Is Incompressibility a Prerequisite for Steady Flow in Fluid Mechanics?

[cepheid]

In fluid mechanics, one hears the term steady, incompressible flow. Steady flow refers to the fact that the overall flow pattern (the flow lines, which indicate the paths of individual particles within the fluid) do not change with time. Incompressible flow refers to the fact that the fluid density does not change with time (the mass of the fluid cannot be "compressed" into a smaller volume, for instance).

My question regards the alternatives to this pair of conditions. I understand that unsteady flow is possible in an incompressible fluid (after all, why wouldn't it be?). More perplexing to me is whether steady flow is possible in a compressible fluid. If the density of the fluid changes with time, then it seems to me the flow is not steady, because the paths of particles (or fluid elements, if you like) passing a certain point could not be the same as their predecessors'. Is incompressibility a prerequisite for steady flow?

 

[PBRMEASAP]

It can be compressible steady flow as long as the density field changes with position but not with time.

 

[cepheid]

I see. That makes sense. Density changing with time means that the flow pattern is changing (particles are moving around as fluid that was once less dense becomes compressed, or vice versa). So the flow cannot be steady. And it is not necessarily true that what goes in (to a tube, for example) must come out. Because the total mass in the tube need not be constant.

If the density only changes with position, then it still must be true that the mass flow rate in equals the mass flow rate out...it just means that the same mass coming out of the tube at the other end may take up a different volume.

Am I interpreting you correctly?

 

[PBRMEASAP]

Exactly. In the steady flow, it is necessary that the flow of mass into a fixed volume equals the flow out. Otherwise the density of that volume (fixed in space) would be changing with time. So if the density of the fluid changes with position, its velocity will have to change with position to keep the mass flow rate constant.

Even in steady flow, if you follow a particular fluid element along its path, you will see its density change with time because its volume changes. But the density at a point fixed in space remains constant with time.

 

[arildno]

Compresssible steady flow can typically occur (and be designed for) in the nozzle of a rocket engine where the gas fuel is expelled through.

By clever designing, the compressibility of the gas, along with the geometry of the nozzle cause the gas to be expelled with an immense velocity.

 

[Clausius2]

Compresssible steady flow can typically occur (and be designed for) in the nozzle of a rocket engine where the gas fuel is expelled through.

.

Hmm No No No, Arildno, too wrong...

Next time see the Space Shuttle launching. Do you really think the flow through a rocket nozzle is steady?. The turbulence makes it to be extremely unsteady and far away from steadiness of mean values also.

I think you need a new fluid mech book, arildno. When is your birthday? I'll send you one as a gift.

 

[arildno]

Hmm No No No, Arildno, too wrong...

Next time see the Space Shuttle launching. Do you really think the flow through a rocket nozzle is steady?. The turbulence makes it to be extremely unsteady and far away from steadiness of mean values also.

I think you need a new fluid mech book, arildno. When is your birthday? I'll send you one as a gift.

Oh dear..

 

[cepheid]

Whatever you do, don't send him Fluid Mechanics by Frank M White. Awful book...