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I am trying to understand Navier-Stokes fluid equations and I have two questions (these are very primitive questions, I am still trying to understand basic concepts)

1) all books say "assume that density is constant...assume fluid is incompressible", but this makes me confused: observing any smoke one can see some parts are completely opaque, some are particularly opaque and some are nearly transparent, so density does change in space and not constant as the books suggest?!

2) is pressure a function of density? I mean, the more you "compress" gas the more dense (i.e. density increases per unit volumes) hence the more pressure? so one can say pressure is just a function of density? Another way to ask my question: is it possible to increase pressure without increasing density? they "look" the same to me!!

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  • $\begingroup$ Have you considered the possibility that you are not dealing with a gas, but rather a liquid? How much does the density of water change when you apply pressure to it (to the extent that it matters in a fluid dynamics situation)? $\endgroup$ – Chet Miller Mar 14 '17 at 16:57
  • $\begingroup$ I agree with you about liquids, but I understand these Navier-Stokes equations are for both liquids and gasses? $\endgroup$ – Khaled Mar 14 '17 at 17:08
  • $\begingroup$ Gases are not incompressible, so the density in the NS equations would have to take into account the effects of temperature and pressure. In this case, the density would be calculated from the equation of state. $\endgroup$ – Chet Miller Mar 14 '17 at 17:12
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When the book says "Assume the density is constant" and "assume fluid is incompressible" they aren't saying that you should always assume that for fluids.

They are saying when those assumptions are made then these equations apply. Those assumptions greatly help to simplify the problem so that it is easier to solve.

Whenever solving a problem like this where the equation is based off assumptions, it's always good practice (and necessary for a thorough analysis) to look at your assumptions and see if they hold for your conditions.

Gasses are very compressible, so they don't work well with incompressible fluid equations. Liquids are usually far less compressible. Increasing water pressure has no appreciable effect on it's volume, so we say it's approximately incompressible.

As far as uniform density goes, with incompressible fluids it's a lot less of an issue than with gasses. In the case of smoke it's two gasses mixing, so the density of smoke will vary quite a bit. Mixing makes density far more complicated, so these equations will work best with a fairly isotropic fluid.

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  • $\begingroup$ Thanks a lot for your explanation. What me got confused is that I've read quickly couple books introductions on Fluid Simulation and all of them say a Fluid is either gas or liquid, then they mention NS equations which assume uniform density! I understand from your reply and the previous one that I am mistaken, i.e. gas are compressible (which also makes sense) :) But I wonder why the books I've read start with an assumption that is valid only for liquids and not gases? I think I need to read further as I must have missed something out... $\endgroup$ – Khaled Mar 14 '17 at 18:52
  • $\begingroup$ @Khaled Just look at the solutions to the NS equations when we make those assumptions. It's pretty complicated as is. Adding compressibility would make it extremely complicated. They basically give that so you can do some fluid dynamics without a bunch more math knowledge. $\endgroup$ – JMac Mar 14 '17 at 18:58
  • $\begingroup$ It is often valid to assume that gases are incompressible, when the flow speeds are much less than the speed of sound. The criterion that the Mach number is less than $0.3$ is often used. $\endgroup$ – alephzero Mar 15 '17 at 1:40

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