sfrac|∇ ∙ ''τ''|''ρ''}})}}, accounts for the diffusion of vorticity due to the viscous effects.
* The term {{math|∇ × '''B'''}} provides for changes due to external body forces. These are forces that are spread over a three-dimensional region of the fluid, such as [[gravity]] or [[electromagnetic force]]s. (As opposed to forces that act only over a surface (like [[drag coefficient|drag]] on a wall) or a line (like [[surface tension]] around a [[Meniscus (liquid)|meniscus]]).

=== Simplifications ===
* In case of [[conservative force|conservative body forces]], {{math|1=∇ × '''B''' = 0}}.
* For a [[barotropic|barotropic fluid]], {{math|1=∇''ρ'' × ∇''p'' = 0}}. This is also true for a constant density fluid (including incompressible fluid) where {{math|1=∇''ρ'' = 0}}. Note that this is not the same as an [[incompressible flow]], for which the barotropic term cannot be neglected. 
** This note seems to be talking about the fact that conservation of momentum says {{math|1=<math>\frac{D \rho}{D t} + \rho (\nabla \cdot \mathbf u) = \frac{\partial \rho}{\partial t} + \mathbf u \cdot \nabla \rho + \rho (\nabla \cdot \mathbf u) = 0</math>}} and there's a difference between assuming that ρ=constant (the 'incompressible fluid' option, above) and that <math>\nabla \cdot \mathbf u = 0
</math> (the 'incompressible flow' option, above). With the first assumption, conservation of momentum implies (for non-zero density) that <math>\nabla \cdot \mathbf u = 0
</math>; whereas the second assumption doesn't necessary imply that ρ is constant. This second assumption only strictly requires that the time rate of change of the density is compensated by the gradient of the density, as in:{{math|1=<math>\frac{\partial \rho}{\partial t} = - \mathbf u \cdot \nabla \rho</math>}}. You can make sense of this by considering the ideal gas law  {{math|1=''p'' = ''ρRT''}} (which is valid if the Reynolds number is large enough that viscous friction becomes unimportant.) Then, even for an adiabatic, chemically-homogenous fluid, the density can vary when the pressure changes, e.g. with Bernoulli.
* For [[inviscid]] fluids, the viscosity tensor {{mvar|τ}} is zero.

Thus for an inviscid, barotropic fluid with conservative body forces, the vorticity equation simplifies to
: <math>\frac{D}{Dt} \left( \frac{\boldsymbol \omega}{\rho} \right) = \left( \frac{\boldsymbol\omega}{\rho} \right) \cdot \nabla \mathbf u </math>

Alternately, in case of incompressible, inviscid fluid with conservative body forces, 
: <math>\frac{D \boldsymbol \omega}{Dt} = \frac{\partial \boldsymbol \omega}{\partial t} + (\mathbf u \cdot \nabla)\boldsymbol \omega = (\boldsymbol \omega \cdot \nabla) \mathbf u </math><ref>{{cite book |last1=Fetter |first1=Alexander L. |last2=Walecka |first2=John D. |title=Theoretical Mechanics of Particles and Continua |date=2003 |publisher=Dover Publications |isbn=978-0-486-43261-8 |page=351 |edition=1st}}</ref>

For a brief review of additional cases and simplifications, see also.<ref>{{cite web| first=K.&nbsp;P.| last=Burr |title=Marine Hydrodynamics, Lecture 9| website=MIT Lectures| url=https://ocw.mit.edu/courses/mechanical-engineering/2-20-marine-hydrodynamics-13-021-spring-2005/lecture-notes/lecture9.pdf}}</ref> For the vorticity equation in turbulence theory, in context of the flows in oceans and atmosphere, refer to.<ref>{{cite web | first=Richard L. | last=Salmon | title=Lectures on Geophysical Fluid Dynamics, Chapter 4 | website=Oxford University Press; 1 edition (February 26, 1998) | url=http://pordlabs.ucsd.edu/rsalmon/chap4.pdf}}</ref>

==Derivation==
The vorticity equation can be derived from the [[Navier–Stokes equations|Navier–Stokes]] equation for the conservation of [[angular momentum]]. In the absence of any concentrated [[torque]]s and line forces, one obtains:

:<math>\frac{D \mathbf{u}}{D t} = \frac{\partial \mathbf{u}}{\partial t} + \left(\mathbf{u} \cdot \nabla\right) \mathbf{u} = -\frac{1}{\rho} \nabla p + \frac{\nabla \cdot \tau}{\rho} + \frac{\mathbf{B}}{\rho}</math>

