This might be an incredibly simple question, but I haven't been able to figure it out on my own. I'm an engineer by trade, so please forgive my unfamiliarity with vector calculus. I'm interested in the inviscid, incompressible Navier-Stokes equation and how it relates to the Bernoulli equation. The form of the NS equation I'm interested in is as follows:
$$ \frac{\partial {\mathbf u} }{\partial t} + ({\mathbf u} \cdot \nabla ){\mathbf u} = -\frac{1}{\rho} \nabla p + {\mathbf g} $$
which can relatively easily be manipulated into the Bernoulli equation if the flow is assumed to be steady-state.
$$ \frac{p}{\rho} + \frac{1}{2}\| {\mathbf u} \| ^2 +gh = const. $$
The convective term of NS $({\mathbf u} \cdot \nabla ){\mathbf u}$ turns into the velocity term in Bernoulli, $\frac{1}{2}\| {\mathbf u} \| ^2$. This is obviously a very important part of the Bernoulli equation. However, if the fluid is incompressible, we have the continuity equation which states that $\nabla \cdot {\mathbf u}=0$. Since the dot product is commutative, doesn't this imply that $({\mathbf u} \cdot \nabla ){\mathbf u} = {\mathbf 0}$? Obviously this is not the case, I am just wondering where I am going wrong, since vector calc is not my strength.
