Bernoulli's principal and law of conservation of energy

[SystemTheory]

The simple question here is that suppose the fluid inside the first tube has inertia twards the right (the tube is not infinitely long, i.e it has a limited amount of fluid in it) will all the fluid easily pass out of the thinner cross section following Bernoulli's equation?

Yes, assuming the fluid is ideal and there is zero gravity. Of course a real fluid is not ideal.

I'm questioning the theory (of Bernoulli's equation)

How can static pressure of an ideal fluid store energy? The microscopic particles in an ideal fluid have kinetic energy. No energy or momentum is lost when the particles collide within an ideal fluid. The kinetic theory of ideal gases is a good place to study this conceptual model.

When a force acts at one end of an ideal fluid, it sees the total mass of the fluid, so the acceleration in many cases is relatively small. The piston accelerates quite slowly. The pressure is transmitted instantaneously in an ideal fluid and therefore Bernoulli's law is a reasonable approximation anyway. This is like the voltage spike on an inductor with zero starting current flow, a long pipe full of fluid is like an inductor in a circuit analogy when applying the pressure-voltage analogue.

 

[russ_watters]

I absolutely do not understand what is the significance of the assumption you are talking about regarding the main question.

In the first place, how do you come to the topic of assumption when when I'm questioning the theory? An assumption absolutely does not have to do anything here...we're taking an imaginary case and I'm not comparing it to any real life scenario.

When you said this in the OP:

...considering an ideal fluid in this situation...

...you were invoking the incompressible flow simplifying assumption (as well as inviscid, steady, lossless, and no heat transfer assumptions). How could you not know that? Didn't you know that there is a version of Bernoulli's equation that does not assume incompressible flow? Did you not read the wiki on the subject?

This is why I said in post #29 that the entire key to this thread is that you don't understand the purpose and scope of simplifying assumptions...

1) Bernoulli's principal* assumes an ideal fluid and the fact that pressure can store energy...how can an ideal fluid store energy in terms of static pressure?

This is the only problem with Bernoulli's equation I'm having...

I answered this directly, already: the incompressible flow form of Bernoulli's equation only assumes no change in density. You are equating the incompressible flow assumption with an assumption that static pressure can't store energy. It does not assume that static pressure can't store energy. You can see that in the equation!

*And agin, Bernoulli's principle DOES NOT assume an ideal fluid. A particular form of Bernoulli's equation might, though.

I'm questioning the theory (of Bernoulli's equation)

You need to stop doing that. You need to stop with making these diagrams and inventing situations to try to prove physical theories wrong. You don't understand Bernoulli's principle anywhere near well enough to challenge it and you're meandering around with lines of thought that don't have anything to do with Bernoulli's principle. This is why you are having so much trouble here (and why people are getting frustrated with you): You are not really trying to learn.

Since you clearly didn't read it, here's the opening two paragraphs from the wiki on the subject. Read them!

In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.[1][2] Bernoulli's principle is named after the Dutch-Swiss mathematician Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738.[3]

Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is valid for incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases) moving at low Mach numbers. More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).

The simple question here is that suppose the fluid inside the first tube has inertia twards the right (the tube is not infinitely long,

The phrase "inertia towards the right" is gibberish. I don't think you understand what the word "inertia" means...

...perhaps the word you are looking for is momentum?

i.e it has a limited amount of fluid in it)

Bernoulli's principle doesn't work in empty pipes, only full pipes.

...will all the fluid easily pass out of the thinner cross section following Bernoulli's equation?

What does "easily" mean?

That line of questioning is incoherent.

 

[russ_watters]

When we are manipulating circuits we assume the wires to be a superconductor and then analyze the theory.

That's a great example of exactly what I'm talking about: it is a simplifying assumption.

The assumption does not have a significance when we're talking everything in paper...

Oh, it absolutely does! If the resistance of the wire is high enough, it will affect the accuracy/usefulness of the calculations. For example, if the wires are 10 miles long (ie, power lines), the resistance can't be ignored. If the circuit is a flashlight, they can.

Understanding what simplifying assumptions are available, which ones to use and when you can use them is a critical skill for "doing" science.

 

[dE_logics]

Yes, assuming the fluid is ideal and there is zero gravity. Of course a real fluid is not ideal.
How can static pressure of an ideal fluid store energy? The microscopic particles in an ideal fluid have kinetic energy. No energy or momentum is lost when the particles collide within an ideal fluid. The kinetic theory of ideal gases is a good place to study this conceptual model.

