# What causes pressure and is it isotropic in a moving fluid?

• Fountains
In summary, the concept of pressure in a fluid is defined as the force per unit area at a fluid/solid interface. It is caused by the exchange of momentum between the fluid molecules and the boundary. However, there is also organized motion of the molecules within the fluid that contributes to the overall momentum flux. This motion is not isotropic and affects the differential momentum balances. The term "static pressure" is often used interchangeably with pressure, and it describes the isotropic part of the stress tensor. The equation rho*g*h is often used to calculate pressure in a liquid at a certain depth, but it does not take into account the tangential velocity at the surface, which also contributes to pressure. This is because it acts as a constant
Fountains
I'm studying fluid dynamics and we just had a lecture about the momentum equation. We started the lecture by talking about pressure in terms of molecules moving across a hypothetical surface element and carrying their momentum with them (in both directions). There are 2 things confusing me about this:

1) we said the pressure is isotropic, but given I'm studying fluid dynamics, with this description it seems pretty clear that the pressure will point in the direction of the fluid velocity, and in a direction normal to that there will be (on average) no molecules moving in that direction so what would cause the pressure?

2) When deriving the momentum equation, we then also included the effect of momentum entering and leaving our arbitrary volume, but this seems to be exactly what we described pressure to be at the start of the lecture. So I think I have misunderstood the root cause of pressure: eg. is it particles colliding or just moving between regions and transferring momentum?

I have been thinking about this in frustration for hours now so any help would be appreciated!

There are two parts to the motion of the molecules in the fluid. For the most part, the different molecules are moving randomly in all different directions, and this gives rise to the isotropic feature. But, in addition to this, there is organized motion of the molecules that averages out to what we observe macroscopically as the "velocity of the fluid." This motion is not isotropic, and gives rise to the additional momentum flux in the differential momentum balances that is not isotropic.

Hi,

About pressure in a fluid. Pressure is a property at a point in a fluid. Some define pressure using a fluid/solid boundary or interface.
Suppose you have water in a cubical container on a table. On one side (say the right side) fluid water molecules impact the boundary
and rebound. Momentum exchange happens. That exchange per unit of area (force per area) is pressure. Defined at a fliud/solid interface.

Now, consider the horizontally left side of the water/container. This is a boundary also. Water momentum and area are pressure again.
So we have pressure defined at the left interface and right interface. Next we conclude pressures exists as a property of the fluid for
all points between the walls and at all points in fluids.

Also be careful about the word "static." Static can mean "not moving." It is better to think os static as "undisturbed."
Hence motion at constant velocity is "static."

http://www.thermospokenhere.com/wp/02_tsh/B0600___wilma_static/wilma_static.html

JP

SirCurmudgeon said:
Hi,

About pressure in a fluid. Pressure is a property at a point in a fluid. Some define pressure using a fluid/solid boundary or interface.
Suppose you have water in a cubical container on a table. On one side (say the right side) fluid water molecules impact the boundary
and rebound. Momentum exchange happens. That exchange per unit of area (force per area) is pressure. Defined at a fliud/solid interface.

Now, consider the horizontally left side of the water/container. This is a boundary also. Water momentum and area are pressure again.
So we have pressure defined at the left interface and right interface. Next we conclude pressures exists as a property of the fluid for
all points between the walls and at all points in fluids.

Also be careful about the word "static." Static can mean "not moving." It is better to think os static as "undisturbed."
Hence motion at constant velocity is "static."

http://www.thermospokenhere.com/wp/02_tsh/B0600___wilma_static/wilma_static.html

JP
Even if the fluid is not moving at constant velocity, the term “static pressure” is often used in place of of the word pressure. Both terms are used interchangeably, and describe the isotopic part of the stress tensor.

Thanks! I think I understand the concept now in terms of kinetic theory, like this https://en.wikipedia.org/wiki/Kinetic_theory_of_gases#Pressure_and_kinetic_energy , but I'm having trouble squaring this with the equation rho*g*h often given for the pressure in a liquid at depth h: the second equation implies that the pressure at the surface would be zero, but for a non-viscous fluid their may still be tangential velocity at the surface causing pressure right (by kinetic theory) so there should still be pressure just due to the molecules having energy? Is the issue just that this acts like a constant which is added everywhere so can be ignored given we often talk only about pressure gradients?

Fountains said:
Thanks! I think I understand the concept now in terms of kinetic theory, like this https://en.wikipedia.org/wiki/Kinetic_theory_of_gases#Pressure_and_kinetic_energy , but I'm having trouble squaring this with the equation rho*g*h often given for the pressure in a liquid at depth h: the second equation implies that the pressure at the surface would be zero, but for a non-viscous fluid their may still be tangential velocity at the surface causing pressure right (by kinetic theory) so there should still be pressure just due to the molecules having energy? Is the issue just that this acts like a constant which is added everywhere so can be ignored given we often talk only about pressure gradients?
Sorry. I don't get the gist of this question. Are you talking about a free surface where the liquid is in contact with the air?

Chestermiller said:
Sorry. I don't get the gist of this question. Are you talking about a free surface where the liquid is in contact with the air?
Yes sorry-perhaps this is not so related to fluid dynamics-it isn't in my lecture notes but it just seemed to keep coming up whenever I looked up pressure!

