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}

Read more about this topic:  Vorticity

Famous quotes containing the words mathematical and/or definition:

    “It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.”
    —Henry David Thoreau (1817–1862)

    “It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
    —Jane Adams (20th century)