Representing the properties of a fluid mathematically is a
complex business as different "bits" of fluid occupy different
points in space at different times (and the properties of interest
*themselves* are likely to change in space or time or
both)

There are two generally accepted descriptions of fluid properties as a function of position and time:

In an **Eulerian description** the fluid motion is described
by calculating all of the necessary properties (e.g. density,
pressure, velocity, etc) as functions of space (i.e. "fixed"
positions in x,y,z space) and time. The flow itsef is characterized
by what happens at fixed points in space as fluid flows past these
points.

In a **Lagrangian description** the fluid motion is described
by following individual fluid particles and noting how all relevant
fluid properties change as a function of time as seen by the fluid
particle being "followed".

If "B" represents a particular fluid property (like temperature):

Eulerian description: $B = B(x,y,z,t)$

Lagrangian description: $B = B_A(t)$ or $B = B(x_{A_o}, y_{A_o}, z_{A_o}, t)$

where fluid particle "A" can be identified by either its "name" (A) or its initial coordinates $(x_{A_o}, y_{A_o}, z_{A_o})$

Eulerian vs. Lagrangian description of temperature field:

Explain the difference between an Eulerian and Lagrangian viewpoints

If you throw a feather into the air and track its velocity will this give you a(n) Eulerian/Lagrangian description of the air's velocity? What if you were to measure the rotational speed of a bunch of windmills in a field?

If you assume that the feather is being moved as a *passive
tracer* then you will obtain a Lagrangian description of the air
velocity. That is, you will get

$\displaystyle{\vec{v}(t) = \vec{v}(x_o,y_o,t)}$

where $\vec{v(t)}$ is the time evolution of the velocity of the fluid that passed through point $(x_o, y_)$ (the initial location of your feather).

The windmills, on the other hand, will give you an Eulerian description including a full velocity field whereby the velocity at a bunch of locations $(x,y)$ are know at all (observed) values of $t$, giving $\displaystyle{\vec{v}(x, y, t)}$.