... decrease in pressure ? ...

[zetetic56]

I have found related information but I have not found anywhere on the Internet where this question is directly addressed.

A more dense body is falling in a less dense column of fluid.

Does the pressure from fluid on the walls of the container decrease?

( My guess is "yes".  As the body falls the fluid is displaced upwards and so the fluid moves. )

Thank you : )

[ Edit: I should have said " does the pressure from the fluid on the walls decrease when the body is falling as opposed to before the body falls and the fluid is static ? " ]

Edited June 6, 2023 by zetetic56

[studiot]

4 hours ago, zetetic56 said:

I have found related information but I have not found anywhere on the Internet where this question is directly addressed.

A more dense body is falling in a less dense column of fluid.

Does the pressure from fluid on the walls of the container decrease?

( My guess is "yes".  As the body falls the fluid is displaced upwards and so the fluid moves. )

Thank you : )

[ Edit: I should have said " does the pressure from the fluid on the walls decrease when the body is falling as opposed to before the body falls and the fluid is static ? " ]

 

Yes your reasoning is essentially correct.

But it will be a much smaller effect than I expect you are expecting.

Remember that the fluid column moves upwards about the diameter of the ball in the same time as the ball falls the whole depth of the colum.

Ie the fluid velocity relative to the walls is much slower than the ball's velocity relative to the fluid.

 

 

 

 

[zetetic56]

1 hour ago, studiot said:

But it will be a much smaller effect than I expect you are expecting.

Interesting.

Would it be so small as to be negligible?

If someone had the right equipment could the decrease in pressure be detected or measured?

Thank you.

 

[Genady]

If the tube's diameter is almost equal to the ball's diameter, the velocity of the fluid near the ball's equator might be quite high, I think.

[zetetic56]

If there is movement of the column of fluid above and below the falling body (the fluid at "a" and "c"), then I could see that being very slow.  I could see the possibility that the change and decrease in the pressure on the container from the fluid could be negligible.  (?)

However, it seems that the movement of the fluid around the body and through the narrow spaces between the body and the fluid container would be faster (the fluid at "b").  It seems like the change and decrease in pressure on the container from the fluid here would be less likely to be so small as to be negligible here (at "b") if the space is sufficiently narrow.  (?)

Thank you both for your replies and getting me to think about the different points along the column of fluid and the different changes that perhaps/probably occur.

[ Edit: I showed the fluid at "a" moving straight up and i should have showed it swirling around after getting displaced by the falling body.  And I might be totally wrong about showing the fluid moving at "c" at all.  The fluid below the falling body perhaps gets a bit compressed but it probably doesn't really move until the falling body reaches it and it gets displaced upward around the falling body (which is "b"), right?  Apologies. ]

Edited June 6, 2023 by zetetic56

[zetetic56]

I think my previous "a" "b" "c" stuff is probably more confusing than helpful (and just flat out wrong in places).

Now this is what I am thinking: Just as a matter of pure geometry, if the width of the falling object is the same as the width of the space between the object and the container, then the speed of the falling body will be the same as the speed of the rising displaced fluid around that body.  As the same volume moves down the same volume must move up.

(And if the widths of the falling body and the space between the body and the container are different from one another then the speed of the falling body and rising fluid will be different from one another, with the one with the greater width moving slower and the other one with narrower width moving faster.)

Right?

Edited June 6, 2023 by zetetic56

[Genady]

18 minutes ago, zetetic56 said:

I think my previous "a" "b" "c" stuff is probably more confusing than helpful (and just flat out wrong in places).

Now this is what I am thinking: Just as a matter of pure geometry, if the width of the falling object is the same as the width of the space between the object and the container, then the speed of the falling body will be the same as the speed of the rising displaced fluid around that body.  As the same volume moves down the same volume must move up.

(And if the widths of the falling body and the space between the body and the container are different from one another then the speed of the falling body and rising fluid will be different from one another, with the one with the greater width moving slower and the other one with narrower width moving faster.)

Right?

IMO, you should compare cross-section areas rather than widths.

[zetetic56]

3 minutes ago, Genady said:

IMO, you should compare cross-section areas rather than widths.

IMO you are right.

I was thinking in terms of my 2 dimensional drawing.  3D is correct.

Thank you : )

[zetetic56]

When a more dense body falls in a less dense fluid the fluid is displaced upwards and moves around the body and will also end up moving relative to the container walls.

