Vorticity

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 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.

Mathematically, the vorticity ${\displaystyle {\boldsymbol {\omega }}}$ is the curl of the flow velocity ${\displaystyle \mathbf {v} }$:

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.

The dynamics of vorticity are fundamentally linked to drag through the Josephson-Anderson relation.

Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by ${\displaystyle {\boldsymbol {\omega }}}$, defined as the curl of the velocity field ${\displaystyle \mathbf {v} }$ describing the continuum motion. In Cartesian coordinates:

We may also express this in index notation as ${\displaystyle \omega _{i}=\varepsilon _{ijk}{\frac {\partial v_{k}}{\partial x_{j}}}}$. In words, the vorticity tells how the velocity vector changes when one moves by an infinitesimal distance in a direction perpendicular to it.

In a two-dimensional flow where the velocity is independent of the ${\displaystyle z}$-coordinate and has no ${\displaystyle z}$-component, the vorticity vector is always parallel to the ${\displaystyle z}$-axis, and therefore can be expressed as a scalar field multiplied by a constant unit vector ${\displaystyle {\hat {z}}}$: