Vorticity-velocity formulation of the stokes problem with variable density and viscosity

We investigate the vorticity-velocity formulation of the stationary, two-dimensional Stokes problem in the case of variable density and viscosity. The analysis is presented in the low Mach number regime, where the density is independent of spatial variations of the pressure. We introduce a variational framework and prove the equivalence of the vorticity-velocity and velocity-pressure formulations in appropriate functional spaces. We then derive a weak formulation for the Stokes equations in vorticity-velocity form. Finally, when the spatial variations of the density and of the viscosity are small enough, we prove the existence and uniqueness of the solution to the Stokes problem in both vorticity-velocity and velocity-pressure forms.