Vorticity Equation

The vorticity equation of fluid dynamics describes evolution of the vorticity (the curl of the velocity) of a particle of a fluid as it moves with its flow:

\begin{align}
\frac{d\vec\omega}{dt} &= \frac{\partial \vec \omega}{\partial t} + (\vec v \cdot \vec \nabla) \vec \omega \\
&= (\vec \omega \cdot \vec \nabla) \vec v - \vec \omega (\vec \nabla \cdot \vec v) + \frac{1}{\rho^2}\vec \nabla \rho \times \vec \nabla p + \vec \nabla \times \left( \frac{\vec \nabla \cdot \tau}{\rho} \right) + \vec \nabla \times \vec B
\end{align}

where the total time derivative operator, is the velocity vector, is the fluid's density, is the pressure, is the viscous stress tensor and represents the external body forces. The equation is valid in the absence of any concentrated torques and line forces, for a compressible Newtonian fluid.

In the case of incompressible (i.e. low Mach number) and isotropic fluids, with conservative body forces, the equation simplifies to the vorticity transport equation

 {d\vec{\omega} \over dt} = \vec{\omega} \cdot \nabla \vec{v} + \nu \nabla^2 \vec{\omega}

where is the kinematic viscosity and is the Laplace operator.

Read more about Vorticity Equation:  Physical Interpretation, Derivation, Tensor Notation

Famous quotes containing the word equation:

    A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.
    Norman Mailer (b. 1923)