Incompressible Viscous Flows

For an incompressible fluid, the continuity equation and the Navier-Stokes equation are given as

Using a vector identity, Equation (2) may be restated as

Define vorticity

and taking curl of (3) we find

Noting that curl of gradient is zero and

Equation (5) may be restated as

Equation (7) is the vorticity transport equation. It shows that in addition to being convected and diffused, vorticity is also generated by the first on the right hand side of Equation (7) by a vortex stretching mechanism.