What's the pressure acting on this fluid?

In summary, as an incompressible fluid flows through a pipe with decreasing cross sectional area, the velocity increases. This is necessary for the mass/volume continuity equations to be true. The increasing velocity indicates a net pressure on the fluid, which can be thought of as potential energy being converted to kinetic energy. This difference in pressure can be better understood as a compression of the fluid molecules, leading to a minute density gradient. This is similar to a chain of balls attached by springs, where the balls accelerate as the springs compress and then break. In the case of a pipe, the force imbalance between upstream and downstream creates this compression and ultimately leads to the increase in velocity.
  • #1
Marmoteer
8
0
I understand that as an incompressible fluid flows through a pipe with decreasing cross sectional area the velocity increases. This must happen for the mass/volume continuity equations to be true. Since the velocity is increasing though there has to be a net pressure on the fluid right? My question is - what's the best way to think about this pressure? Is it like pressure "buildup" because the flow is restricted? Or is there a better way to interpret it?
 
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  • #2
Think of it more like potential energy, and as the velocity increases, more is converted to kinetic energy (dynamic pressure) so the static pressure falls. Indeed, the units of pressure can be rewritten as energy per volume.
 
  • #3
Marmoteer said:
I understand that as an incompressible fluid flows through a pipe with decreasing cross sectional area the velocity increases. This must happen for the mass/volume continuity equations to be true. Since the velocity is increasing though there has to be a net pressure on the fluid right? My question is - what's the best way to think about this pressure? Is it like pressure "buildup" because the flow is restricted? Or is there a better way to interpret it?

Well... The fluid being incompressible has a problem right there.. In deep oceans, the pressure in of water is WAY bigger. Pressure is indeed a energy density (joules per cubic meter). So the problem arises: how can it be that the energy density is higher if the molecules are not closer together? Note that water is not hotter deeper, so it cannot be that the particles have gained kinetic energy. So it must be the potential energy... The potential energy density. That means that the molecules are closer together the deeper we go...

Indeed, molecules have what is called intermolecular bond stiffness: this can be modeled as springs between the molecule..

A little iterative thinking may give a clue:

A molecule is being pulled by the earth: mg
But it does not accelerate, so this mg is balanced.. By what? Well by the molecule right under it: so there is an mg upward.

So there is a repulsive force between the first molecule and the one just below it, that force has a magnitude of mg.

According to Newton: a force is a mutual interaction between two bodies: so if the lower molecule pushes the upper one with mg, the second one is pushed down by the first molecule, by an amount mg - note that this interaction is ELECTRIC: a repulsive interaction.

So the second molecule must experience a force 2mg, because the first molecule is pushing down on it with mg, but the Earth is also pulling it the second molecule down with the amount of mg.

But that 2mg down MUST be balanced by a 2mg upward, because there is no acceleration.

That 2mg upwards is caused by the 3rd molecule.

How can the 3rd molecule be pushing the second one upwards by the amount of 2mg?

Well, it has got to be closer to it then first molecule is to it...

Similarly, the 3rd molecule is experiencing the force 2mg down, due to the mutual repulsion with the second molecule.

But gravity is also acting on it, so there is 3 mg downward for the 3rd molecule...

That means that the 4th is pushing 3mg upward, it can do that by being a TINY TINY bit closer to the 3rd molecule than the 2nd molecule is... etc...

So it becomes pretty clear that fluids cannot be incompressible in reality.

This compression seems to be horrifically small, practically not too measurable, but it has to be there, otherwise it makes no sense that the energy density is higher and it is also nonsense to explain, why the deeper molecules can act on the upper ones with such a huge force.

This also explains why the pressure is the same at in a certain depth level, even when the column of water is not straight: the molecules are a tiny tiny amount closer each other.

Oh, and as far as the pipe is concerned:

you can think of it this way: the potential energy that the molecules HAD is now changing into the kinetic energy of the molecules: they are step-by-step being accelerated away from each other, by tiny amounts: so the pressure drops: the potential energy density drops. What you see is that water is just flowing faster.

At those points, the thinner the tube gets, there is a MINUTE density gradient in water, hardly measurable, but this is the way it works. This is the fundamental difference between water flow and electric current: water molecules do "push each other", and they potential energy relative to EACH OTHER is turning to kinetic energy of the individual molecules.

In the electric current, the electric filed pushes the free electrons in a wire. The field is made by the surface charge gradient: charges on the SURFACE of the wire, whereas the water molecules do really accelerate each other. It is like having many balls attached to each other with easily breakable spring: push it so the chain get shorter and the spring is compressed. Now let it go: the balls accelerate one after another while the spring is going to a relaxed position and then breaks because the balls are getting too far.

This analogy has many problems of course... But all in all: the essence is similar.
 
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  • #4
Marmoteer said:
I understand that as an incompressible fluid flows through a pipe with decreasing cross sectional area the velocity increases. This must happen for the mass/volume continuity equations to be true. Since the velocity is increasing though there has to be a net pressure on the fluid right? My question is - what's the best way to think about this pressure? Is it like pressure "buildup" because the flow is restricted? Or is there a better way to interpret it?
If the velocity is increasing, then there has to be a force imbalance acting on the parcels of fluid to make them accelerate. The force upstream (pressure times area) minus the backward force exerted by the converging channel wall on the fluid is higher than the backward force downstream (pressure times area), so the fluid accelerates.

Chet
 
  • #5


I would like to clarify that the pressure acting on a fluid is a force per unit area that is exerted by the fluid on its surroundings. In the scenario described, the pressure on the fluid is indeed increasing as it flows through a pipe with decreasing cross-sectional area. This is due to the conservation of mass, which states that the mass of a fluid must remain constant as it flows through a confined area.

To better understand this pressure, it is helpful to think of it as a result of the fluid's energy. As the fluid flows through a restricted area, its kinetic energy increases, resulting in an increase in velocity. This increase in velocity leads to a decrease in pressure, as described by Bernoulli's principle. However, the overall pressure on the fluid remains constant, as the fluid is still confined within the pipe.

In terms of interpretation, the pressure acting on the fluid can be seen as a measure of the force that the fluid exerts on its surroundings. It can also be thought of as a result of the fluid's energy and its interaction with its surroundings. Therefore, the best way to think about this pressure is as a dynamic force that is constantly changing as the fluid flows through the pipe.
 

1. What is fluid pressure?

Fluid pressure is the force per unit area exerted by a fluid on the surface of an object. It is typically measured in units of pressure, such as pascals (Pa) or pounds per square inch (psi).

2. How is fluid pressure calculated?

Fluid pressure is calculated using the formula P = F/A, where P is the pressure, F is the force exerted by the fluid, and A is the area over which the force is distributed. This formula is known as Pascal's law.

3. What factors affect fluid pressure?

The factors that affect fluid pressure include the density of the fluid, the depth or height of the fluid, and the acceleration due to gravity. The shape and size of the container holding the fluid can also impact pressure.

4. How does fluid pressure change with depth?

In a static fluid, the pressure increases with depth due to the weight of the fluid above. This is known as hydrostatic pressure. The increase in pressure with depth is directly proportional to the density of the fluid and the acceleration due to gravity.

5. Why is fluid pressure important?

Fluid pressure plays a crucial role in many natural phenomena and technological applications. It is essential in understanding fluid dynamics, such as in the flow of fluids through pipes or channels. It also helps explain the behavior of gases and liquids in various situations, such as in weather patterns and the functioning of hydraulic systems.

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