Pascal´s Principle in Compresssible Fluids

• I
• DaTario
In summary, Pascal's Principle states that, at a given location in a fluid, the pressure acts equally in all directions. However, this principle does not account for how pressure varies from location to location. It is valid for incompressible fluids, but for compressible fluids, there may be higher fluctuations in pressure. Additionally, it may take some time for the pressure to equilibrate in a compressible medium, as pressure changes propagate at the speed of sound.
DaTario
Hi All,

When explaining Pascal´s Principle I had some bad time related to its application in compressible fluids. Correlation functions (forces in two points separated in space at the same time) came to my mind but at the end the doubt persists.

Is Pascal´s Principle valid for compressible fluids?
If not entirely, is it just a matter of higher fluctuations but in average it is valid?

Best wishes,

DaTario

DaTario said:
Hi All,

When explaining Pascal´s Principle I had some bad time related to its application in compressible fluids. Correlation functions (forces in two points separated in space at the same time) came to my mind but at the end the doubt persists.

Is Pascal´s Principle valid for compressible fluids?
If not entirely, is it just a matter of higher fluctuations but in average it is valid?

Best wishes,

DaTario
Pascal's Principle says that, at a given location in a fluid, the pressure acts equally in all directions. That means that, if you place a tiny element of surface area at a given location in a fluid and vary its orientation, the force acting on the element of surface area will not change. Pascal's Principle doesn't say anything about how the pressure varies from location to location. So, yes, for an incompressible fluid, Pascal's Principle still applies.

"A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid."

I think this text suggests that it is not a purely local statement. It has to do with comparison / correlation between pressures at different positions, hasn´t it?

P.S. I was addressing the question of compressible fluids, not incompressible.

Best Regards, Chestermiller.

DaTario

DaTario said:
"A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid."

I think this text suggests that it is not a purely local statement. It has to do with comparison / correlation between pressures at different positions, hasn´t it?
That's not my understanding of Pascal's Principle. However, the statement will be correct for an incompressible fluid. It will not be correct for a compressible fluid in a gravitational field.
P.S. I was addressing the question of compressible fluids, not incompressible.
Oops. I meant compressible in what I answered.

It seems that from the isotropy you mentioned one can arrive to this nonlocal property.

DaTario said:
It seems that from the isotropy you mentioned one can arrive to this nonlocal property.
Why don't you just do an analysis for the case of a constant mass of ideal gas in a gravitational field that satisfies the equation ##dp/dz=-\rho g##, where ##\rho=\frac{pM}{RT}## (where the gas is in a closed container) and see what you get it you change the volume? See if the pressure changes uniformly or not.

Stevin´s Theorem is consistent with Pascal´s Principle, but in a stationary regime. Once you move the piston I believe the pressure takes some time (in compressible fluids) to increase accordingly.

DaTario said:
Stevin´s Theorem is consistent with Pascal´s Principle, but in a stationary regime. Once you move the piston I believe the pressure takes some time (in compressible fluids) to increase accordingly.
I mean when the system re-equilibrates.

For the case of an ideal gas in a container, the pressure at the bottom minus the pressure at the top is given by:
$$p_B-p_T=\frac{mg}{A}$$where A is the cross sectional area of the container and m is the mass of gas. If we increase the temperature in the container (to raise the pressure everywhere) this relationship won't change, and, if we lower the top of the container (to decrease the volume and increase the pressure everywhere) this relationship won't change. So, in a way, for these cases, the pressure does increase uniformly.

Chestermiller said:
For the case of an ideal gas in a container, the pressure at the bottom minus the pressure at the top is given by:
$$p_B-p_T=\frac{mg}{A}$$where A is the cross sectional area of the container and m is the mass of gas. If we increase the temperature in the container (to raise the pressure everywhere) this relationship won't change, and, if we lower the top of the container (to decrease the volume and increase the pressure everywhere) this relationship won't change. So, in a way, for these cases, the pressure does increase uniformly.

Very good approach. But we must bear in mind that this equation describes stationary regimes.
Thank you,
DaTario

Of course a pressure change is not going to be instantaneous in a compressible medium (which is every medium in reality). Pressure changes at one point will propagate through the container at the speed of sound in all directions. Eventually it will reach equilibrium again (fairly quickly).

What is Pascal's Principle?

Pascal's Principle, also known as the principle of transmission of fluid-pressure, states that when pressure is applied to a confined fluid, the pressure is transmitted equally in all directions and acts with equal force on all areas of the container.

How does Pascal's Principle apply to compressible fluids?

Pascal's Principle applies to compressible fluids in the same way as it does to incompressible fluids. The pressure applied to a compressible fluid will be transmitted equally in all directions, but the volume of the fluid may change in response to the applied pressure.

What is an example of Pascal's Principle in compressible fluids?

A common example of Pascal's Principle in compressible fluids is the use of hydraulic systems. In a hydraulic system, a compressible fluid, such as oil, is used to transmit pressure from one point to another. When pressure is applied to one end of the system, it is transmitted to the other end, allowing for mechanical work to be done.

How is Pascal's Principle used in engineering and technology?

Pascal's Principle is used in various engineering and technological applications, such as hydraulic systems, hydraulic presses, and hydraulic brakes. It allows for the amplification of force and the efficient transmission of pressure over long distances.

What are the limitations of Pascal's Principle in compressible fluids?

While Pascal's Principle is a useful concept, it does have some limitations. It assumes that the fluid is incompressible, which is not always the case. It also does not take into account factors such as fluid viscosity and friction, which can affect the transmission of pressure in real-world situations.

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