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I'd like answers both in the more intuitive side an on the more precise side.

Thinking of water as "cubes" of water, for example, would allow pressure in the z axis to be independent of the y or x axis. Choosing other "shapes", like a triangular prism, yield different results. Will this fact, then, be dependent on the format of the molecules of the liquid ?

I've heard of the rotational symmetry of liquids. What is its precise statement ? Why is it true ?

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  • $\begingroup$ what about divergence of the pressure field? it will allow infinitesimal cubes. I'm waiting to see the explanation of this. good question. $\endgroup$
    – Aftnix
    Jul 11, 2012 at 18:42
  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/18255/2451 $\endgroup$
    – Qmechanic
    Jul 11, 2012 at 19:06

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The pressure in three different directions is indeed independent for materials that are composed of cubes or other fixed shapes. But these materials are called solids, not liquids.

By definition, a liquid is a material without any regular crystallic or otherwise periodic structure. A liquid is composed of randomly arranged molecules that are as close to each other as the repulsive forces allow – the latter property distinguishes liquids from gases. Liquids and gases are two subgroups of a larger group called fluids.

When a milliliter of liquid is at rest, it means that in this milliliter of material, the molecules are randomly ordered and randomly moving so that their center-of-mass remains at rest. But when it's so, the only "force-related" quantity by which one milliliter of this liquid differs from another is the density or any function of it, such as the pressure. Because there's only one density, there's only one pressure.

Because liquids are not composed of fixed cubes but of chaotic molecules. a new molecule added to a volume of liquid with excessive pressure may escape in any direction. Whatever direction it chooses, the density (in molecules per unit volume) will be reduced to the appropriate value so the pressure may drop, too.

The independence of the fluids' pressure on the direction is known as Pascal's law and you may read an independent explanation at Wikipedia:

http://en.wikipedia.org/wiki/Pascal%27s_law

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  • $\begingroup$ this relation between pressure and density on liquids is interesting ... Could you elaborate a bit ? $\endgroup$
    – josinalvo
    Jul 11, 2012 at 19:44
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    $\begingroup$ Hi, it's called the "equation of state". For ideal gases, you know it in the simple form $pV=nRT$ which really means that $p$ is proportional to $\rho$ at a fixed temperature. For liquids, the dependence is sharper - a small change of the density requires a huge change of the pressure. Pressure is ultimately coming from the molecules hitting the walls, trying to escape from a vessel. The more molecules you have, the more they hit (not proportionality in general). Also, if they're too crowded, the repulsive force between the molecules gets tranferred to the force from molecules to the walls. $\endgroup$ Jul 12, 2012 at 9:58
  • $\begingroup$ How do you conclude centre of mass is at rest ? $\endgroup$
    – Isomorphic
    Jun 30, 2014 at 21:57
  • $\begingroup$ I haven't used anything of the sort. Pressure - and density (nonrelativistically) - is evaluated in the rest frame of the macroscopic piece of liquid/gas, so whatever collective motion there is simply subtracted. $\endgroup$ Jul 1, 2014 at 3:40
  • $\begingroup$ I'm thinking of a bunch of marbles on the floor in a tight cluster. Push in at one point, and the random disorder of the marbles causes the force to be distributed everywhere and marbles go flying in every direction. Whereas... if you had a tightly paced cluster of six-sided dice, the force would transmit along the faces of the dice to the opposite side. $\endgroup$ Apr 3, 2018 at 20:14
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The answer easily follows from the definition of pressure itself. It is the force that the particles apply on the sides of the container per unit of distance. Particles do have dimensions, but for what concerns the calculations their dimensions are irrelevant when compared to the sides of the container (although little modifications can apply, if you use any other state equation rather than the law of perfect gases). If you assume that the particles randomly move, then there will be the same amount of particles (in average) hitting the unit side of the container in any direction, therefore the pressure will be the same on any side independent of what happens elsewhere.

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Choosing other "shapes", like a triangular prism, yield different results.

Whatever, be the shape of the prism you chose, the result shall always be the same. Here is an example with a right-angled triangular prism of water (of infinitesimal volume, such that it is practically a point). Let's take the horizontal forces acting on the prism first.

enter image description here

$ F_c = F_bsin \theta$

$ P_cA_c = P_b A_bsin \theta$

$ P_cA_bsin\theta = P_b A_bsin \theta$

$P_c = P_b$

If you take the vertical forces, you will get $ P_a = P_b$; thus forces acting on the prism of water are of the same magnitude in all directions.

My explanation is built upon the fact that the net force acting on any imaginary plane inside a fluid is always is normal to it.

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