Consider a fluid falling down a tube of uniform cross-section with an initial velocity at the top of the tube.
Pressure at the top and bottom of the tube is maintained constant and is equal to the pressure outside.
Find it's(i.e that of the particles at the front) velocity after having fallen a height .
Now, the paradox arises when we consider an ideal fluid falling solely under the influence of gravity and no factors like viscosity or surface tension are considered. The fluid is incompressible.
Use Bernoulli's theorem to find the new velocity. Clearly, the fluid will move faster as it reaches the bottom.
Use the equation of continuity. As the cross-sections at the top and the bottom are the same, the velocities must also be the same as the fluid is incompressible.
So which argument, if any, is valid here?
Also, what happens when the fluid is not ideal, but say, just normal water:
In this case, what I believe will happen is that the stream of water will become narrower as it reaches the bottom, just like what happens to water coming out of a tap.
I would like to discuss the following:
1)Is this an example where the scope of applicability of either Bernoulli's theorem or the equation of continuity is exceeded? If so, why and what is their actual scope?
2)While an ideal fluid can't exist, is it, as a purely mathematical construct, consistent with the equations used to define it?
3)What happens in real life? i.e what factors come into play? What happens if we assume a fluid that has surface tension but no viscosity(though I don't think that is possible as the same kind of forces between particles underlie both phenomena)? Can surface tension by itself resolve the paradox? If so, is some crude mathematical analysis possible?
4)I do understand that there will be some confusion as to treat this as a physics problem or a mathematics problem. For one thing, I have my doubts as to whether constant pressure can be maintained at the top and bottom (as mentioned). I would definitely like to hear every physical argument, but what I would really like to know is why the math/equations are not working simultaneously, the assumption again being, that a self-consistent model for an ideal fluid exists.
I remember discussing this problem with Raghav a while back and we reached the aforementioned consensus for the case of a real fluid.
I was reluctant to post this, but of late, discussions on physics have seen a lot of activity and thus, I'm hoping that this discussion will also gain some traction.
P.S: I think that this is closely related to D'Alembert's paradox
This might also be applicable.
Thanks in advance :)