# What is the physical fundamentals of Pascal's law

Pascal's law or the principle of transmission of fluid-pressure (also Pascal's Principle) 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. - Wikipedia

On one hand, "incompressible" suggests that intermolecular interaction plays an important role in Pascal's law.

But when we consider ideal gas, while doesn't obey the Pascal's Principle, will arrive at the "equally transmitted" state when given enough relaxation time. This suggests that the "equal trasmission" arise from statistical behavior of molecules. "Incompressible" merely suggests zero relaxation time.

However, if we think of molecules as hard spheres whose random oscillation will be damped by other molecules, so only directional motion will occur, Pascal's law remains true.

So, what's the physical interpretation of Pascal's law? Is it a statistical law of collective behavior, or is it caused by the repulsive interaction between molecules? Thanks!

• Why doesn't an ideal gas obey Pascal's principle? In an ideal gas (at equilibrium) the pressure is everywhere the same, so it seems to me that it obey's Pascal's principle. The restriction to incompressible fluids just means equilibrium is attained instantaneously because the speed of sound is infinite. Aug 14, 2015 at 9:25
• @JohnRennie Yeah that's actually what I meant. My take on Pascal's law is that if it takes time to achieve equilibrium, the Pascal's law doesn't hold strictly
– arax
Aug 14, 2015 at 16:12
• Well no fluid is incompressible, and all fluids have a finite speed of sound. Both are a function of the bulk modulus, so are you really asking about the microsocopic origin of the bulk modulus? If so, that's a rather different question. Aug 14, 2015 at 16:23
• @JohnRennie Oh actually your comment answers my question - Pascal's law comes from the nature of fluids to counteract exerted pressure, which includes both the intermolecular repulsive interaction (as in liquids) and the statistic aspect (as in ideal gas)
– arax
Aug 14, 2015 at 17:26
• "Why doesn't an ideal gas obey Pascal's principle?" @JohnRennie in the form stated above—which I find a bit confusing—you have to have the density of the fluid remain the same. Because otherwise the initial differences due to variations in altitude would change as you compressed the working fluid. I'm sure we could form a generalized version of the principle which allowed for density variation, but it would need careful wording. Jul 9, 2016 at 19:03

Pascal's Law merely states that, at a given spatial location in a fluid, pressure acts equally in all directions. This means that, if a small "test surface" were situated at a given location within a fluid, the force per unit area acting on that test surface would be independent of the orientation of the test surface. Pascal's Law doesn't say anything about how pressure varies from location to location in a fluid.

The variation of pressure within a flowing or static inviscid fluid from location-to-location is described by Euler's equations. The variation of pressure within a flowing or static viscous fluid from location-to-location is described by the Navier-Stokes equations.

• What you 've written here is the way I understand the principle. However, the wording given above (which is not the one I'm used to) does address pressure differences in a limited way by saying that what every equilibrium pressure differences that existed to begin with (which must mean those due to variation in altitude) remains the same as you vary the absolute pressure. In essence it is a claim of constant density. Jul 9, 2016 at 19:06