1. Part 5:Vorticity

2. Definition of the vorticity:
Microscopic interpretation:
The vorticity represents the rotation of a parcel of fluid, or the small angular momentum of this parcel.

3. The dynamics equation for the vorticity is obtained by taking the curl of the Navier-Stokes equation:
In this class we should consider only the case of a fluide of uniform density at first order. Since
It follows immediately that,
Following a fluid parcel it can be recast as:
Vortex stretching

5. For an inviscid flow with no external forces or forces that derives from a potential, the vorticity equation reduces to:
The only changes of the vorticity of a fluid parcel arise through vortex stretching. In that case, if the vorticity is zero at some particular time to, then it remains zero for t>to.

6. Irrotational flows
In this section we will consider a particular class of inviscid flows that have no vorticity, the so-called irrotational or potential flows. A typical application would be the flow around a obstacle under the inviscid assumption, a problem we already considered in the previous section but only using handwaving arguments. We shall now find the analytical solution.

7. We call irrotational flows the flows that satisfy:
Recalling that the curl of the gradient of any function is identically zero:
Irrotational flows always derive from a potential
The three unknowns of the velocity components have been replaced by a single scalar function. In the limit of incompressibility, the potential is governed by a Poisson equation:
The boundary conditions will close this equation. The reduction of the unknowns and the Poisson structure of the equation makes analytical and numerical approaches particularly powerful.
Compressibility can be taken into account, some considerations are presented for instance in

8. Of particular interest to us in this section are inviscid flow, if we also assume that the external forces derive from a potential, the vorticity equation reduces to:
Hence, if the inviscid flow is irrotational at t=to, it will remain irrotational at t>to.