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| {{Continuum mechanics|cTopic=Fluid mechanics}}
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| The '''vorticity equation''' of [[fluid dynamics]] describes evolution of the [[vorticity]] {{vec|''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 [[velocity field]]).
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| The equation is:
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| :<math>\begin{align}
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| \frac{d\vec\omega}{dt} &= \frac{\partial \vec \omega}{\partial t} + (\vec v \cdot \vec \nabla) \vec \omega \\
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| &= (\vec \omega \cdot \vec \nabla) \vec v - \vec \omega (\vec \nabla \cdot \vec v) + \frac{1}{\rho^2}\vec \nabla \rho \times \vec \nabla p + \vec \nabla \times \left( \frac{\vec \nabla \cdot \tau}{\rho} \right) + \vec \nabla \times \vec B
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| \end{align}</math>
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| where ''d''/''dt'' the [[total time derivative]] operator, also denoted by in capital D notation as ''D''/''Dt'', {{vec|''v''}} is the [[velocity field]], ''ρ'' is the local fluid [[density]], ''p'' is the local [[pressure]], ''τ'' is the [[viscous stress tensor]] and {{vec|''B''}} represents the sum of the external [[body force]]s. The first source term on the right hand side represents [[vortex stretching]].
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| The equation is valid in the absence of any concentrated [[torque]]s and line forces, for a compressible [[Newtonian fluid]].
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| In the case of [[compressibility|incompressible]] (i.e. low [[Mach number]]) and [[isotropic]] fluids, with [[conservative force|conservative]] body forces, the equation simplifies to the '''vorticity transport equation'''
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| :<math>
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| {d\vec{\omega} \over dt} = (\vec{\omega} \cdot \nabla) \vec{v} + \nu \nabla^2 \vec{\omega}
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| </math>
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| where ''ν'' is the [[viscosity|kinematic viscosity]] and ∇<sup>2</sup> is the [[Laplace operator]].
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| ==Physical Interpretation==
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| * The term ''d{{vec|ω}}/dt'' on the left-hand side is the [[substantive derivative|material derivative]] of the vorticity vector {{vec|''ω''}}. It describes the rate of change of vorticity (that is, the [[angular acceleration]]) of the fluid particle. This change can be attributed to [[steady state flow|unsteadiness]] in the flow (∂{{vec|''ω''}}/∂''t'', the ''unsteady term'') or due to the motion of the fluid particle as it moves from one point to another ({{vec|''v''}} ∙ ({{vec|∇}}{{vec|''ω''}}), the ''[[convection]] term'').
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| * The term ({{vec|''ω''}}∙ {{vec|∇}}) {{vec|''v''}} on the right-hand side describes the stretching or tilting of vorticity due to the velocity gradients. Note that {{vec|∇}}{{vec|''v''}} is an order-2 [[tensor]] with nine components.
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| * The term {{vec|''ω''}}({{vec|∇}} ∙ {{vec|''v''}}) describes [[vortex stretching|stretching of vorticity]] due to flow compressibility. It follows from the Navier-Stokes equation for [[continuity equation|continuity]],namely
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| ::<math>\frac{\partial \rho}{\partial t} + \vec \nabla \cdot(\rho \vec v) = 0 </math>
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| :or
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| ::<math> \vec \nabla \cdot \vec v = -\frac{1}{\rho} \frac{d \rho}{dt} = \frac{1}{V} \frac{dV}{dt} </math>
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| :where ''V'' = 1/''ρ'' is the [[specific volume]] of the fluid element. One can think of {{vec|∇}} ∙ {{vec|''V''}} as a measure of flow compressibility. Sometimes the negative sign is included in the term.
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| * The term (1/''ρ''<sup>2</sup>){{vec|∇}}''ρ'' × {{vec|∇}}''p'' is the [[Baroclinity|baroclinic term]]. It accounts for the changes in the vorticity due to the intersection of density and pressure surfaces.
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| * The term {{vec|∇}} × ({{vec|∇}} ∙ ''τ''/''ρ''), accounts for the diffusion of vorticity due to the viscous effects.
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| * The term {{vec|∇}} × {{vec|''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]]).
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| === Simplifications ===
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| * In case of [[conservative force|conservative body forces]], {{vec|∇}} × {{vec|''B''}} = 0.
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| * For a [[barotropic|barotropic fluid]], {{vec|∇}}''ρ'' × {{vec|∇}}''p'' = 0. This is also true for a constant density fluid (including incompressible fluid) where {{vec|∇}}''ρ'' = 0. Note that this is not the same as an [[incompressible flow]], for which the barotropic term cannot be neglected.
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| * For [[inviscid]] fluids, the viscosity tensor ''τ'' is zero.
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| Thus for an inviscid, barotropic fluid with conservative body forces, the vorticity equation simplifies to
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| : <math>\frac{d}{dt} \left( \frac{\vec \omega}{\rho} \right) = \left( \left( \frac{\vec\omega}{\rho} \right) \cdot \vec \nabla \right) \vec v </math>
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| Alternately, in case of incompressible, inviscid fluid with conservative body forces,
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| : <math>\frac{d \vec \omega}{dt} = (\vec \omega \cdot \vec \nabla) \vec v </math>
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| ==Derivation==
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| 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
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| :<math>
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| \frac{d \vec v}{d t} = \frac{\partial \vec v}{\partial t} + (\vec v \cdot \vec \nabla) \vec v = - \frac{1}{\rho} \vec \nabla p + \vec B + \frac{\vec \nabla \cdot \tau}{\rho}
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| </math>
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| Now, vorticity is defined as the curl of the velocity vector. Taking curl of momentum equation yields the desired equation.
