Vorticity - Mathematical Definition

Mathematical Definition

Mathematically, the vorticity of a three-dimensional flow is a vector field, usually denoted by, defined as the curl or rotational of the velocity field :

$\begin{array}{rcl} \vec{\omega} &=& \nabla \times \vec{v} \;=\; \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\times(v_x,v_y,v_z)\\ &=& \left( \frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z},\; \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x},\; \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \right) \end{array}$

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 z coordinate and has no z component, the vorticity vector is always parallel to the z axis, and therefore can be viewed as a scalar field:

$\vec{\omega} \;=\; \nabla \times \vec{v} \;=\; \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)\times(v_x,v_y) \;=\; \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}$

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