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

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