Why should fluids be confined for Pascal's Law to be applicable

Question

When is Pascal's law about fluid pressure propagation applicable? Is it applicable to a closed circular pipe with a pump rotating the fluid, but not to a tub of water. Most statements require only confinement of the fluid.

Or is the question irrelevant ? This law is no longer used or taught ?

Here is my full story about it.

You do not have to read it all to answer. Your choice.

The question and its context

Being a bit surprised by a dismissal on this site of the applicability of Pascal's Law because the fluid was not confined (well, it was an ocean), I took my favorite browser and asked: Pascal Law, then I took the answers in order, more or less. I do skip youtube.

According to Wikipedia (quoting a book):

Pascal's law or the principle of transmission of fluid-pressure is a principle in fluid mechanics that states that pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid such that the pressure variations (initial differences) remain the same.

which is confirmed by the Wiktionnary:

The law that states that a confined fluid transmits externally applied pressure uniformly in all directions

Then according to NASA (who seems to equate fluid with liquid):

Pascal's law states that when there is an increase in pressure at any point in a confined fluid, there is an equal increase at every other point in the container.

The Q&A system of the Physics Dept at University of Illinois at Urbana-Champaign agrees (quoting a book):

Pascal's Law (also called Pascal's Principle) says that "changes in pressure at any point in an enclosed fluid at rest are transmitted undiminished to all points in the fluid and act in all directions."

insisting further that it must be "a closed container".

And the sciencefair site confirms for the layman:

Pascal's Law states that when you apply pressure to confined fluids [...], the fluids will then transmit that same pressure in all directions within the container, at the same rate.

While the Hyperphysics site insist for the professional with this statement of Pascal's Principle:

Pressure is transmitted undiminished in an enclosed static fluid.

To be fair, the Free Dictionnary defines Pascal's law as follow:

The principle that external static pressure exerted on a fluid is distributed evenly throughout the fluid. Differences in static pressure within a fluid thus arise only from sources within the fluid (such as the fluid's own weight, as in the case of atmospheric pressure).

Maybe because it is not a science site.

My main question is: Why should the fluid be enclosed ? (for the law to be applicable)

A side question is, why do most statements ignore compressibility ? Well, Pascal did too, afaik, but he was in a much different context. Also we should remember that not all fluids are liquids. Forgetting that fact may lead to death by drowning, or by diving in empty pools.

My own clarification of the issue

Maybe I should not answer myself, so as to better encourage other answers. But then, why pretend I am too stupid to answer it when I can actually prove that I am ?

As I said, I would like some clarification on this, and check my own understanding, which is not quite in agreement with the above definitions.

If this law is considered in a uniform gravity field, the requirement of incompressibility is needed only to preserve a constant density, so that the pressure difference between different heights in the liquid does not change. If correct, this also means that the incompressibility requirement can be lifted regarding the transmission of pressure change between two points at the same height.

Note that if pressure change is small, there is little compression and density variation so that the incompressibility requirement can be lifted as first approximation. That was, I guess, the situation of Pascal as he was analysing atmospheric pressure and its variation under limited height changes.

In free fall, the weight of the fluid no longer matters, and the compressibility should not matter either. However the fluid has to be confined if a non-zero pressure is to be applied. It is actually more complex, as there may be surrounding pressure from the environment, or surface tension in various cases (e.g., a bubble of gas in a liquid). Also, enclosure may be necessary to prevent mixing of the liquid or gaz with surrounding medium, though that does not require a rigid enclosure, and is not related to Pascal's law itself.

However, I do not see the need for the confinement constraint in a gravity field. A container is certainly needed, as is usual for liquids (especially when it is hot coffee), but nothing requires that it be a closed confinement. The top can perfectly well be open to atmospheric pressure in the case of a liquid, or only separated by a very light membrane for gas. This may not be convenient for achieving high pressure in some devices, but that is an engineering issue that has nothing to do with Pascal's law.

This is even true for gas. After all, the law is originally based on a study of Earth atmosphere which can hardly be considered as enclosed.

Of couse, the fluid is considered only in those parts of space where it is present before and after the pressure change.

To conclude, my understanding is that the confinement may be needed to actually achieve a change of pressure somewhere in the fluid, in engineering applications of the law but confinement has nothing to do with Pascal's law itself.

Why then ?

Why this agreement on requiring an irrelevant condition for the applicability of Pascal's law.

The question should possibly be another one. What is meant by requiring that the fluid be confined or enclosed ?

Maybe the answer is given by the sciencefair site which (if I did not miss anything) is the only one to actually define what it means:

contained in a flexible yet leak-proof enclosure so that it can't flow out

I will not dispute this disputable definition, but the intent seem clearly to prevent dynamic flow, to have a static system. Whatever the actual intent, that is the only explanation I found.

Hyperphysics is very professional and to make the point absolutely sure uses both belt and suspenders: "an enclosed static fluid". The University of Illinois has the same worry about decency: "an enclosed fluid at rest".

As I understand the problem, it would be wiser to only use the correct qualifier, and mention "static fluid" or "fluid at rest", rather than inappropriate qualifiers that create much confusion and misunderstanding, particularly in an educational context, as shown by the example that initially motivated this question.

