Vorticity

In <a href="/facts/Continuum_mechanics/vI7LpFDk">continuum mechanics</a>, vorticity is a <a href="/facts/Pseudovector/c3b5GgWD">pseudovector</a> (or axial vector) <a href="/facts/Vector_field/ccFNuPPp">field</a> that describes the local <a href="/facts/Rotation/yChkz49x">spinning</a> motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the <a href="/facts/Fluid_dynamics/y7uzSCfg">flow</a>. It is an important quantity in <a href="/facts/Fluid_dynamics/y7uzSCfg">the dynamical theory</a> of <a href="/facts/Fluids/tMQ46Tmz">fluids</a> and provides a convenient framework for understanding a variety of complex flow phenomena, such as the formation and motion of <a href="/facts/Vortex_ring/N5IpqxrV">vortex rings</a>.
Mathematically, the vorticity 
 
 
 
 
 ω
 
 
 
 {\displaystyle {\boldsymbol {\omega }}}
 
 is the <a href="/facts/Curl_(mathematics)/GuS8dc7o">curl</a> of the <a href="/facts/Flow_velocity/amIT4Ht8">flow velocity</a> 
 
 
 
 
 v
 
 
 
 {\displaystyle \mathbf {v} }
 
:

ω
        
        ≡
        ∇
        ×
        
          v
        
        
        ,
      
    
    {\displaystyle {\boldsymbol {\omega }}\equiv \nabla \times \mathbf {v} \,,}

where 
 
 
 
 ∇
 
 
 {\displaystyle \nabla }
 
 is the <a href="/facts/Nabla_operator/0xEmQeBo">nabla operator</a>. Conceptually, 
 
 
 
 
 ω
 
 
 
 {\displaystyle {\boldsymbol {\omega }}}
 
 could be determined by marking parts of a continuum in a small <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighborhood</a> of the point in question, and watching their relative <a href="/facts/Displacement_(geometry)/jkmEfS3C">displacements</a> as they move along the flow. The vorticity 
 
 
 
 
 ω
 
 
 
 {\displaystyle {\boldsymbol {\omega }}}
 
 would be twice the mean <a href="/facts/Angular_velocity/2n8JljAT">angular velocity</a> vector of those particles relative to their <a href="/facts/Center_of_mass/n4Lr9GAz">center of mass</a>, oriented according to the <a href="/facts/Right-hand_rule/LJe0Qqh4">right-hand rule</a>. By its own definition, the vorticity vector is a <a href="/facts/Solenoidal/9ohQ6clr">solenoidal</a> field since 
 
 
 
 ∇
 ⋅
 
 ω
 
 =
 0.
 
 
 {\displaystyle \nabla \cdot {\boldsymbol {\omega }}=0.}

In a <a href="/facts/Two-dimensional_flow/84CMyYu3">two-dimensional flow</a>, 
 
 
 
 
 ω
 
 
 
 {\displaystyle {\boldsymbol {\omega }}}
 
 is always perpendicular to the plane of the flow, and can therefore be considered a <a href="/facts/Scalar_field/8MtPTGob">scalar field</a>.
The dynamics of vorticity are fundamentally linked to drag through the Josephson-Anderson relation.

Vorticity open-in-new

Vorticity