I am still trying to understand Navier-Stokes equations. I understand the equations in general, BUT there is one aspect I still cannot digest: The equations assume that density is constant (okay, I understand why, because if it varies, then the derivation of the equations would be more difficult), HOWEVER, what I don't understand is : how it is possible to change pressure without changing density? meaning we use the negative gradient of P to represent forces acting on a blob of fluid, but is it even possible to change P without changing density? I understand that the ONLY way to increase pressure at a one point in space is via increasing the density of the smoke at this point?? Is there other way to change pressure without changing density? Maybe I am missing another definition of pressure that does not depend on density!
Just one last note, I understand if the fluid is liquid then changing its density is not "easy", but for gases, I "think" this is rather doable...
Thanks in advance :)
Answer
In the case of incompressible fluids such as liquids, pressure changes do not change the density of the fluid very much.
And in the case of gases, and if you can treat your fluid as an ideal gas it can be written as
$$\rho=\frac{P}{RT}$$
So you can see by properly adjusting $T$ to compensate for changes in $P$ you can hold density, $\rho$ constant
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