Vorticity Equation -- from Eric Weisstein's World of Physics

V v

Vorticity Equation

Apply to the Navier-Stokes equations with .

Exchanging time and space derivatives and noting that

Use the vector identity

to write

Now use the vector identity

The continuity equation for an incompressible fluid requires

so this reduces to

and

Note that this equation requires that if at any instant ,

for all subsequent time. Therefore, if a fluid particle has no initial vorticity, it can never acquire it. Now, if , then

and we can define a scalar velocity potential such that

Velocity Potential