PX285 - H5 - fluid element

a fluid element is a parcel of a fluid, which can be visualised by some shape, eg: a cube
it moves with a flow and may deform in time, but the same molecules remain in the parcel and its mass remains the same

two approaches to fluid parcels:
- lagrangian
- eulerian

lagrangian

a lagrangian fluid element moves with the fluid in such a way, into or out of the element
constant mass
the velocity depends only on time

the displacement:

\vec{s} = \vec{r} (t) - {\vec{r}}_{0}

therefore, the velocity:

\vec{u} = \frac{d}{d t} (\vec{r} (t) - {\vec{r}}_{0}) = \frac{d x}{d t} \hat{i} + \frac{d y}{d t} \hat{j} + \frac{d z}{d t} \hat{k}

the lagrangian coordinate system is fixed with a fluid

eulerian

it is fixed in space and does not move with the flow
the velocity:

\vec{u} = \vec{u} (t, \vec{r})

the flow can come into this element or go out
so, the eulerian coordinate system remains the same, whereas the lagrangain system deforms with the fluid
eg:
- "a cold wind is coming from scotland" - lagrangian
- "in coventry, the wind is $10$ ms $^{- 1}$ " - eulerian