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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. ]When is a given flow approximately incompressible? We can derive a nice criterion by playing a little fast and loose with density approximations....

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)? ..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]

Any thoughts?

Casey

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# Fluid Mechanics'Deriving' Incompressible Flow Criteria

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