Navier-Stokes equations

The incompressible Navier-Stokes equations are
$\rho {\frac {D\mathbf {u} }{Dt}}=-\nabla p-\rho g{\hat {\mathbf {k} }}+\mu \nabla ^{2}\mathbf {u} \,,$
$\nabla \cdot \mathbf {u} =0\,,$
for an adiabatic fluid, a prognostic equation for density is
${\frac {D\rho }{Dt}}=\kappa \nabla ^{2}\rho \,,$
where diffusion (the right hand side) is typically neglected.