Marmoteer said:
I understand that as an incompressible fluid flows through a pipe with decreasing cross sectional area the velocity increases. This must happen for the mass/volume continuity equations to be true. Since the velocity is increasing though there has to be a net pressure on the fluid right? My question is - what's the best way to think about this pressure? Is it like pressure "buildup" because the flow is restricted? Or is there a better way to interpret it?
Well... The fluid being incompressible has a problem right there.. In deep oceans, the pressure in of water is WAY bigger. Pressure is indeed a energy density (joules per cubic meter). So the problem arises: how can it be that the energy density is higher if the molecules are not closer together? Note that water is not hotter deeper, so it cannot be that the particles have gained kinetic energy. So it must be the potential energy... The potential energy density. That means that the molecules are closer together the deeper we go...
Indeed, molecules have what is called intermolecular bond stiffness: this can be modeled as springs between the molecule..
A little iterative thinking may give a clue:
A molecule is being pulled by the earth: mg
But it does not accelerate, so this mg is balanced.. By what? Well by the molecule right under it: so there is an mg upward.
So there is a repulsive force between the first molecule and the one just below it, that force has a magnitude of mg.
According to Newton: a force is a mutual interaction between two bodies: so if the lower molecule pushes the upper one with mg, the second one is pushed down by the first molecule, by an amount mg - note that this interaction is ELECTRIC: a repulsive interaction.
So the second molecule must experience a force 2mg, because the first molecule is pushing down on it with mg, but the Earth is also pulling it the second molecule down with the amount of mg.
But that 2mg down MUST be balanced by a 2mg upward, because there is no acceleration.
That 2mg upwards is caused by the 3rd molecule.
How can the 3rd molecule be pushing the second one upwards by the amount of 2mg?
Well, it has got to be closer to it then first molecule is to it...
Similarly, the 3rd molecule is experiencing the force 2mg down, due to the mutual repulsion with the second molecule.
But gravity is also acting on it, so there is 3 mg downward for the 3rd molecule...
That means that the 4th is pushing 3mg upward, it can do that by being a TINY TINY bit closer to the 3rd molecule than the 2nd molecule is... etc...
So it becomes pretty clear that fluids cannot be incompressible in reality.
This compression seems to be horrifically small, practically not too measurable, but it has to be there, otherwise it makes no sense that the energy density is higher and it is also nonsense to explain, why the deeper molecules can act on the upper ones with such a huge force.
This also explains why the pressure is the same at in a certain depth level, even when the column of water is not straight: the molecules are a tiny tiny amount closer each other.
Oh, and as far as the pipe is concerned:
you can think of it this way: the potential energy that the molecules HAD is now changing into the kinetic energy of the molecules: they are step-by-step being accelerated away from each other, by tiny amounts: so the pressure drops: the potential energy density drops. What you see is that water is just flowing faster.
At those points, the thinner the tube gets, there is a MINUTE density gradient in water, hardly measurable, but this is the way it works. This is the fundamental difference between water flow and electric current: water molecules do "push each other", and they potential energy relative to EACH OTHER is turning to kinetic energy of the individual molecules.
In the electric current, the electric filed pushes the free electrons in a wire. The field is made by the surface charge gradient: charges on the SURFACE of the wire, whereas the water molecules do really accelerate each other. It is like having many balls attached to each other with easily breakable spring: push it so the chain get shorter and the spring is compressed. Now let it go: the balls accelerate one after another while the spring is going to a relaxed position and then breaks because the balls are getting too far.
This analogy has many problems of course... But all in all: the essence is similar.