Bernoulli principle and fluid particle

[tonyjk]

Hello,
Bernoulli principle describes the flow of a fluid for steady, incompressible flow along a streamline. But it is said for a particle of a fluid along a streamline. My question is a particle of fluid refers to a molecule or a group of molecules?

Thank you

 

[Chestermiller]

It applies to a small parcel of fluid comprised of many, many, many molecules.

 

[tonyjk]

So each point of a streamline represents a small volume of molecules?

 

[Chestermiller]

So each point of a streamline represents a small volume of molecules?

Yes. They call it a "particle" because the fluid is being treated as a continuum. But, it's really a small volume of many (statistically averaged) molecules.

 

[tonyjk]

Hello again,
Just last question:

Here they are taking a fluid particle along a stream, can we say inside of the "big" particle, we have many particles because we have many points inside the "big" particle? and when we take a surface and we want to calculate the pressure force on it, I know that the particles of fluid are causing the pressure, but for exemple at a infinitesimal area (or a point like in a streamline) of the surface, is the small volume of fluid(containing many molecules) causing the pressure?

 

[Chestermiller]

Yes. The pressure is also associated with large numbers of molecules. Just as yourself, how many molecules are contained inside 1 cubic mm of a gas at atmospheric pressure and room temperature (let along a liquid). Why don't you calculate it and see what number you come up with?

Chet

 

[tonyjk]

This is an infinitesimal volume of fluid, does all this cube apply a pressure on a surface or only one of the x-direction surfaces is applying it?

 

[Chestermiller]

This is an infinitesimal volume of fluid, does all this cube apply a pressure on a surface or only one of the x-direction surfaces is applying it?

The surrounding fluid applies pressure and viscous stresses on all 6 faces of the cube, and, by Newton's 3rd law, the fluid inside the cube applies equal and opposite pressure and viscous stresses on the surrounding fluid at all 6 faces.

Chet

 

[tonyjk]

The surrounding fluid applies pressure and viscous stresses on all 6 faces of the cube, and, by Newton's 3rd law, the fluid inside the cube applies equal and opposite pressure and viscous stresses on the surrounding fluid at all 6 faces.

Chet

Great but the surrounding fluid is also considered as a cube thus the pressure is only applied by the surface of the cube? the molecules on the surface have a volume? That's why I am confused.

 

[Chestermiller]

Great but the surrounding fluid is also considered as a cube thus the pressure is only applied by the surface of the cube?

The pressure and stress exist throughout the cube. This analysis is just looking at the values at the surface of the cube.

the molecules on the surface have a volume? That's why I am confused.

You need to start thinking about solids, liquids and gases as continua, rather than as being comprised of individual molecules. The continuum approach averages the behavior over large numbers of molecules. You're certainly not going to be considering the forces and motion of each individual molecule, are you? Taking this approach allows you to get away from that.

Chet

 

[tonyjk]

The pressure and stress exist throughout the cube. This analysis is just looking at the values at the surface of the cube.
Chet

Yes I know that inside the cube we have pressure and stress. But what I am confused that the pressure on a surface (any surface) is due to the cube of molecules or it is due to the surface of the cube? or we cannot know?

 

[Chestermiller]

Yes I know that inside the cube we have pressure and stress. But what I am confused that the pressure on a surface (any surface) is due to the cube of molecules or it is due to the surface of the cube? or we cannot know?

In this development, we are looking at situations where the mean free path of the molecules is very small compared to the linear dimensions of the cube. And we are looking at the forces exerted by the fluid outside the cube on the surfaces of the cube (i.e., and thence transmitted to the fluid within the cube). Since the mean free path is very small, the forces on the outer surfaces of the cube by the surrounding fluid are primarily the result of the molecules outside the cube that are in close proximity to the surfaces.

Chet

 

[tonyjk]

In this development, we are looking at situations where the mean free path of the molecules is very small compared to the linear dimensions of the cube. And we are looking at the forces exerted by the fluid outside the cube on the surfaces of the cube (i.e., and thence transmitted to the fluid within the cube). Since the mean free path is very small, the forces on the outer surfaces of the cube by the surrounding fluid are primarily the result of the molecules outside the cube that are in close proximity to the surfaces.

