Vorticity

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In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate^[1]), as would be seen by an observer located at that point and traveling along with the flow. It is an important quantity in the dynamical theory of fluids and provides a convenient framework for understanding a variety of complex flow phenomena, such as the formation and motion of vortex rings.^[2][3]

Mathematically, the vorticity ${\displaystyle {\boldsymbol {\omega }}}$ is the curl of the flow velocity ${\displaystyle \mathbf {v} }$:^[4][3]

${\displaystyle {\boldsymbol {\omega }}\equiv \nabla \times \mathbf {v} \,,}$

where ${\displaystyle \nabla }$ is the nabla operator. Conceptually, ${\displaystyle {\boldsymbol {\omega }}}$ could be determined by marking parts of a continuum in a small neighborhood of the point in question, and watching their relative displacements as they move along the flow. The vorticity ${\displaystyle {\boldsymbol {\omega }}}$ would be twice the mean angular velocity vector of those particles relative to their center of mass, oriented according to the right-hand rule. By its own definition, the vorticity vector is a solenoidal field since ${\displaystyle \nabla \cdot {\boldsymbol {\omega }}=0.}$

In a two-dimensional flow, ${\displaystyle {\boldsymbol {\omega }}}$ is always perpendicular to the plane of the flow, and can therefore be considered a scalar field.