Vorticity

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Revision as of 04:42, 13 September 2005

Vorticity is a vector field variable which is derived from the velocity vector. Mathematically, it is defined as the curl of the velocity vector

$\omega := \textrm{curl}(u) = \nabla \times u$

In tensor notation, vorticity is given by

$\omega_i = \epsilon_{ijk} \frac{\partial u_k}{\partial x_j}$

where $\epsilon_{ijk}$ is the alternating tensor. The components of vorticity in Cartesian coordinates are

$\omega_1 = \frac{\partial u_3}{\partial x_2} - \frac{\partial u_2}{\partial x_3}$

$\omega_2 = \frac{\partial u_1}{\partial x_3} - \frac{\partial u_3}{\partial x_1}$

$\omega_3 = \frac{\partial u_2}{\partial x_1} - \frac{\partial u_1}{\partial x_2}$

This can be obtained by using determinants

$\omega = \begin{vmatrix} \hat{e}_1 & \hat{e}_2 & \hat{e}_3 \\ \frac{\partial}{\partial x_1} & \frac{\partial}{\partial x_2} & \frac{\partial}{\partial x_3} \\ u_1 & u_2 & u_3 \end{vmatrix}$

where $\hat{e}_1, \hat{e}_2, \hat{e}_3$ are the unit vectors for the Cartesian coordinate system.

Physical Significance

Related Pages

Vorticity transport equation

Vorticity

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Revision as of 04:42, 13 September 2005

Physical Significance

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