2. Formulation of incompressible Navier-Stokes equations in terms of current-vorticity (current-vortex)

In the previous sections, we saw the formulation of the Navier-Stokes equations in velocity-pressure form. This formulation complicates the resolution of the equations because of the constraint $\mathrm{div}\left(\overrightarrow{U}\right)=0$ which translates the conservation of mass. In this section, we will transform the Navier-Stokes equations into a form that no longer includes this constraint. Such a formulation is made possible by the introduction of vorticity using the vortex vector $\overrightarrow{\omega }$ defined by :