# Fluid Mechanics'Deriving' Incompressible Flow Criteria

Here we go....

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

]When is a given flow approximately incompressible? We can derive a nice criterion by playing a little fast and loose with density approximations....
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:
..In essence, we wish to slip the density out of the divergence in the continuity equation and approximate a typical term as
$$\frac{\partial{}}{\partial{x}} (\rho u)\approx\rho\frac{\partial{u}}{\partial{x}} \qquad (1)$$

This is equivalent to the strong inequality

$$|u\frac{\partial{\rho}}{\partial{x}}|\ll |\rho\frac{\partial{u}}{\partial{x}}| \qquad (2)$$

or

$$|\frac{\delta\rho}{\rho}|\ll|\frac{\delta V}{V}| \qquad (3)$$
I am completly baffled as to how we went from (1) to (2) ...let alone from (2) to (3)?
Any thoughts?

Casey

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tiny-tim
Homework Helper
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)

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?'

arildno