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I’m not sure I 100% understand the buoyant force on the atomic scale. I know the force is due to pressure exerted on the object, due to the volume of the fluid that the object displaces. I’m under the understanding that the reasoning for this is that the displaced fluid pushed against the walls of its container, which in turn apply a normal force to the fluid, which is transferred to the object against the force of gravity.

If this isn’t the case, then if someone could try to explain to me why displacing a volume of the fluid exerts a force on the object?

Anyway, say there is a large body of water falling through space, with a force applied to it due to gravity. The fluid has no container, and accelerates downwards forever. If an object were placed inside the the fluid, would the fluid exert a buoyant force against the pull of gravity?

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The buoyant force is equal to the WEIGHT of the fluid that an object displaces. For an object volume of "V", where "V" is the volume of the object that is submerged in a given fluid, this equates to:

$F_B = \rho V g$.

Note that "g" is in this equation. In free fall, the effective value of "g" is 0. Therefore, the answer to your question is "no", a fluid in free fall will not exert a buoyant force on an object submerged in it. For this very reason, you will see video footage of water globules inside the space station with bubbles inside of them, and the bubbles do not leave the water because there is no buoyant force on them to force them out of the water phase.

For the case where a container of water is sitting on the surface of the earth, the container does experience a force due to gravity, which causes a vertical pressure gradient in the water column. In this case, a bubble in the water column experiences a higher pressure on the bottom of the bubble relative to the top of the bubble, and this causes a net force in the upward direction. Hence, the bubble rises until it is expelled from the water column.

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I know the force is due to pressure exerted on the object,

Right.

due to the volume of the fluid that the object displaces.

Wrong. It's not due to the volume displaced. Imagine the following situation. You have an empty container, open on top. Place a solid block in the bottom, then pour some water within, covering the block. There has been fluid displaced? There is buoyant force?

It's true that buoyancy equals the weight of displaced fluid, but the cause is in differences of pressure because of gravity. If gravity disappears (free fall) buoyancy disappears too.

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