# Why should the force due to the pressure at a point be same in all direction?

Note: I have asked multiple questions under the same title because they are closely related and also because i think that answer to each of these questions will give a better understanding ab

1. Firstly what is the actual definition of a static fluid? Does it mean that every particle of the fluid is at rest? I think this is wrong because the pressure exerted by liquids is due to their random motion(is it?).
2. Now, coming to the actual question, this is how wikipedia explains why force due to the pressure at a point is equal in all directions.

If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal.

Now consider this example of this block.

For it to be in equilibrium it is not necessary that the forces on it are equal (in magnitude) in all directions. It is sufficient for the forces in the opposite directions to be equal in magnitude.

A key concept for understanding fluid mechanics is ergodicity. Informally, when you assume ergodicity, you assume that all of the little transient details about the system wont affect the final outcome, because the statistical behaviors are all that will matter (the actual definition is more nuanced than that, but that informal definition should be enough for answering this particular question). In the case of your static fluid, you are correct that the particles in the fluid are not at rest. They're bouncing around with thermal energy, sometimes at quite high velocities. However, in a static fluid, if you look at average motion, its 0. For every particle that's traveling to the left, it's statistically likely that there's a nearby particle traveling to the right.

As you noticed, this is the cause of pressure on objects. IF you look at any boundary on a fluid, you'll have fluid particles bombarding it from one side (with some statistically expected velocities), and no fluid particles on the other side, so you can easily see how this can impart impulse, which can be related to pressure.

In your equilibrium example, Wikipedia is not perfectly correct in their statement, but they're very close. In many fluid situations, you have the force of gravity pulling down on all fluid particles. To be in equilibrium, the pressures on the bottom side of your cube actually have to be slightly higher than the pressures on the top. The difference is exactly enough to counteract the extra downward force that the cube of particles applies due to the force of gravity on its mass. Now if the cube is really small, the difference here becomes vanishingly small. However, it's nonzero, and if you are integrating pressures, those little tiny differences can add up. In fact, when you look at the concept of buoyancy, you'll find those tiny differences add up to exactly the buoyancy force.

This minor detail aside, the pressures in all directions on that cube must be nearly equal. The reason for this is that the system will equalize itself if they are not equal. Consider a case where the left and right forces are substantially lower than those on the top and bottom (and assume constant temperature to make the description easier). Nothing stops the particles from flowing into and out of the cube, and the rate at which they flow is proportional to the pressure in that little region of fluid. In this example, lots of particles will flow into the cube from the top and bottom, very few will flow in from the left and right. Meanwhile, the same number of particles will flow out of the cube in all direction, based on the pressure in the cube. This will raise the pressure in the cube until the net movement of particles coming in from the top and bottom is equal to the net movement of particles going out through the left and right.

This will generate a small flow, redistributing particles from the top and bottom through the cube into the left and right regions. This process continues until the pressures are all brought into statistical equilibrium. At this point, there is no statistically significant movement of particles (they're all moving around equally in all directions), so the fluid is now "static." This is why the pressures in all directions are nearly equal in a static fluid: because the only state where the net fluid flow is 0 is when these pressures are all nearly equal!

@Cort Ammon has given a good answer from molecular point of view. I will just add a few remarks from the continuum point of view. Continuum, as you must know, is a model which treats matter as if it were infinitely divisible. So there are no atoms and molecules in this model.

You speak of fluid particle. In books on fluid mechanics that adopt continuum approach (the only ones I have read), by fluid particle is not meant a molecule, but an infinitesimal blob of fluid (shape is unimportant except for the purpose of making derivations easier). Therefore "fluid is at rest" means "every particle of fluid is at rest", period. For molecular picture, refer @Cort Ammon's answer.

Let us ignore gravity (even if there is gravity, as we take the limit of blob's volume going to zero, its contribution will vanish). Force balance on a stationary cubical blob of fluid shows, as you have said yourself, that pressure on opposite faces are equal. At first glance it seems, as you have said, that it does not prove that pressure on adjacent faces must be equal. To see why this is not the case, recall that, fluid, unlike solid, deforms indefinitely under the action of shear stress. When pressure acting on adjacent faces are unequal, although pressure on opposite faces are equal, a shear stress is created within the body of the fluid particle. Say pressure on top-bottom face is greater than that on left-right face. As a consequence fluid particle will be squashed; it will be compressed in the vertical direction and expand sideways (highly simplified picture).

Mathematical details of how stresses are distributed inside a cubical fluid particle is intricate. If equality of pressure is all you want to see, then choose a (2-D) triangular fluid particle with a right angle, as most elementary books on fluid mechanics do (I recommend Fluid mechanics by F.M. White).

• I always like seeing multiple different approaches to the same question. I found it particularly fascinating to compare and contrast the statistical motion of molecules to equalize pressure with a infinitesimal blob deforming under sheer stress. They're so different... and yet so much the same! Thanks! Aug 25, 2016 at 15:52

The answers all miss an essential factor because the answers do not distinguish between compressible fluids (gases) and noncompressible (fluids), which obviously behave differently. So there is one answer for gases and a radically different one for liquids. Because liquids cannot expand the molecules do not move against the direction of gravity, though they can move laterally. They can only push upward as a reaction to an external force. So this concept of pushing upward due to hydrostatic pressure is superimposed from gas mechanics.