Acceleration as $\frac{D\vec u}{Dt}$ in the case of unsteady flow

Question

In the book of Acheson, it is stated that the material derivative $\frac{D\vec u}{Dt}$ can be considered as an acceleration of a "fluid element" - which is something the author never defined, but my understanding is that a "fluid element" is a physical volume that moves with the streamline it started with such that the fluid particles contained in that volume at any given become the corresponding "fluid element".

However, they also mention that (see exercise 1.8) that when the flow is unsteady, the particle trajectories will not be the same as the streamlines. So, does this mean that the concept of acceleration for a fluid element via $\frac{D\vec u}{Dt}$ is only meaningful and defined for a steady-flow?

Answer 1

You just have to change your "definition" : a "fluid element" is a physical volume that moves along the pathline. (But this is the definition of a pathline !)

It is simply necessary to follow a material particle of fluid between $t$ and $t + dt$ along its trajectory (pathline). the key point is to note that the velocity of a fluid material particle at time $t$ is the velocity field where the particle is at time $t$. And when time changes, the particle changes location. It is therefore necessary to take into account the variations of the velocity field in time but also in space.

What the material derivative shows is that even when the velocity field is not time dependent, a fluid particle moves within the velocity field and therefore can have non-zero acceleration.