Incompressible flow

Remarks

Since the continuity equation reads $D\rho/Dt + \rho \, \nabla \cdot \mathbf{u} = 0$ and $\rho > 0$, the flow is incompressible if and only if the divergence of the velocity is zero, $\nabla \cdot \mathbf{u} = 0$. Equivalently, the flow map preserves the volume of every region of fluid. Example: the plane flow $\mathbf{u} = (x, -y)$ with constant density is incompressible; check: $\partial_x x + \partial_y (-y) = 1 - 1 = 0$. Non-example: the flow $\mathbf{u} = (x, y)$ has divergence $2$, so it is not incompressible; the fluid spreads out and its density must fall. Incompressible does not mean constant density: a stratified ocean with density varying in depth can still flow incompressibly.