What is the advective acceleration

[pkrdsb]

What is the "advective acceleration"

I am taking a course on Classical Mechanics and the current topic is fluid in motion. Here I have come across a so called advective acceleration: $\mathbf{v} \cdot \nabla$

I know that it is not the same as $$\begin{equation} \nabla\, \mathbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} \end{equation}$$

But, how should one see the math behind the expression (I will not ask about the physical meaning before the understanding of the math is in place)?

(My first post, hope I got the post placement right)

In advance, thanks.

 

[AlephZero]

But, how should one see the math behind the expression (I will not ask about the physical meaning before the understanding of the math is in place)?

I think that's the back-to-front way to understand it.

In fluid mechanics there are two basic ways to describe the physics.

(1) The way Euler did it: you choose a point in space, and describe how the velocity, accleration, etc. of the fluid at that point varies with time.

(2) The way Lagrange did it: You choose a particle of fluid, and describe how the velocity acceleration, etc of that particular particle varies with time.

The derivatives etc in (2) are called "material derivatives", or "advective derivatives" (which is a sensible name if you know enough Latin to understand what "advective" means, but otherwise is just another pointless name to learn IMO), or several other names.