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!!
Best Answer
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.