Does energy or force provide a better understanding of fluid dynamics?

[tillitea]

When water moves from a larger pipe into a smaller pipe it speeds up which I think everyone will agree. Is the pressure of the faster moving water in the smaller tube greater than the same area/volume of water in the larger pipe? Many refer to the force equation and Bernoulli's principle. Some say the pressure coming out of the smaller tube is less than the same area or volume of water in the larger tube. I'm not sure how this can be unless the remaining volume of water in the larger pipe surrounding the area or volume of water in question places pressure on the that volume of water or if there is some other reason.
I'd rather be hit by a truck moving at 5 mph than the same truck moving at 50 mph. Then again, I'd rather not be hit by a truck at all.

 

[mfig]

"Is the pressure of the faster moving water in the smaller tube greater than the same area/volume of water in the larger pipe?"If we look at a simple case where there is no change in elevation and where Bernoulli's equation holds, we can tell right away.

##p_1 + \frac{\rho v_1^2}{2} = p_2 + \frac{\rho v_2^2}{2}##

Assume ##v_2>v_1## to find the impact on the relative sizes of the pressures. If ##\rho## is constant, then we can solve for the pressure at location 2 as

##p_2 = p_1 + \frac{\rho}{2} \left(v_1^2 - v_2^2\right ) ##

Since ##v_2>v_1##, the term in parenthesis is less that zero. Thus ##p_1>p_2##.

 

[jbriggs444]

Assume no friction. Water entering the small tube from the large tube must speed up. What force can be responsible for this? Water leaving the small tube and re-entering a larger tube must slow down. What force could be responsible for this?

The pressure in a stream of water that has already come out of a hose does not have anything to do with how much it hurts when it hits you. The pressure in that "free stream" is zero. It is the velocity of the stream and the fact that it takes force to deflect it that makes it hurt.

 

[tillitea]

Thank you for the reply. To me it seems more natural for the water moving faster to have more pressure than the same volume of water moving slower. But I can see how that thinking is not always true especially with water in pipes. Can you tell me in words why the volume of water in the large pipe has more pressure. Is there more pressure on the volume of water or more forces acting on it? Sorry, I am not savy with the physics/equations.

 

[Tazerfish]

Thank you for the reply. To me it seems more natural for the water moving faster to have more pressure than the same volume of water moving slower. But I can see how that thinking is not always true especially with water in pipes. Can you tell me in words why the volume of water in the large pipe has more pressure. Is there more pressure on the volume of water or more forces acting on it? Sorry, I am not savy with the physics/equations.

Ok if you want an answer without equations i will give it a shot.
Intuitively you might say: the water that is quick will have more pressure.
I can experience a high velocity fluid jet (like from an air compressor) hitting my skin.
In this case the high pressure is not intrinisically there.
It arises when the jet STOPS on your skin.(and gets deflected afterwords)
Imagine a pipe with a small diameter followed by a large diameter pipe and again followed by a small diameter pipe.
When the quick water enters it will slow down significantly. The slowing can only happen if something pushes it back.
That pushing happens due to the pressure gradient.
Just imagine something sitting in a fluid with a pressure gradient.
One side of the object will experience a larger force that the other side.That will make it accelerate.
When entering a low pressure zone you speed up while entering a high pressure zone makes you slow down.

Another way to think about this:
You have to REALLY understand and internalize that pressure distributes itself in any flow significantly slower than the speed of sound.(so pretty much everytime)
If a constriction is placed into a flow that does not only change the flow there but everywhere in the flow.
The water in front of the pipe "knows" that there is a constriction so the pressure is higher there.
Once it enters the smaller pipe it gets "shoved" out of the high pressure zone and merrily continues on it's way.
To me it is not weird that the pressure inside a gas tank is higher than the pressure in a tube leading out of it.(if there is flow)
What is still extremely counterintuitive to me, is that when you apply higher pressure to one end of the tube down below
YOU GET LOWER PRESSURE IN THE MIDDLE THAN AT BOTH ENDS.
While it mathematically makes sense the thought that by blowing into one end of a straw shaped like this you could actually create LOWER pressure inside it is weird as hell.

 

[mfig]

Can you tell me in words why the volume of water in the large pipe has more pressure.

If you have a good sense for the general conservation of energy, you can get a good sense of this situation. Look at the units of each term in the Bernoulli equation for an ideal fluid:

## p_1 + \frac{\rho v_1^2}{2} + \rho g h_1= p_2 + \frac{\rho v_2^2}{2} + \rho g h_2##

The units are energy/volume (## J/m^3 ##). So it is useful think of pressure as an internal energy storage term for the fluid. Thus what this equation is telling you is that the energy per unit volume of a fluid can be found divided between gravitational potential energy, kinetic energy and pressure - but the total energy does not change from location 1 to location 2. Given a constant elevation, if we transform some of the fluid's energy into velocity such that ## v_2 > v_1##, then by conservation of energy there will have to be a decrease in the energy storage term (pressure) so that the sum of all the terms remains the same at location 2 as it was at location 1.