Is this analysis correct or did I miss a point ? If I err, can you tell me where ?

In a widely ignored question (I am very good at that) that only managed to gather one downvote, I was trying get reactions or opinions on the issue of precision in science. I did not mean quantitative but rather qualitative precision, precision in concepts and hypothesis rather than mathematical formulae.

I guess this is just another example, and I did not have to look for it: it found me.

All contributions to understanding this are answers.

Answer 1

Fluid power began with hydraulics and the fluid being water. Water cannot be compressed. Though other fluids can be compressable and some even into gasious states, some remain non-commpressable.

External force or pressure means anything protruding from the outside in being the pushing force creating the pressure. We need non-compressable fluids to make the hydraulics and other fluid powers to work in the ways in which they do today.

Keep in mind that even if the verbage of fluid power were to change, that the principles will continue to work the same as they exist today. To alter the verbage now and in todays society, would only confuse people more in this already confusing world.

Ask yourself this; is it worth the fight to chage laws of theroys for one mans different interiptations? Also, is this why our constituion is so insanely ammended because everyone seems to find a way to change it to how they see it to be?

I think you're over thinking this thing buddy. It's just meant to be used as a basic tool in understanding how fluid power works.

Thanks

Joe

Answer 2

At first glance, I will suggest that Pascal's law indeed does not address the matter of internal pressure gradients (and variations) within this 'ideal confined incompressible' fluid.

As you've indicated, the internal pressures may change, even though any height remains the same; unless I misread you, this is the same thing as saying that the fluid is confined. Its heights do not change.

So what you've described is a special case, that is a subset of Pascal's Law, dealing with the effects of energy-transfer in a non-compressible (or nearly-non-compressible) fluid, e.g. induced zeta-potential, etc.

Answer 3

Update

• This is an old post, but after a recent notification I've delved into this topic again. The key here is to look at the definition when it says "all points":

  Pascal's Law (also called Pascal's Principle) says that "changes in pressure at any point in an enclosed fluid at rest are transmitted undiminished to all points in the fluid and act in all directions."

  Now let's look at the following figure. In (A) we simply have the fluid "resting". Assume that point A and B pressures are zero, i.e., in "vacuum". If you apply a force in point A to change the pressure at that point to be $P$ using e.g. a piston, the fluid will move upwards in point B, until a liquid column is created such that the pressure generated by its height makes the system static again (situation (B)). In this case we can see that point B pressure remains zero since it's only exposed to vacuum.

On the other hand, as depicted in situation (C), if you placed a piston at point B and then make it fixed, you would see that the increment of pressure, $P$, in point A would transmit that pressure undiminished to point B (you could test this: if instead of fixing the piston you held it yourself, you would notice that a counter pressure $P$ would be required to be applied in point B to keep the system static).

In summary, it is not true that a change in pressure at any point (e.g. point A) in an open fluid system at rest is transmitted undiminished to all points (e.g. point B).

In response to Babou: You could argue that instead of the fixed piston you would add a liquid column yourself, but that is something external to the fluid system we are analyzing, like the piston at B. Remember that the principle applies to the fluid itself and not the hypothetical external objects you use.

Old Answer (just read the update above)

Pascal used a syringe that allowed leakage of fluid, to demonstrate that increasing pressure at one point would increase pressure at all points. Although water is leaking, as Babou said, we can still consider the fluid enclosed, where, instead of holes (that allow leakage) we could use sensors to mesure pressure.

_{(source: ucla.edu)}

Watch a Wolfram demonstration of the syringe here

It is useful to assume the container must be leak-proof. Why? If fluid left the container while changing pressure in point A (for example pressing the piston) then the heights between point A and B would be different and they must be equal in order to prove that pressure on A and B are equal and then proving that by raising pressure on one of the points then the pressure on the other point will raise equally.

This is because $\Delta{p} = \rho g \Delta z $ (you can see in this document how to derive this formula). If the difference between the heights are zero ($\Delta z = 0$) then $\Delta p = 0$ thus we conclude $p_A = p_B $.Now we can visualize that if I double the pressure on A then I also double the pressure on B, meaning the pressure was indeed transmitted through the tube.

Image from: http://pascalteam.hu/en_pascal_law.php

EDIT: Babou after reading your comments I did some research and found this interesting document stating Pascal's law as follow:

"Pressure is defined as force per unit area. Can pressure be increased in a fluid by pushing directly on the fluid? Yes, but it is much easier if the fluid is enclosed. The heart, for example, increases blood pressure by pushing directly on the blood in an enclosed system (valves closed in a chamber). If you try to push on a fluid in an open system, such as a river, the fluid flows away. An enclosed fluid cannot flow away, and so pressure is more easily increased by an applied force. What happens to a pressure in an enclosed fluid? Since atoms in a fluid are free to move about, they transmit the pressure to all parts of the fluid and to the walls of the container. Remarkably, the pressure is transmitted undiminished. This phenomenon is called Pascal’s principle, because it was first clearly stated by the French philosopher and scientist Blaise Pascal (1623–1662): A change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container."