I understand that liquids are understood to be virtually incompressible, and so when a force over an area is applied to a liquid in a fixed volume container, the pressure change is even throughout the fluid—otherwise the internal pressure differences would cause movement and compression. However, what I don't understand is why the fluid pressure increases by the amount of applied pressure, completely independent of the fluid volume. It seems intuitive to me that the applied pressure would be "averaged" out throughout the fluid, leading to less pressure change (though constant throughout) for a larger fluid volume. Can anyone explain why this isn't the case?
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$\begingroup$ Draw a free-body diagram of some liquid at the boundary. If the pressure weren't equal on each side, that region would start to accelerate, which isn't observed in a static fluid. This can be repeated for any region to show that the pressure must be equal at any horizontal level (or any point, if gravity is ignored). Does this clarify? $\endgroup$– ChemomechanicsCommented Feb 11 at 22:26
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No liquid is fully incompressible. Lets say for sake of argument, that it takes $1000 N \times$ Area of piston to compress a cylinder containing a litre of liquid. Now if we want to compress 10^6 litres of the liquid using the same piston, we would have to move the piston 10^6 mm or 1 km to raise the pressure of the larger volume of the liquid by the same amount.
Energy is force times displacement and it takes $10^6$ more energy to compress the larger volume of liquid. If we limit ourselves to only moving the piston 1 mm, we would have to increase the area of the piston by $10^6$ and since pressure is force times area, it would require $10^6$ more force, so it is the force or the energy that is "averaged out".