And so, it seems, like the pressure from the fluid on the container walls will decrease and there will be less pressure on the walls from the fluid than when the fluid was static.

[ And thank you studiot for suggesting that my simplistic reasoning here is "essentially correct". ]

And the same (or so it seems to my reasoning) would occur when a less dense body rises in a more dense fluid.  As the fluid is displaced (or falls) downwards around the rising body it also moves relative to the container walls and so the pressure from the fluid on the container walls would decrease and become less than when the fluid was static.

And my guess is that this decrease when a body rises or falls would be detectable and noticeable, and if there was some sort of pressure gauge on the container walls this change in pressure would be noticeable and not just negligible.  But, right now, that is just my guess.

Question: Is there a decrease in pressure on the container walls from the fluid when a body rises or falls within the fluid (as opposed to the amount of pressure from the fluid when it is static), and, if so, would this decrease more than negligible?

Thank you guys for considering my question.  : )

 

 

[zetetic56]

Up to this point my two questions seem to have fairly obvious answers and are not very interesting questions.

This was leading to a (I think) more interesting question, but I wanted to establish that I had gotten the underlying physical dynamics correct first.

1. When a body rises in a column of fluid, is there a decrease in pressure on the container walls?

and

2. Is this decrease in pressure not just negligible?

My guess is that the answers to those two questions are "yes".  But, again, I could be wrong.

If they are "yes", then here is what I think is the more interesting question:

There are two containers of fluid with a wall between them (or, same thing said differently, there is one container of fluid with a wall in the middle).

The wall between them (orange) is moveable.  It can move back and forth horizontally.

There is a submerged buoyant body at rest in one of the containers.  The columns of fluid are the same height on both sides of the moveable wall.

The less dense body is allowed to rise in the more dense fluid.  It is, again, on one side of the moveable wall.

If

1. The pressure on one side of the wall decreases

and 

2. If this decrease in pressure is not negligible

then

3. Does/will the the wall move?

(The movement of the wall could be considered frictionless for the purposes of this thought experiment, or the amount of friction could be considered to be very small and less than the force from the difference in pressures on the two sides of the wall.)

This is my main question.  And this is what (I think) is the interesting question.

[ If the answer is "yes" and the wall moves, then the height of the one column of fluid with the submerged body will rise while the height of the other column of fluid will fall.  And this is what (I think) is interesting (if this is what will occur). ] 

---

I realize I am a guest in your home.

I do appreciate you letting me post my questions here.  And I do appreciate your time and the feedback you have given me.

If there is no interest in replying to another one of my questions, I understand.

Thank you all for your time and feedback you have given me (on this question as well as on my other questions I have posted previously).

Thank you and take care : ) !!!

[Genady]

Do you need a complication of the body moving up and down? Will your question change if there is no body, but instead you simply create a vertical stream of water on one side of the wall, e.g., by making a circulation through an external pipe? 

[zetetic56]

1 hour ago, Genady said:

Do you need a complication of the body moving up and down? Will your question change if there is no body, but instead you simply create a vertical stream of water on one side of the wall, e.g., by making a circulation through an external pipe? 

I think it would be the same question.

Before the pump is turned on the fluid is static on both sides, and there is the same level of water on both sides of the moveable wall.

The water is then circulated on one side of the wall, while the water remains static on the other side of the wall.

The question is: will the wall move?

And my guess is that the answer is "yes" (assuming no friction, or assuming a small enough amount of friction, with the moving wall).

And so, it seems like we would end up with my same question.

Would this scenario cause the water level to fall on the one side of the wall and rise on the other side of the wall?

It seems to be the same basic question.  Perhaps its easier to think of it differently than how I first conceived of it.

: )

 

 

[Genady]

Or, like this:

Yes, it seems that you are correct. The pressure on the right side of the wall is lower than on the left. The wall will move to the right and the water level on the right will rise.

[zetetic56]

1 hour ago, Genady said:

Yes, it seems that you are correct. The pressure on the right side of the wall is lower than on the left. The wall will move to the right and the water level on the right will rise.

 

Thank you for confirming my suspicions.

The logic seems pretty straight forward: there will be a difference in pressure on the two sides of the wall, and so the wall will move to the right, and so the column of fluid on the left will get lower and the column of fluid on the right will get higher.  It seems pretty simple and its my guess that this is what would happen.

