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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):


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):


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.

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This question may appear as a trivial waste of time, but the compressibility and confinement issues are found in most statements of Pascal's Law on the web, and are also misused in (disputable) contributions to physics.SE. So I would really welcome comments, even very short ones, or answers. – babou Sep 5 '13 at 9:43
I think this is a great, valuable base-question... and I think you should edit the format so that it is more answer-able. Open speculation / crit-feedback is not very germane here. But again, Great Thinking – New Alexandria Sep 5 '13 at 19:58

3 Answers 3

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.



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A physical law works the same, no matter how we describe it. When the wording of its description is wrong, it is wrong no matter what opinion you may have regarding changing the wording of a law, which is not the same for a physical and for a political law. For a political law, the wording cannot be wrong because the wording is the law, whatever it describes. For a physical law, the wording can be wrong, when not describing exactly what the physical law actually is. My post is concerned with the fact that Pascal law is misused by some people because it is too often incorectly stated. – babou Oct 9 '14 at 17:08

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.

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I am not sure I am following you. I am not considering a subset of the law. I am saying that the confinement hypothesis is unneeded for the statement of the law as found in Wikipedia or on the NASA site, or even in some contributions to physics.SE. Actually this law simply states that the pressure gradient is independent of the actual pressure if the fluid is incompressible. This is not very surprising since this gradient magnitude is $g\rho$, the density $\rho$ being constant. (thanks for the comment) – babou Sep 5 '13 at 21:53
I do not understand what height you mean that could be viewed as confinement. As a thought experiment, consider a volume cut out in the fluid at rest, and take it out to place it in a different context where a hole of the same size has been cut, and that is such that the fluid remains at rest in the new context. The only thing that may change is that different pressure will be applied to the boundary surface of your selected volume. The law says that the pressure will change everywhere by the same amount in the selected volume. Where is the confinement? The context may be the same fluid. – babou Sep 7 '13 at 9:23

Pascal used a serynge 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.


Watch a Wolfram demonstration of the syringe here

Is usefull 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:

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


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Your reasonning in the 2nd experiment supports also an open container. You remove the weight in A and replace it by a higher column of liquid. Then the container is open. Nothing says that the amount of liquid must be constant. Alternatively, you could measure pressure increase at points in the liquid that are far enough below the plates, so that liquid level does not matter even when removing weight A. All that matters is that there is some action that somehow modifies pressure in one point, even when the point is in the middle of the liquid. The change is transmitted everywhere. – babou Nov 24 '14 at 11:44
Regarding the Pascal syringe, I suspect it should rather be considered enclosed, with negligible leaks. The leaks introduce fluid dynamics phenomena that lower the pressure. These are tiny holes supposed to have negligible effect, and intended only a pressure measuring sensors. But you can do a similar experiment with an open container such as the second one without weight A, having pressure gauges or tiny holes on the sides to measure pressure. And you modify the system by changing the weight in B, or adding liquid in A. – babou Nov 24 '14 at 11:58
I've edited my answer, have a look – 21Brunoh Nov 24 '14 at 20:43
Also, remember that Pascal's principle can be translated into the mathematical form: $\Delta{p} = \rho g \Delta z$. This formula is derived assuming that the fluid is incompressible ($\rho$ is almost constant, but could be a function of distance, $\rho (z)$ for example) and is static, so that if we imagine a cube under water, then the pressure along the $x$ direction wouldn't change. This is a consequence of the balance of the forces acting on the two parallel faces of the cube, perpendicular to the $x$ axis – 21Brunoh Nov 24 '14 at 21:53

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