Navier-Stokes equations

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The incompressible Navier-Stokes equations are

\rho  \frac{D \mathbf{u}}{D t}  = - \nabla p + \rho \mathbf{g} + \mu \nabla^2 \mathbf{u} \, ,

\nabla \cdot \mathbf{u} = 0 \,,
for an adiabatic fluid. A prognostic equation for density is

\frac{D \rho}{D t} = \kappa \nabla^2 \rho \, .