- #1
Saladsamurai
- 3,020
- 7
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 completely baffled as to how we went from (1) to (2) ...let alone from (2) to (3)?
Any thoughts?
Casey
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
[tex]\frac{\partial{}}{\partial{x}} (\rho u)\approx\rho\frac{\partial{u}}{\partial{x}} \qquad (1)[/tex]
This is equivalent to the strong inequality
[tex]
|u\frac{\partial{\rho}}{\partial{x}}|\ll |\rho\frac{\partial{u}}{\partial{x}}| \qquad (2)
[/tex]
or
[tex]
|\frac{\delta\rho}{\rho}|\ll|\frac{\delta V}{V}| \qquad (3)
[/tex]
I am completely baffled as to how we went from (1) to (2) ...let alone from (2) to (3)?
Any thoughts?
Casey