Transitioning between Lagrangian and Eulerian fluid variables I'm having trouble understanding the motivation between wanting to transition between the two descriptions.
I figure that switching from Eulerian to Lagrangian fluid variables is useful if we want to recover a particle's trajectory maybe? Is there any other reason to do so? and then what are we measuring by switching from Lagrangian to Eulerian?
 A: To illustrate the answer, let's quickly revisit the basic concepts behind each formulation.
For some arbitrary continuum material (solid or fluid), we could consider a reference configuration of the material $\mathscr{B_0}$ at some time $t_0$, and associate every point in the material at that configuration with some position coordinate $\mathbf{X}$. We thus label every differential material element in the continuum by its position in the configuration $\mathscr{B_0}$.
We can track the material element labeled $\mathbf{X}$ during the temporal evolution of the system through a mapping that provides the spatial coordinates $\mathbf{x}$ of that material element. This mapping is of the form $\mathbf{x} = \mathbf{x}(\mathbf{X},t)$. This is the basis for the Lagrangian formulation.
We can invert this mapping in principle in order to track which material element $\mathbf{X}$ is occupying some point in space $\mathbf{x}$, leading to a mapping of the form $\mathbf{X} = \mathbf{X}(\mathbf{x},t)$. This is the basis of the Eulerian formulation.
As a result, the Lagrangian formulation tracks a specific differential material element while the Eulerian formulation tracks a specific differential spatial element. We can use the Eulerian formulation to understand how the properties of a flow change with respect to time, while the Lagrangian formulation furnishes methods to understand how elements in the flow change over time.
The Lagrangian formulation is useful when describing effects such as Taylor dispersion and viscoelastic fluids, in both cases because tracking the evolution of elements in the flow provides more information about the relevant physics than describing the flow itself.
