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When deriving the hydrostatic equilibrium for a static fluid, it is said that the net force on a fluid element dV of the fluid must be 0 because it is at rest (on the average, of course). There is a downwards force acting on the fluid element due to the weight of the water above it, and also there is another downwards force due to the weight of the fluid element itself. Therefore, in order for there to be equilibrium, there must be an upwards force on the fluid element that exactly equals the total downwards force. However, I am very confused about the origin of this force.

An upwards force is mandatory for there to be equilibrium, there's no doubt about that. However, what is the origin of this upwards force?

For the downwards force for example, the origin is reasonable, it is due to the weight. But for the upwards force, is the reason due to "collisions" of the liquid molecules below on the fluid element or what? I've read that this upwards force is caused because the pressure below the fluid element is greater than the pressure acting on top of the fluid element, but I am still not able to get why the fact that there is more pressure justifies that there is an upwards force. Also, the reason for the buoyant force is this difference in pressures, if I am not wrong, but then what is the reason for the buoyant force?. Also, another way to think about this situation is that the upwards force is due to the normal force, just like a solid object standing on the earth has a normal force, but this doesn't explain too much about the origin of the upward force. The normal force on a solid object on earth is due to the electric repulsion of the molecules of the block and the molecules of the ground of the earth, since they are being "pushed together" due to the weight of the solid object, and this causes the repulsion. I wonder if this is the same reason for the upwards force in a static fluid scenario, but I'm not sure.

I hope that someone can help clear my confusion, I would appreciate it a lot.

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    $\begingroup$ The force exerted on a part of the fluid by the surrounding fluid is due to the fluid's pressure. Microscopically, the pressure is due to electromagnetic forces exerted between particles in the fluid. $\endgroup$
    – d_b
    Feb 13 at 22:04

2 Answers 2

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You can think about it this way- the fluid element is pushing on bottom element, so by the 3rd newtons law it feels reaction force keeping it in equilibrium. Hope this helps

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  • $\begingroup$ Yes, as this answer says the upward force is a reaction force arising from the volume element attempting to push the fluid below it downwards, and in equilibrium there is no motion of the volume element. Just like a rock stationary on the ground has an upwards force on it from the ground. And as @d_b says the microscopic reason for the force is due to electromagnetic forces between molecules in the fluid (or in the ground) that increase as you attempt to compress the fluid or ground (liquids are not very compressible). $\endgroup$
    – John Darby
    Feb 13 at 22:46
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Trying to analyze the atoms and molecules in a substance when your doing a simple mechanics problems is always a bit tricky, and I don't find them so helpful. For example, if I have a block sitting on a surface, and I pull on it with a certain force, would you try to envision all the molecules inside the block being pulled and pushed by electromagnetism? I certainly don't; I instead just think of it as a mathematical construct: it has this mass and this force, dividing is the computation that results in acceleration, multiply acceleration and time is what gives velocity, etc. It's better to abstract this stuff away and just keep in the back of your mind that empiricism has shown that these mathematical formulas and constructions happen to make good predictions of the real world.

That being said, if you still hold fast to trying envision everything, I would say the following. The liquid cannot be thought of like gasses; it isn't a bunch of molecules floating around in free space. Instead, it is a bunch of molecules linked together, but the linkings aren't so rigid, meaning it is easy to deform this substance. Anyways, when you put water in a beaker, what happens is that the weight of the above water presses the lower molecules into the floor of the beaker, but the beaker pushes back up; it resists this. The bottom molecules cannot be rammed through the bottom of the beaker. This normal force then gets transmitted to the next "layer" of molecules and then the next, and it propagates all the way up to the top. This is where the upwards force comes from. So when you take a small element of water, it is being crushed by the weight above it, but it cannot be pushed farther downwards because there is a column of water beneath it, and beneath that column of water is the beaker floor. Just think about the water being really incompressible, and that gives a good mental model.

With regard to buoyancy, you are correct in the fact that it is caused by pressure differences, but I like to reason about buoyancy like this: if we enclose any arbitrary shape of water with a thin membrane, the upwards force on that must be the weight of the enclosed volume of water. Water remains static if no one touches it. Therefore, if we put a basketball inside the water, it doesn't matter if we think about it like spherical object made out of plastic or a spherical membrane enclosing more water. There could be anything inside the basketball. The only thing that matters is its shape. In the thin membrane case, the force is simply the weight of displaced water because water remains static. Therefore, the force on the basketball must be the weight of the displaced water.

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