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The pressure at a point inside a static fluid is same in all directions because the the collisions of particles take place isotropically.

However, at the same time pressure increases with depth. So downward pressure must not be equal to upward pressure at that point.

This two statements are contradictory and I am really confused.

What is the problem in my understanding of the topic?

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    According to Wikipedia pressure is a scalar. They write "It is incorrect (although rather usual) to say "the pressure is directed in such or such direction". The pressure, as a scalar, has no direction." – Marius Ladegård Meyer Aug 01 '23 at 09:55
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    @MariusLadegårdMeyer Pressure is the result of forces, which do have a direction. The fact that pressure is independent of direction is an emergent phenomenon. The reason why pressure is direction independent is the thing which this question is effectively asking about. – R.M. Aug 01 '23 at 21:58
  • @R.M. that might be correct, but I parsed the question as "here are two contradictory statements, which statement is incorrect?" and one statement mentioned "downward pressure" and "upward pressure". To me this statement is like saying "temperature decreases with altitude, so upward temperature must not equal downward temperature". My comment just points this out, which is also why it's a comment and not an answer. – Marius Ladegård Meyer Aug 02 '23 at 05:14
  • In a point, which part is lower than some other part? – Jens Aug 02 '23 at 10:35
  • Possible duplicates: https://physics.stackexchange.com/q/31822/2451 , https://physics.stackexchange.com/q/18255/2451 and links therein. – Qmechanic Aug 02 '23 at 12:17

2 Answers2

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Suppose that we consider an element of fluid, density $\rho$, of sides $\delta x,\,\delta y$ and $\delta z$ as shown in the diagram below.

enter image description here

The arrows adjacent to pressure labels, $P_0$ and $P_0+\delta P$, indicate the directions of the forces on the element which have magnitudes, $P_0\,\delta x\,\delta y$ and $(P_0+\delta P)\,\delta x\,\delta y$ due to the pressure at $z$ and $z+\delta z$ respectively.

If the element is assumed to be in equilibrium the net force on it must be zero,

ie $P_0\,\delta x\,\delta y - (P_0+\delta P)\,\delta x\,\delta y-\delta x \,\delta y\,\delta z \,\rho\,g=0 \Rightarrow \dfrac {\delta P}{\delta z} = - \rho \,g$

and if $\delta z \to 0$ then $\dfrac {dP}{dz} = - \rho \,g$ which is called the hydrostatic equation.

I imagine that you were considering a point, which has no dimensions, in the fluid, where you could say that the pressure was $P_0$ and there was also a rate of change of pressure with respect to height, $z$.

Farcher
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The issue you are having is a problem of scale. You are completely correct that if you were to take a small “box” of the fluid from the bulk, the pressure exerted on the upper and lower portions of the fluid would not be equal, but only nearly equal. But in the limit that this fluid “box” becomes infinitesimally small, so too does the difference in the pressure between the upper and lower portions. In your case, you refer to a “point,” which has no dimensionality and therefore you are utterly incapable of discerning any pressure differential. Of course you are right that the pressure is isotropic when looking side to side, as the gravity experienced should be the same. Interestingly, if you zoom in to a molecular scale you will run into the interesting question of whether it even makes sense to talk about pressure anymore, as the interactions now appear much more local and you can only talk about an average pressure. Just a bit of food for thought.

I also think that the point made in an earlier comment is important to reiterate. Pressure is a scalar, nor a vector. This means that pressure depends on the magnitude of the force exerted on some unit area, not what direction you are measuring that force from. More technically, you would have to specify axes and define the stress tensor if you wished to do a more rigorous analysis of some directionally dependent phenomena. But in your case it is perfect consistent to define the pressure at any given point in space and be done with it.

Matt Hanson
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  • IMO it certainly makes no sense to talk about "pressure" on the molecular/atomar scale: at this scale, all of this is simply electromagnetism (i.e. one of the four fundamental forces), at least in normal scales of energy (not talking about neutron stars here). I guess the step from that to macro fluids is extremely complicated (including stuff like brownian motion etc.). – AnoE Aug 02 '23 at 09:41
  • I couldn’t agree more. The best you could do is get some sort of “average” forces via the Hellman-Feynman theorem and try to match them up with some sort of area, but that feels extremely bizarre. But it’s also fun to ponder at what scale you make the jump from molecules where this is nonsense to a bulk fluid where it makes perfect sense. – Matt Hanson Aug 02 '23 at 12:22