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I'm studying fluid dynamics and we just had a lecture about the momentum equation. We started the lecture by talking about pressure in terms of molecules moving across a hypothetical surface element and carrying their momentum with them (in both directions). There are 2 things confusing me about this:

1) we said the pressure is isotropic, but given I'm studying fluid dynamics, with this description it seems pretty clear that the pressure will point in the direction of the fluid velocity, and in a direction normal to that there will be (on average) no molecules moving in that direction so what would cause the pressure?

2) When deriving the momentum equation, we then also included the effect of momentum entering and leaving our arbitrary volume, but this seems to be exactly what we described pressure to be at the start of the lecture. So I think I have misunderstood the root cause of pressure: eg. is it particles colliding or just moving between regions and transferring momentum?

I have been thinking about this in frustration for hours now so any help would be appreciated!

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  • $\begingroup$ Related physics.stackexchange.com/questions/31822/… $\endgroup$
    – Farcher
    Jan 26 '18 at 22:06
  • $\begingroup$ There are two parts to the motion of the molecules in the fluid. For the most part, the different molecules are moving randomly in all different directions, and this gives rise to the isotropic feature. But, in addition to this, there is organized motion of the molecules that averages out to what we observe macroscopically as the "velocity of the fluid." This motion is not isotropic, and gives rise to the additional momentum flux in the differential momentum balances. $\endgroup$ Jan 26 '18 at 22:55
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The reason for your confusion is that there is a difference between static and dynamic pressure. Static pressure is the isotropic component, formed by particles moving through some arbitrary boundary. Static pressure is measured in a frame that's moving with the fluid. Dynamic pressure is an expression of the net kinetic energy density of the fluid, and represents the contribution from the bulk motion of the fluid.

In Bernoulli's equation, ignoring potential energy, the sum of static and dynamic pressure along any streamline is a constant, called the stagnation pressure.

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  • $\begingroup$ thank you so much! We haven't learnt about that and this is exactly the answer I was looking for! $\endgroup$
    – user294388
    Jan 26 '18 at 21:47
  • $\begingroup$ No problem! This semantic distinction isn't always made clear in textbooks, but it's an important one. $\endgroup$ Jan 26 '18 at 21:48

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