Chet

so we can say it is like the surface of the outer fluid cube right?

 

[tonyjk]

I think I understood now. What confused me before is the first image that I attached. It is normal that the fluid "behind" or surrounding the surface will cause the pressure. It can be the surface of the cube with molecules behind the surface it depends of the mean free path

 

[Chestermiller]

so we can say it is like the surface of the outer fluid cube right?

I don,t know what "the outer fluid cube" means.

 

[tonyjk]

I don,t know what "the outer fluid cube" means.

sorry , I was confused.
If we have a fluid(Static or dynamic), and for example we pick a point in this fluid. This point is represented by a very small cube (or small volume) we can say that the average property of the parcel(pressure for exemple) is the property of this point?

Because usually we take a parcel and we study the forces acting on it so we can know the behaviour of a fluid. But sometimes we take a point of a fluid and we represent it by a parcel that has constant properties (velocity, pressure etc...) that's why it I was confused

 

[boneh3ad]

sorry , I was confused.
If we have a fluid(Static or dynamic), and for example we pick a point in this fluid. This point is represented by a very small cube (or small volume) we can say that the average property of the parcel(pressure for exemple) is the property of this point?

It really depends on the scale. For a typical fluid mechanics analysis, one assumes that any of the flow length scales involved are orders of magnitude larger than the mean free path of the fluid, and therefore it can be treated as a continuum (this is characterized by a dimensionless number called the Knudsen number). This let's us wholly ignore the action of individual particles or consider Brownian motion in any way. It greatly "simplifies" the analysis. In a continuous medium, the properties at a point are still the result of the molecules buzzing around, but you don't really have to worry about that to get "the answer".

Now, when you just pick a point in a continuous fluid, you don't have to use a small control volume to think about it. The pressure at that point is just the pressure at that point. When you do want to draw a small control volume, then the pressure at the center of the volume is not necessarily just the average throughout the volume depending on the size of the volume and the nature of the flow. On the other hand, as you shrink the size of the control volume, the average properties will approach those at the center, until you get infinitesimally small and this essentially becomes true. That is sort of the point of differential volume elements.

 

[tonyjk]

Now, when you just pick a point in a continuous fluid, you don't have to use a small control volume to think about it. The pressure at that point is just the pressure at that point. .

Ok but can we say that this point is an infinitesimal volume? because for example at this point you have a velocity or pressure... of what? It should of something physical. Maybe it is the average of an infinitesimal volume properties right?

 

[boneh3ad]

Ok but can we say that this point is an infinitesimal volume? because for example at this point you have a velocity or pressure... of what? It should of something physical. Maybe it is the average of an infinitesimal volume properties right?

I am not really sure what you by "of what" here. It would be the velocity or the pressure of the fluid.

 

[Chestermiller]

Once you accept the continuum model, the pressure and velocity of the fluid become point functions of spatial position.

 

[tonyjk]

I am not really sure what you by "of what" here. It would be the velocity or the pressure of the fluid.

Ok but a fluid consists of many molecules. we cannot say that a point represents a point in a fluid. physically this point represent what? a very small volume of molecule? or a point inside a parcel? or we can say it is like a volume dv-->0?

 

[boneh3ad]

Ok but a fluid consists of many molecules. we cannot say that a point represents a point in a fluid. physically this point represent what? a very small volume of molecule? or a point inside a parcel? or we can say it is like a volume dv-->0?

That's precisely what we've been talking about. If you can assume the fluid is a continuum, you can completely ignore the fact that it is made of many molecules in the analysis. Physically, a point in a continuum represents a single point in a continuous medium. If you really want to treat it as a collection of moving particles, then you need to abandon the typical fluid mechanical constructs and start studying statistical mechanics and the Boltzmann equation. Otherwise, you will have to come to grips with the fact that one of the fundamental tenets of fluid mechanics is that in most cases the fluid may be treated as a continuum and a point simply represents a single point in a continuous medium. The Bernoulli equation, the Navier-Stokes equations, the continuity equation, and all of that other fun, essential collection of mathematical constructs that allow us to study fluids depend on the continuum assumption being valid.

 

[tonyjk]

Okay I understood now thank you