Give the physical interpretation of the substantial derivative?

Answer 1

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.

Answer 2

The substantial derivative represents the rate of change of a property at a specific location as a fluid element moves with the flow. It includes both the convective transport and the intrinsic changes in the property within the fluid element.