# Incompressible vs constant density fluid flow

1. Dec 24, 2012

### tomwilliam2

This may sound like a basic question, but it's just to get it clear:

When describing fluid flows, does the term "incompressible" mean exactly the same thing as "constant density"?

I was under the impression that if a fluid cannot be compressed, then its density must remain constant for any fixed volume of fluid. My textbook intriguingly says "For an incompressible fluid (including the case of constant density), divergence is zero." This wording seems to suggest you can have an incompressible fluid which does not have constant density...
Can anyone elucidate?

2. Dec 24, 2012

It depends on your definition of "constant". An incompressible fluid flow must have the material derivative of density equal to zero, so in a sense it is "constant density". It does mean that the individual time and spatial derivatives need not be zero, however.

According to the continuity equation, an equivalent statement to the material derivative being zero is that the divergence of the flow field is zero. That is a more easily measurable and computable quantity, so it is often just used as the definition of incompressibility for ease of use and to avoid confusion with various definitions of "constant".

3. Dec 24, 2012

### tomwilliam2

Thanks, that's very useful.

4. Dec 24, 2012

### Studiot

Your textbook is quite right to be more general.

1) Incompressible flow mechanics is usually about liquids, however even with gas dynamics, some treatments regard the gas as incompressible for sufficient accuracy.

2) In chemical engineering for instance a liquid may possess a solute concentration gradient and be 'incompressible' at all concentrations, but the density will vary with concentration.

go well