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), as would be seen by an observer located at that point and traveling along with the flow. It is an important quantity in fluid dynamics and provides a convenient framework for understanding a variety of complex flow phenomena, such as the generation of lift on wings. Mathematically, the vorticity ω {\displaystyle {\boldsymbol {\omega }}} is the curl of the flow velocity v {\displaystyle \mathbf {v} } :
ω ≡ ∇ × v , {\displaystyle {\boldsymbol {\omega }}\equiv \nabla \times \mathbf {v} \,,}
where ∇ {\displaystyle \nabla } is the nabla operator. Conceptually, ω {\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 ω {\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 ∇ ⋅ ω = 0. {\displaystyle \nabla \cdot {\boldsymbol {\omega }}=0.}
In a two-dimensional flow, ω {\displaystyle {\boldsymbol {\omega }}} is always perpendicular to the plane of the flow, and can therefore be considered a scalar field. The dynamics of vorticity are fundamentally linked to drag through the Josephson-Anderson relation.
Mathematical definition and properties Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by ω {\displaystyle {\boldsymbol {\omega }}} , defined as the curl of the velocity field v {\displaystyle \mathbf {v} } describing the continuum motion. In Cartesian coordinates:
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