# Hydrostatics: What remains the same in a fluid?

## Main Question or Discussion Point

I am looking into hydrostatics, but am now very confused about what has to remain constant in an incompressible fluid. I initially thought that pressure has to be the same all throughout the fluid, and that this is the reason why you can use water or oil when raising a car- you apply a small force to a small area, and for the pressue to be the same as the fluid comes up on the other side there is a greater force exerted by this fluid on raising the car because the 'arm' has a greater cross-sectional area. So then the pressure would be the same throughout the pipe, but the force would not be.

However thinking about it in terms of work, when you have an incompressible fluid in a pipe which is initially wider but then it gets narrower, then the speed of the fluid in the narrower portion of the pipe must be greater. I was wondering about where the extra kinetic energy comes from, and I found this answer :
"That means that its kinetic energy is increasing. Where is it getting the energy from? The answer is that it can only do so if the pressure in the narrower pipe is lower than in the wider pipe."

Now I am really confused: is it pressure or force that is transmitted through an incompressible fluid, or neither?

Thank you :)

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A.T.
what has to remain constant in an incompressible fluid
Density?

Density?
But does the force/pressure/some combination of these not have to be the same?

Orodruin
Staff Emeritus
Homework Helper
Gold Member
To expand a bit: By definition of incompressibility, the constant quantity is density. The fluid, for a static solution, must still obey the relation $dP = \rho g dh$. An implication of this is that if you increase the pressure at one point of the fluid, it must increase everywhere as a result, giving you the same increase in force per area.

That being said, the force on a larger area will be greater. However, if you press down in the small area, the surface of that area moves down more than the surface of the large area moves up and energy id neither gained or lost. The principle is similar to a lever.

• 21joanna12
Bystander
Homework Helper
Gold Member
what has to remain constant in an incompressible fluid
Look at the definitions of the two underlined words.

Okay so I see that pressure will vary depending on height, and the force will be different depending on the cross sectional area at the same height (where pressure is the same), but can pressure change in a horizontal pipe? In my original post, one of the things that confused me was when fluid was going from a pipe that was wider and then narrower, then the pressure had to decrease so that there was a net force and work was done on the fluid. Is this not a closed system then? Where does the energy come from? When it comes to gravity, when the kinetic energy increases it is because potential energy is converted into kinetic, both of the object and the earth as they experience equal and opposite forces. Where is the 'opposite force' in this system. and where does the energy come from?

Chestermiller
Mentor
In the case of a horizontal pipe, it's an F = ma kind of thing. If the velocity downstream is higher than the velocity upstream, the fluid must be accelerating. For this to happen, you have to apply a higher pressure upstream than downstream. The forces acting on the fluid in the "control volume" between the upstream and downstream locations are the upstream pressure times the upstream area minus the downstream pressure times the downstream area. There is also an axial force contribution from the pressure at the converging wall acting on the fluid. This must all be equal the rate of change of momentum of the fluid contained between the upstream and downstream locations.

Chet