In a real fluid there are losses associated with the energy transformations between terms and other kinds of losses, but this gives you a way to think about what is happening.

 

[Tazerfish]

If you have a good sense for the conservation of energy, you can get a good sense of this situation. Look at the units of the Bernoulli equation:

## p_1 + \frac{\rho v_1^2}{2} + \rho g h_1= p_2 + \frac{\rho v_2^2}{2} + \rho g h_2##

The units are energy/volume. So it is useful think of pressure as an energy of storage term. Thus you see that what this equation is telling you is that the energy of a fluid can be found divided between gravitational potential energy, kinetic energy and pressure - but the total energy does not change from location 1 to location 2. So if we transform some of the fluid's energy into velocity, by conservation of energy there will have to be a decrease in energy of storage term (pressure) if we are at constant elevation.

Just on a side note i prefer to think about this in terms of forces.
I also prefer the derivations of Bernoullis equation using forces.
Energy considerations can lead to misconceptions and just show a sort of correlation.

Ps: it might be good to know that pressure can be seen as a form of energy density because
the work you do on an object of Volume V when pushing ti through the pressure differential ## \Delta p ## is
## W=F *s ##
##F= \Delta p* A## and ## s=h##
that gives ## W=\Delta p *A *h ##
## W=\Delta p *V##
(Just so you see this isn't pulled out of thin air)

 

[mfig]

I also prefer the derivations of Bernoullis equation using forces.
Energy considerations can lead to misconceptions and just show a sort of correlation.

I prefer to think in terms of energy in fluids generally, not force. But to each his own. I have no idea what you mean by "just show a sort of correlation." Would you mind elaborating on that? Since Bernoulli's equation can be derived from an application of conservation of energy, it seems like more than a mere correlation to me.

 

[Delta2]

You can think of pressure as some sort of "internal potential energy" the fluid has. When the velocity of the fluid increases , and thus its kinetic energy increases, this increase in kinetic energy can come either from an external source that adds energy like a turbine or from decreasing the "external" potential energy due to Earth's gravitational field, or from decreasing the "internal potential energy" aka pressure.

 

[Tazerfish]

I prefer to think in terms of energy in fluids generally, not force. But to each his own. I have no idea what you mean by "just show a sort of correlation." Would you mind elaborating on that? Since Bernoulli's equation can be derived from an application of conservation of energy, it seems like more than a mere correlation to me.

I feel like the considerations of force give a more causal picture.
When considering a Newtons cradle (i don't know if they are always called that/if it doesn't ring a bell just google it)
you can argue using conservation of energy and momentum, that what actually happens is the only viable option.
Dropping two balls must "repell" the two opposite balls.It is the only solution that does not violate conservation of momentum
and has (reasonably)elastic collisions /kinetic energy conservation.
This might be stupid and just personal but i feel like energy considerations often just tell you what and not really how.
If you went the extra mile and maybe considered how the balls compress and then shove the other balls away you tend to better grasp the bigger picture.
In this case and some others this is mathematically probably quite hard and sometimes you might need slow motion footage or simulations to really observe what's happening, but on a conceptual level i feel like you get more out when not soley considering energy.
Though i appretiate how energy considerations can make your life easier and how quickly they can solve problems.( for example. potential energies of stuff in not linear electric or graviational fields.)

However i haven't specifically answered your question:
In case of the bernoulli equation there are a lot of misconceptions.People often don't realize when you can and can not apply it.
Just look at the regular explanation of why wings generate lift.(The path length difference one)
It seems like a correlation because with lower pressure comes with higher velocity and vice versa(ignoring external forces) there isn't really any causality in the equation(at least no obvious one).When thinking about forces i tend to look more behind the equation and it is more natural to me.

And energy is just fine when considering straight pipes. No worries.
But another simple problem: Why does a sheet of paper go up when you blow over it ?
Here many explain it falsely. They think the high fluid velocity somehow causes lower pressure which would suck the paper up.
That is because they only remember the equation, and don't remember that it was only valid if the total energy density was the same everywhere throughout the flow.
It obviously isn't.When considering the forces you first examine the flow.
You will find it curves along the surface.(i will just take that as an axiom) Then you can deduce that in order for the fluid stream to curve there must be a pressure gradient.And because of this necessary gradient you have lower pressure on top of the surface.That the flow speeds up at that point is more of a side effect imo.
So you can fairly easily understand and approximate the pressure drop due to a curved surface(and subsequently curved flow).
That is actually something that turns up A LOT. Wings /Umberellas in wind /the paint dispenser in some airbrushs/flow around a spinning ball etc.

Sure you could measure the pressure on top of the sheet and you can measure the pressure on top of a wing.
From that you could also calculate the speeds given some initial conditions but would you really know much about the rest of the flow ?
That is why i personally don't like to rely much on energy considerations
Sorry for getting emotional about this crap and spamming the chat.
I hope it provided some interesting insight.