PX285 - J1b - bernoulli's principle from navier-stokes equation

\begin{aligned} ρ (\vec{u} \cdot \vec{\nabla}) \vec{u} & = - \vec{\nabla} P - ρ g z k \\ ρ g z \hat{k} & = \vec{\nabla} (ρ g z) \\ ∴ ρ (\vec{u} \cdot \vec{\nabla}) \vec{u} & = - \vec{\nabla} (P + ρ \vec{g} z) \end{aligned}

\begin{matrix} ρ \vec{u} \cdot (\vec{u} \cdot \vec{\nabla}) \vec{u} = - \vec{u} \cdot \vec{\nabla} (P - ρ \vec{g} z) \\ \vec{u} \cdot \vec{\nabla} (P + ρ g z + ρ \frac{u^{2}}{2}) = 0 \\ P + ρ g z + ρ \frac{u^{2}}{2} = constant \end{matrix}

where $(\vec{u} \cdot \vec{\nabla}) \vec{u} = \vec{\nabla} u^{2} / 2$