When a force acts at one end of an ideal fluid, it sees the total mass of the fluid, so the acceleration in many cases is relatively small. The piston accelerates quite slowly. The pressure is transmitted instantaneously in an ideal fluid and therefore Bernoulli's law is a reasonable approximation anyway. This is like the voltage spike on an inductor with zero starting current flow, a long pipe full of fluid is like an inductor in a circuit analogy when applying the pressure-voltage analogue.

Ok...I'm sort of satisfied with the fact that it has energy in terms of K.E. of each molecule...but you know the real doubt still persists.

There are 2 things that manifest high K.E of molecules -

1) High temperature

2) High pressure.

Now, depending on the type of fluid it might be that at high pressure the amount of energy stored in the fluid is high or low...but for an ideal fluid it's 0, although the K.E of the molecules is very high.

So the K.E is not a direct measure of the energy stored in the fluid...I think. Cause it can also happen that under certain amount of pressure the molecules do not gain certain amount of K.E as compared to other fluids.

It does not assume that static pressure can't store energy.

Okay...I see.

So we assume an ideal fluid just to put ρ instead of m...but in reality since fluids can store energy in terms of pressure and that is quiet significant, we do not assume P as 0...or when it comes to P, we do not assume the fluid as ideal.

I conclude that at certain places the unmodified Bernoulli's equation assumes an ideal fluid, and at some places a real one.I think that solves the issue...thanks a lot, it could not have been cleared without bringing in this perspective of the assumption.However there's still a small doubt left -

So, suppose the fluid gains an inertia, and then the force stops acting; then the fluid will flow out of such a pipe -

without any more force application?

That means, if there is a continuous force application as in this case -

There will be a constant acceleration?

 

[russ_watters]

However there's still a small doubt left -

You are misusing the word "inertia", so the question is meaningless (perhaps you mean "momentum"?). Please rephrase.

Also, Bernoulli's principle and equation don't deal with partially empty pipes, nor does it deal with unsteady (accelerating) flow. So your question doesn't seem to me to have anything to do with Bernoulli's principle.

 

[rcgldr]

So, suppose the fluid gains an inertia, and then the force stops acting; then the fluid will flow out of such a pipe

Except that once there is no inwards force from the pistons, the pressure drops to zero. Each piston need to exert some exact amount of force to maintain a constant mass flow rate within the pipe. As you pointed out earlier, this occurs when the there is no net work done on the fluid.

 

[dE_logics]

You are misusing the word "inertia", so the question is meaningless (perhaps you mean "momentum"?). Please rephrase.

Also, Bernoulli's principle and equation don't deal with partially empty pipes, nor does it deal with unsteady (accelerating) flow. So your question doesn't seem to me to have anything to do with Bernoulli's principle.

I meant momentum.

What actually do you mean by partially empty?...vertically (i.e the whole cross section of the pipe is not filled) or horizontally (the whole pipe is not filled along it's length, but filled along it's cross section)?

Except that once there is no inwards force from the pistons, the pressure drops to zero. Each piston need to exert some exact amount of force to maintain a constant mass flow rate within the pipe. As you pointed out earlier, this occurs when the there is no net work done on the fluid.

So, the answer is no, there has to be a constant force application to make all the fluid flow out...but it's not a function of how much force...even the slightest force will be good enough to make all the fluid flow out.

 

[Phrak]

1. If I take a piston and change the pressure of an ideal, incompressible, nonviscous fluid then no work is done on the fluid.

2. Changing the pressure on this fluid does no work, whether it is in motion or not.

Any logical or physical errors so far?

 

[dE_logics]

1. If I take a piston and change the pressure of an ideal, incompressible, nonviscous fluid then no work is done on the fluid.

2. Changing the pressure on this fluid does no work, whether it is in motion or not.

Any logical or physical errors so far?

That problem has been solved...this is an extended issue.

Maybe I'll start a new thread on this.

 

[russ_watters]

What actually do you mean by partially empty?...vertically (i.e the whole cross section of the pipe is not filled) or horizontally (the whole pipe is not filled along it's length, but filled along it's cross section)?

