# How can mass diffuse out of a lagrangian control volume when it always tracks the same mass?

I am trying to understand how mass conservation in fluid flow works and how it relates to lagrangian and eulerian control volumes.

The way I understand eulerian control volumes is that they are stationary and feel the change in time at that constant location and lagrangian control volumes follow a parcel of fluid that contains the same particles. I also understand the basic idea behind the material derivative:

$$\frac{Df}{Dt} = \nabla f \cdot \vec{v} + \frac{\partial f} {\partial t}$$

aka the total change of the quantity in time is the one experienced due to motion through space with a certain velocity plus the change at that location.

What I don't get is how this ties exactly with with conservation and the lagrangian and eulerian framework. Is the material derivative refering to a lagrangian or eulerian control volume or is it relating them to each other?

In my class notes it says that mass is in general not conserved in an eulerian control volume but it always is in a Lagrangian control volume (since it always contains the same particles) and that this means that conservation implies $$\frac{Df}{Dt} = 0$$

Does this statement about the material derivative being equal to zero refer now to an eulerian control volume? How can it refer to a Lagrangian control volume, when it implies mass exchange and hence different particles?

I am further confused, because later on in these notes it is implied that there is mass exchange in lagrangian control volumes. The example of advection diffusion systems with the governing equation $$\frac{\partial f} {\partial t} = \nabla \cdot (D \nabla f) - \nabla \cdot (f \vec{v})$$

is studied and a particle methods algorithm to simulate them is explained. They state that the computational particles in the method correspond to Lagrangian control volumes and store extensive quantities. It is then stated that from the perspective of the particle $$\frac{Df}{Dt} = \nabla \cdot (D \nabla f)$$ $$\frac{dx}{dt} = \vec{v}$$

Does this mean that the change in mass a particle experiences is only due to diffusion, since it flows through the field with velocity $$\vec{v}$$ and does not experience any advective flux? If so, is there any diffusion across the boundary of the Lagrangian control volume? Doesn't that mean not the same mass particles are tracked?

This post is super confused and long since I myself am quite confused. Assume you are explaining to someone without foundational knowledge.