The first ∂/∂t term is called V the local derivative. The second
~· ∇ term is called the convective derivative. In steady flows,
∂/∂t =0, and only the convective derivative
The substantial derivative has a physical meaning: the rate of
change of a quantity (mass,
energy, momentum) as experienced by an observer that is moving
along with the flow. The
observations made by a moving observer are affected by the
stationary time-rate-of-change
of the property (∂f/∂t), but what is observed also depends on
where the observer goes as
it floats along with the flow (v · ∇f). If the flow takes the
observer into a region where, for
example, the local energy is higher, then the observed amount of
energy will be higher due
to this change in location. The rate of change from the point of
view of an observer floating
along with a flow appears naturally in the equations of
change.