The "substantial derivative" (or material derivative) of a property $\…

Question

The "substantial derivative" (or material derivative) of a property $\phi$ is:

Accepted Answer

B. {'text': '$D\phi/Dt = \partial\phi/\partial t + \mathbf{u}\cdot\nabla \phi$ — combines local rate of change AND advection following the fluid particle', 'label': 'B'}

Substantial derivative D/Dt converts between Eulerian (fixed-frame) and Lagrangian (particle-following) views. Two contributions: local change at a fixed point AND change because we are moving with the fluid (advective).

Answer

{'text': '$D\phi/Dt = \partial\phi/\partial t$ (Eulerian)', 'label': 'A'}

Answer

{'text': '$D\phi/Dt = \nabla \phi$', 'label': 'C'}

Answer

{'text': '$D\phi/Dt = \phi^2$', 'label': 'D'}