Work - Energy Principle Application to Fluid Flow

[Arjan82]

This isn't about logic really, it is about understanding and correctly applying scientific principles. Why is it important if fluid flow is steady? That doesn't change that these fluid elements aren't the same. What principle in physics can back this claim?

Actually there is also a form of the Bernoulli equation which includes times derivatives.

 

[russ_watters]

Problem for me is as follows. Fluid has different velocity on two cross sections. This requires acceleration. Acceleration requires force. This force has something to do with pressure difference on two cross sections. I am sure we are on the same page here.

Yes.

...net force acting on that volume of fluid should cause its center of mass (COM) to accelerate, but that doesn't happen in fluids...

I don't know why you would think that's true. It's absolutely not. Fluids get forces applied and experience accelerations. Newton's laws requires it.

This isn't about logic really, it is about understanding and correctly applying scientific principles.

That's logic.

Why is it important if fluid flow is steady? That doesn't change that these fluid elements aren't the same. What principle in physics can back this claim?

Bernoulli's Principle.

Bernoulli found that given a certain set of starting assumptions, a certain mathematical constant remained constant at different times and locations in a fluid.

 

[Dario56]

Actually Newton's second law is not about acceleration but about momentum. The 'real' 2nd law of Newton states that the change in momentum is equal to a force.

Remember that change in momentum = d(mv)/dt = mdv/dt + vdm/dt = sum of all forces. But for solids usually dm/dt is zero and dv/dt is of course acceleration. This results in the famous F=ma. But for fluids the term dm/dt is not necessarily zero.

So, in a case of a fluid you take a control volume stationary in space (this is actually not necessary, but makes integration a lot easier) then you apply Newton's second law by stating that the momentum flow into the volume (over it's boundaries) is equal to the momentum out of that volume plus the force applied on that volume (for example if one of the boundaries of that volume is a solid wall, but note that the pressure integrated over a boundary over which fluid flows is also a force which needs to be taken into account). If you apply this correctly than you end up with the Navier-Stokes equations.

This is actually Cauchy equation. Navier - Stokes are than derived for isotropic and Newtonian fluids. This is all correct, but it doesn't answer my question if common derivation of Bernoulli's equation via work - energy principle is in fact wrong.

For stationary flow, dm/dt is zero. Mass of the fluid accumates nowhere.

 

[russ_watters]

This is all correct, but it doesn't answer my question if common derivation of Bernoulli's equation via work - energy principle is in fact wrong.

Not to be too pointed here, but you're claiming something is wrong with an established physics principle. Given a choice of where the problem might lie, I wouldn't pick Bernoulli or his principle. It's not a coincidence that the derivation produces Bernoulli's principle and that Bernoulli's principle works. It works because the equivalence he discovered is real.

 

[Arjan82]

This is actually Cauchy equation. Navier - Stokes are than derived for isotropic and Newtonian fluids. This is all correct, but it doesn't answer my question if common derivation of Bernoulli's equation via work - energy principle is in fact wrong.

For stationary flow, dm/dt is zero. Mass of the fluid accumates nowhere.

Are we getting picky here? But you are absolutely right, it's the Cauchy equation. And of course for stationary flow dm/dt is zero, all d/dt terms are zero, that's the definition of stationary flow...

Looking at accelerating centers of masses of a body of water is a bit of a crude way to look at a fluid flows. But I think it is still correct when you do not allow any fluid to cross the control volume (at least in an average sense, things get very messy very quickly when turbulence is involved, also, when the control volume deforms in such a way that it becomes self-intersecting or disjoint, things get very messy indeed...). Why would that view be wrong*? when there is no flow crossing the boundaries the COM is perfectly well defined at all instances and you can talk about its acceleration. You can also perfectly well talk about forces acting on your control volume. So I don't see why that view is wrong.

*)[edit] at least for engineering purposes, I mean, the analysis in the video posted earlier is really valid for plug flow, which doesn't really exist in reality. But then again, for Bernoulli it is necessary to assume inviscid flow, so you cannot have any boundary layers anyway.

 

[Dale]

It is true that forces aren't summed in these derivations.

Then that objection doesn’t apply.

However, to apply work - energy principle,

This derivation doesn’t apply the work - energy principle.

This is very frustrating. I asked you for an example of a derivation that you object to. Any derivation that exemplified the problems. You posted a link to the derivation in Wikipedia. But so far that derivation does not do anything that you are complaining about.

At this point this thread is useless. I am going to close it. You are free to post a new thread on this topic. When you do so, you need to link to a specific derivation. You should provide exact quotes from the derivation highlighting the problematic statements. That would be a solid basis for a productive thread.