If the fluid is air, the air will spread out to fill the pipe due to its own pressure. This will happen whether you are applying a force to it or not. The far end can't just stay a vacuum.

If the fluid is water and there is air at the end of the pipe, the water will spill out and the air will move backwards to fill the pipe.

Also, this "force" has to come from somewhere. If you have a piston, for example, and you stop moving the piston, the air or water will have to stop too, otherwise you'll generate a vacuum where it meets the piston.

So, the answer is no, there has to be a constant force application to make all the fluid flow out...but it's not a function of how much force...even the slightest force will be good enough to make all the fluid flow out.

Ehh, water in an unfilled pipe with air on each end will gain momentum that will carry it for a while if you ram it with a piston (then somehow remove the piston and let air in that end). But again, these things have bear no relationship to the scenarios described by Bernoulli's principle and your description is not detailed enough for us to know what your scenarios are really trying to describe. You can't just draw an arrow and say there is a force. You need to be more specific about what is causing the force and how it can just go away.

I really have no idea what you are trying to describe with that first diagram. For the second diagram, the system will accelerate and quickly reach an equilibrium with steady flow, based on the applied pressure equalling the total pressure required to move the fluid at a certain velocity through the small pipe.

 

[dE_logics]

Ok, just for simplicity morph the original question by a bit, with only 1 image -

And assume that we have another piston towards the narrower cross section just to not allow the fluid to 'spill', i.e ensure that the whole cross section is filled with the fluid; both the pistons are mass-less.

If, initially the piston towards the LHS applies a force so as to make the fluid gain a momentum, then, after a while, the force application stops, but the pistons keeps pace with the fluid to avoid it's spilling...ensuring that the fluid fills the cross section.

Assuming the length of the narrow cross section is long enough, will all the fluid in the wider cross section move into the narrow cross section without application of any additional force?...i.e solely by virtue of the momentum gained by the fluid?

 

[dE_logics]

I really have no idea what you are trying to describe with that first diagram. For the second diagram, the system will accelerate and quickly reach an equilibrium with steady flow, based on the applied pressure equalling the total pressure required to move the fluid at a certain velocity through the small pipe.

Ok...by this I conclude, it might happen that all the fluid (referring to the above description) will not move to narrower cross section.

 

[SystemTheory]

For a practical application of fluid models go to this link:

http://dynast.net/course/index.html

Course parts for printing:

Modeling and Simulation (This is a link to a 140 page modelsim.pdf. Chapter 5 covers fluid models. The paper itself is an introduction to multidomain system models).

 

[russ_watters]

And assume that we have another piston towards the narrower cross section just to not allow the fluid to 'spill', i.e ensure that the whole cross section is filled with the fluid; both the pistons are mass-less.

First, could you tell me whether we are talking about air or water?

Next, massless or not, the second piston has to be applying a force (there is a pressure due to the presence of the fluid), so you need to be more specific about what you want it to do. How much force, exactly does the second piston apply? A force equal to "F"? Something lower than "F"? It makes a difference.

If, initially the piston towards the LHS applies a force so as to make the fluid gain a momentum, then, after a while, the force application stops, but the pistons keeps pace with the fluid to avoid it's spilling...ensuring that the fluid fills the cross section.

The piston cannot "keep pace" without applying a force. The fluid is under pressure!

Assuming the length of the narrow cross section is long enough, will all the fluid in the wider cross section move into the narrow cross section without application of any additional force?...i.e solely by virtue of the momentum gained by the fluid?

Though your scenario is still being described in an incomplete and contradictory way, I can see is no reason to believe the fluid would completely move through the narrower tube. I can't think of any variant of your problem description where it would.

Do you have a specific real-world scenario that you are trying to understand or is this completely hypothetical?

 

[dE_logics]

For a practical application of fluid models go to this link:

http://dynast.net/course/index.html

Course parts for printing:

Modeling and Simulation (This is a link to a 140 page modelsim.pdf. Chapter 5 covers fluid models. The paper itself is an introduction to multidomain system models).

It's a guide to a software which's proprietary and works only on windaz.

 

[dE_logics]

First, could you tell me whether we are talking about air or water?

Water.

Next, massless or not, the second piston has to be applying a force

The force is applied on the second piston...and the first piston applies the force (initially). The force on the second piston is a consequence.

I think it's impossible to describe this.

So just leave it for now...really appreciated the help.