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| The following identities are useful in derivation of the equation,
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| :<math>\vec \omega = \vec \nabla \times \vec v</math>
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| :<math> \vec v \cdot \vec \nabla \vec v = \vec \nabla (\tfrac{1}{2} \vec v \cdot \vec v) - \vec v \times \vec \omega </math>
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| :<math>\vec \nabla \times (\vec v \times \vec \omega ) = -\vec \omega (\vec \nabla \cdot \vec v) + (\vec \omega \cdot \vec \nabla ) \vec v - (\vec v \cdot \vec \nabla) \vec \omega </math>
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| :<math>\vec \nabla \times \vec \nabla \phi = 0 </math>, where ''ϕ'' is any scalar field.
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| :<math>\vec \nabla \cdot \vec \omega = 0 </math>
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| ==Tensor notation==
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| The vorticity equation can be expressed in [[tensor notation]] using [[Einstein notation|Einstein's summation convention]] and the [[Levi-Civita symbol]] ''e<sub>ijk</sub>'':
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| :<math>\begin{align}
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| \frac{d\omega_i}{dt} &= \frac{\partial \omega_i}{\partial t} + v_j \frac{\partial \omega_i}{\partial x_j} \\
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| &= \omega_j \frac{\partial v_i}{\partial x_j}
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| - \omega_i \frac{\partial v_j}{\partial x_j}
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| + e_{ijk}\frac{1}{\rho^2}\frac{\partial \rho}{\partial x_j}\frac{\partial p}{\partial x_k}
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| + e_{ijk}\frac{\partial}{\partial x_j}\left(\frac{1}{\rho}\frac{\partial \tau_{km}}{\partial x_m}\right)
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| + e_{ijk}\frac{\partial B_k }{\partial x_j}
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| \end{align}</math>
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| ==In specific sciences==
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| ===Atmospheric sciences===
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| 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
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| :<math>\frac{d \eta}{d t} = -\eta \nabla_h \cdot\vec{v}_h - \left( \frac{\partial \omega}{\partial x} \frac{\partial v}{\partial z} - \frac{\partial \omega}{\partial y} \frac{\partial u}{\partial z} \right) - \frac{1}{\rho^2} \vec{k} \cdot ( \nabla_h p \times \nabla_h \rho )</math>
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| Here, ''η'' is the polar (''z'') component of the vorticity, ''ρ'' is the atmospheric [[density]], ''u'', ''v'', and ''ω'' are the components of wind [[velocity]], and ∇<sub>''h''</sub> is the 2-dimensional (i.e. horizontal-component-only) [[del]].
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| ==See also==
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| * [[Vorticity]]
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| * [[Barotropic vorticity equation]]
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| * [[Vortex stretching]]
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| * [[Burgers vortex]]
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| ==References==
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| {{reflist}}
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| *{{citation
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| | author = Utpal Manna and S. S. Sritharan
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| | title= Lyapunov Functionals and Local Dissipativity for the Vorticity Equation in ''L<sup>p</sup>'' and Besov spaces,
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| | journal = Differential and Integral Equations
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| | volume= 20
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| |number =5
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| |year= 2007
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| |pages= 581–598.
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| }}
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| * V. Barbu and S. S. Sritharan, “M-accretive Quantization of the Vorticity Equation", in Semi-groups of Operators: Theory and Applications, edited by A. V. Balakrishnan, Birkhauser, Boston, 2000, pp. 296–303. http://www.nps.edu/Academics/Schools/GSEAS/SRI/BookCH12.pdf
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| * A. M. Krigel, "Vortex evolution", Geophysical, Astrophysical Fluid Dynamics, 1983, '''24''', pp.213-223.
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| {{Refimprove|date=May 2009}}
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| [[Category:Equations of fluid dynamics]]
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Are we one of the 50% - 70% of adults whom suffer from hemorrhoids? If you are, you are almost absolutely here shopping for several efficient hemorrhoid treatment options. And like me you'd probably wish a hemorrhoid all-natural remedy, when possible.
Later I did discover out he had hemorrhoidectomy surgery. At the time I wondered when operation was the greatest hemorrhoid relief for him. I didn't recognize anything regarding hemorrhoids back then.
Start by taking several steps that will stop the hemorrhoids from worsening. This involves using soaps which are dye and perfume free. Rubbing the anal area will create factors worse. Instead, use moistened toilet paper plus blot the area after utilizing the bathroom. After we shower, pat dry gently with a soft towel.
Constipation is the popular cause of hemorrhoid. In purchase to overcome with this constipation, you must change the diet. Before should you employ to consume those instant foods or processed food, then it really is the perfect time for you to stop eating these foods. You have to change your die with those fibrous foods inside order to cure your irregularity. Another good thing to do is to increase a fluid intake. This can furthermore enable you to soften a stool and avoid from straining throughout bowel movement.
There are 2 types of hemorrhoids- internal and external. Both are the result of swollen veins inside the anal region. Internal hemorrhoids is hard to discover because they are not noticeable. You'll only find out later when it starts to bleed. On the alternative hand, exterior hemorrhoids is felt because a hard lump inside the anal opening. These are typically extremely noticeable due to the fact that they are swollen, red, itchy, and pretty painful.
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