If a fluid flow, such as water, is incompressible then the convective derivative term in the material derivative is equal to zero. How then, in a Bernoulli flow where there is an increase in velocity at the narrowing of a pipe, is the convective derivative zero? The local derivative shouldn't matter for a flow that speeds up with respect to position, because the local derivative is time-dependent.
I know that del*v (the divergence of the velocity field) is zero for incompressible fluids. This makes sense because the divergence of the velocity field tells us the rate of change of volume per unit volume of a finite control volume. But how does this relate to things like Bernoulli's equation when fluids are speeding up or slowing down in a manner that is not time-dependent? I have really confused myself over this. Can some explain to me what these derivatives mean in such a scenario?