# Fluid Mechanics'Deriving' Incompressible Flow Criteria

1. Nov 11, 2009

Here we go....

My text attempts to 'derive' an expression that explains when a flow is compressible or not:

Great ... if there's anything I like better than making density approximations, it's playing 'fast and loose' with them.

He then goes on to say:
I am completly baffled as to how we went from (1) to (2) ...let alone from (2) to (3)?
Any thoughts?

Casey

2. Nov 11, 2009

### tiny-tim

Hi Casey!

∂(ρu)/∂x = u∂ρ/∂x + ρ∂u/∂x,

so if u∂ρ/∂x << ρ∂u/∂x, we can ignore it, and then ∂(ρu)/∂x ~ ρ∂u/∂x.

(2) to (3) is simply rearrangement (and changing u to V for some reason which escapes me)

3. Nov 11, 2009

Oh...that darned chain rule! Thanks tiny-tim.

Also, silly question, but why did we change the ∂'s into $\delta$'s ?

Is it because the (∂x)'s 'canceled' and thus it is no longer a derivative, but just a relation between 'changes?'

4. Nov 11, 2009

### arildno

(2) to (3) is based upon the fact that this argument must be repeated for the v and w-components as well, and hence, that the relative infinitesemal change in density must be much less than the relative infinitesemal change in the maximal velocity component, and hence, much less than the relative infinitesemal change in the fluid speed.