Undergrad Pascal´s Principle in Compresssible Fluids

[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

 

[Chestermiller]

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.

 

[DaTario]

"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

 

[Chestermiller]

"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.

 

[DaTario]

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

 

[Chestermiller]

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.

 

[DaTario]

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.

 

[Chestermiller]

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.

 

[Chestermiller]

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.

 

[DaTario]

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

 

[boneh3ad]

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).