Fountains said:
Yes sorry-perhaps this is not so related to fluid dynamics-it isn't in my lecture notes but it just seemed to keep coming up whenever I looked up pressure!
At a free surface with air, the pressure at the boundary is equal to the atmospheric pressure ##p_a##. So the equation should really be ##p=p_a+\rho gh## rather than ##p=\rho gh##. When we write the equation the latter way, what we are referring to is the so-called "gauge" pressure, which is the absolute pressure minus the atmospheric pressure.

Fountains said:
I'm studying fluid dynamics and we just had a lecture about the momentum equation. We started the lecture by talking about pressure in terms of molecules moving across a hypothetical surface element and carrying their momentum with them (in both directions). There are 2 things confusing me about this:

1) we said the pressure is isotropic, but given I'm studying fluid dynamics, with this description it seems pretty clear that the pressure will point in the direction of the fluid velocity, and in a direction normal to that there will be (on average) no molecules moving in that direction so what would cause the pressure?

2) When deriving the momentum equation, we then also included the effect of momentum entering and leaving our arbitrary volume, but this seems to be exactly what we described pressure to be at the start of the lecture. So I think I have misunderstood the root cause of pressure: eg. is it particles colliding or just moving between regions and transferring momentum?

I have been thinking about this in frustration for hours now so any help would be appreciated!

Velocity is frame dependent. The same fluid could be seen as moving at any speed, depending on your frame of reference. If you force a surface though a fluid, then you feel a resisting force that is more than the static fluid pressure at each point.

The pressure of a fluid should be defined in the rest frame of the fluid.

PeroK said:
Velocity is frame dependent. The same fluid could be seen as moving at any speed, depending on your frame of reference. If you force a surface though a fluid, then you feel a resisting force that is more than the static fluid pressure at each point.

The pressure of a fluid should be defined in the rest frame of the fluid.
As a guy with lots of fluid mechanics experience, I don't follow this. Are you saying that fluid pressure exists only at a solid surface?

Chestermiller said:
As a guy with lots of fluid mechanics experience, I don't follow this. Are you saying that fluid pressure exists only at a solid surface?

No, just that in the OP's view of the pressure increasing with velocity, you could increase the fluid pressure simply by looking at the fluid in a different reference frame - one where it is moving very fast, for example.

Or, alternatively, by forcing something through the fluid and calling the resisting force "pressure", which I think is what the OP is thinking.

PeroK said:
No, just that in the OP's view of the pressure increasing with velocity, you could increase the fluid pressure simply by looking at the fluid in a different reference frame where it is moving.

Or, by forcing something through the fluid and calling the resisting force "pressure", which I think is what the OP is thinking.
Oh. That wasn't my interpretation.

Chestermiller said:
Oh. That wasn't my interpretation.

Fountains said:
1) we said the pressure is isotropic, but given I'm studying fluid dynamics, with this description it seems pretty clear that the pressure will point in the direction of the fluid velocity,

But, fluid velocity is frame dependent, was really my only point.

PeroK said:
But, fluid velocity is frame dependent, was really my only point.
I think he was referring to differences in momentum between fluid entering a control volume through one surface and fluid leaving the control volume through another surface.

Chestermiller said:
I think he was referring to differences in momentum between fluid entering a control volume through one surface and fluid leaving the control volume through another surface.

I was just unsure as to whether the velocities were being measured in the frame of the fluid velocity or a rest flame, since it was not stated in my notes, but your first comment made me realize that it was as measured traveling with the fluid, so I think I get that now thanks.
But for future reference, it's "she"!

PeroK

## 1. What is pressure and how is it defined in a fluid?

Pressure is a measure of the force exerted per unit area by the molecules of a fluid. It is defined as the force applied perpendicular to the surface of an object divided by the area over which the force is applied.

## 2. What are the main factors that cause pressure in a fluid?

The main factors that cause pressure in a fluid are gravity, density, and the motion of the fluid. Gravity causes the weight of the fluid to push down on the objects below it, while density determines how closely packed the molecules are and therefore how much force they exert on each other. The motion of the fluid can also contribute to pressure by creating areas of high and low pressure.

## 3. How does the pressure change in a moving fluid?

In a moving fluid, the pressure can change due to the fluid's velocity, direction of flow, and the shape of the object that it is flowing around. As the fluid moves faster, the pressure decreases, and the pressure increases when the fluid slows down. The direction of flow can also affect pressure, with areas of high pressure forming on the front of an object and low pressure on the back. Additionally, the shape of the object can cause variations in pressure.

## 4. Is pressure isotropic in a moving fluid?

No, pressure is not isotropic in a moving fluid. Isotropic means that the pressure is the same in all directions, and in a moving fluid, this is not the case. The pressure can vary in different directions, depending on the fluid's velocity, direction of flow, and the shape of the object it is flowing around.

## 5. How do scientists measure pressure in a moving fluid?

Scientists use instruments called pressure gauges to measure pressure in a moving fluid. These gauges can measure both the magnitude and direction of the pressure, allowing scientists to understand how pressure varies in different areas of the fluid. Some common types of pressure gauges used in fluid mechanics include manometers, Bourdon gauges, and Pitot tubes.

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