# Vorticity transport equation

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The vorticity transport equation governs the evolution of the vorticity. It is obtained by taking the curl of the momentum equation.

$\frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} + \frac{1}{\rho} \frac{\partial p}{\partial x_i} = \frac{1}{\rho}\frac{\partial \tau_{ij}}{\partial x_j}$

For incompressible flow the vorticity transport equation reduces to

$\frac{\partial \omega_i}{\partial t} + u_j \frac{\partial \omega_i}{\partial x_j} = \omega_j \frac{\partial u_i}{\partial x_j} + \nu \Delta \omega_i$