Now, vorticity is defined as the curl of the flow velocity vector; taking the [[Curl_(mathematics)|curl]] of momentum equation yields the desired equation. The following identities are useful in derivation of the equation:
:<math>\begin{align}
               \boldsymbol{\omega} &= \nabla \times \mathbf{u} \\
  \left(\mathbf{u} \cdot \nabla\right)\mathbf{u}
                                   &= \nabla \left(\frac{1}{2}\mathbf{u} \cdot \mathbf{u}\right) - \mathbf{u} \times \boldsymbol\omega \\
  \nabla \times \left(\mathbf{u} \times \boldsymbol{\omega} \right)
                                   &= -\boldsymbol{\omega} \left(\nabla \cdot \mathbf{u}\right) + \left(\boldsymbol{\omega} \cdot \nabla\right) \mathbf{u} - \left(\mathbf{u} \cdot \nabla\right) \boldsymbol{\omega} \\[4pt]
  \nabla \cdot \boldsymbol{\omega} &= 0 \\[4pt]
         \nabla \times \nabla \phi &= 0
\end{align}</math>

where <math>\phi</math> is any scalar field.

==Tensor notation==
The vorticity equation can be expressed in [[tensor notation]] using [[Einstein notation|Einstein's summation convention]] and the [[Levi-Civita symbol]] {{mvar|e<sub>ijk</sub>}}:

:<math>\begin{align}
\frac{D\omega_i}{Dt} &= \frac{\partial \omega_i}{\partial t} + v_j \frac{\partial \omega_i}{\partial x_j} \\
&= \omega_j \frac{\partial v_i}{\partial x_j} - \omega_i \frac{\partial v_j}{\partial x_j}
+ e_{ijk}\frac{1}{\rho^2}\frac{\partial \rho}{\partial x_j}\frac{\partial p}{\partial x_k}
+ e_{ijk}\frac{\partial}{\partial x_j}\left(\frac{1}{\rho}\frac{\partial \tau_{km}}{\partial x_m}\right)
+ e_{ijk}\frac{\partial B_k }{\partial x_j} \end{align}</math>

==In specific sciences==

===Atmospheric sciences===
In the [[atmospheric sciences]], the vorticity equation can be stated in terms of the absolute vorticity of air with respect to an inertial frame, or of the vorticity with respect to the rotation of the Earth. The absolute version is

:<math>
  \frac{d\eta}{dt} = -\eta\nabla_\text{h} \cdot \mathbf{v}_\text{h} - \left(
    \frac{\partial w}{\partial x} \frac{\partial v}{\partial z} -
    \frac{\partial w}{\partial y} \frac{\partial u}{\partial z}
  \right) -
  \frac{1}{\rho^2}\mathbf{k} \cdot \left(\nabla_\text{h} p \times \nabla_\text{h}\rho\right)
</math>

Here, {{mvar|η}} is the polar ({{mvar|z}}) component of the vorticity, {{mvar|ρ}} is the atmospheric [[density]], {{mvar|u}}, {{mvar|v}}, and w are the components of [[wind velocity]], and {{math|∇<sub>h</sub>}} is the 2-dimensional (i.e. horizontal-component-only) [[del]].

==See also==
* [[Vorticity]]
* [[Barotropic vorticity equation]]
* [[Vortex stretching]]
* [[Burgers vortex]]

==References==
{{reflist}}

== Further reading ==
*{{cite journal | first1= Utpal |last1=Manna |first2=S.&nbsp;S. |last2=Sritharan | title= Lyapunov Functionals and Local Dissipativity for the Vorticity Equation in {{mvar|L{{isup|p}}}} and Besov spaces | journal = Differential and Integral Equations | volume= 20 |number =5 |year= 2007|pages= 581–598|doi=10.57262/die/1356039440 |s2cid=50701138 |arxiv=0802.2898 }}
* {{cite book|first1=V. |last1=Barbu |first2=S.&nbsp;S. |last2=Sritharan |chapter={{mvar|M}}-Accretive Quantization of the Vorticity Equation |title=Semi-Groups of Operators: Theory and Applications |editor-first=A.&nbsp;V. |editor-last=Balakrishnan |publisher=Birkhauser |location=Boston |date=2000 |pages=296–303 |chapter-url=http://www.nps.edu/Academics/Schools/GSEAS/SRI/BookCH12.pdf}}
* {{cite journal|first=A.&nbsp;M. |last=Krigel |title=Vortex evolution |journal=Geophysical & Astrophysical Fluid Dynamics |date=1983 |volume=24 |issue=3 |pages=213–223|doi=10.1080/03091928308209066 |bibcode=1983GApFD..24..213K }}

{{More citations needed|date=May 2009}}

[[Category:Equations of fluid dynamics]]
[[Category:Transport phenomena]]