To write physical laws, we must be able to find time derivatives for a fixed point of fluid. For example, the acceleration of a point of fluid is the time derivative of the velocity keeping that point constant.
Derivatives depend on what you hold constant!
Lagrangian/material/substantial time derivatives keep the fluid constant (move along with the fluid):
Example:
Exercise:
What can you say about the pressure field for the flow in the picture? Is the pressure increased on the curved walls to keep the incoming stream from going through the wall?
Calculus:
The additional terms are the convective terms.