Thank you.

---

As an aside:

I proposed this set up as purely a theoretical thought experiment.  If this were to be built the difference in height as the submerged body rises (or falls) would probably be so small so as to not really be noticeable by our human eyes.

But a pump could move the fluid fast.  And this set up could potentially produce a height difference that is noticeable by us human beings.

(I realized I should have the fluid moving down along the wall, instead of up along the wall (the way I first drew it), so that the pump isn't pushing fluid up on the higher side and so its just pressure differences resulting in the differences in height.)

---

 Thank you for continuing to consider my question.  : ) !!!

[zetetic56]

Here is what I think is interesting about the "moveable wall" question.

Up to this point I think my questions are pretty obvious and simple and so I have guesses about all of their answers.

This part of the moveable wall question, however, I do not have a guess at an answer.

Say we have this system, and in this system the wall does not move.

We start out with X amount of energy in the battery.  The pump is run until the battery is drained.  And while the pump is running the fluid moves and there is a decrease in pressure on the right side of the wall.  The wall is fixed in place and so cannot move.

In the end, after the battery is drained, and after the moving water comes to a stop due to friction, we end up with an increase in thermal energy.

The X amount of energy in the battery is gone, and there is an increase in X amount of thermal energy.

All makes sense (to me).

Then the same thing is done again.  But this time the wall can move.  The wall here can move one way.  It can move from left to right, but its kept from moving back right to left.

Just like before, we start out with X amount of energy in the battery.  The pump is run until the battery is drained.  And while the pump is running the fluid moves and there is a decrease in pressure on the right side of the wall.  The wall is not fixed in place and so can move.

Due to the difference in pressures on the two sides of the wall, the wall moves from left to right, and the height of the column of fluid on the left goes down while the height of the column of fluid on the right goes up.

There is an increase in gravitational potential energy.  (The negative gravitational potential energy becomes less negative.)  Say the increase is Y amount of energy.

In the end, just as before, after the battery is drained, and after the moving water comes to a stop due to friction, we end up with an increase in thermal energy.

However, here, in the end, we also have an increase in gravitational potential energy.

And so, the X amount of energy in the battery is gone, and there is an increase in Y amount of gravitational potential energy, and so, for energy to be conserved there must be less of an increase in thermal energy, there must only be an increase of X - Y amount of thermal energy.

This does not make sense to me.

Question: Why would the wall moving due to a difference in pressures on the two sides of the wall and pushing the fluid on the right side of the wall up (as opposed to the wall staying in place when there is a difference in pressures on the two sides of the wall and the fluid on the right side of the wall is not pushed up) cause a decrease in thermal energy in the end?

What is the physical dynamic that would result in less thermal energy being generated when the wall moves and pushes the fluid up as opposed to when the wall remains in place and does not push the fluid up?

Thank you. : )

 

Edited June 9, 2023 by zetetic56

[Genady]

47 minutes ago, zetetic56 said:

Why would the wall moving due to a difference in pressures on the two sides of the wall and pushing the fluid on the right side of the wall up (as opposed to the wall staying in place when there is a difference in pressures on the two sides of the wall and the fluid on the right side of the wall is not pushed up) cause a decrease in thermal energy in the end?

What is the physical dynamic that would result in less thermal energy being generated when the wall moves and pushes the fluid up as opposed to when the wall remains in place and does not push the fluid up?

In the second case, there is a smaller amount of moving fluid to be stopped by friction. This makes a smaller amount of thermal energy.

[zetetic56]

3 hours ago, Genady said:

In the second case, there is a smaller amount of moving fluid to be stopped by friction. This makes a smaller amount of thermal energy.

Thank you for your reply.

When the wall moves to the right versus staying in place, there will be less fluid moving around in the circle but in the narrowing part of the circle this fluid will be moving faster.

----------------

This is what I'm thinking:

An amount of fluid will move around this circle.  And in the end, the decrease in X amount of energy in the battery will end up as X amount of thermal energy.

And here a lesser amount of fluid will move around this circle.  And in the end the same thing will happen here.  The decrease in X amount of energy in the battery will end up as X amount of thermal energy.

----------------

Both of the above systems have smaller sections of "pipe".

The one on the left must end up with overall X amount of energy (since it starts out with X amount of energy in the battery).

And the one on the right must end with with overall X amount of energy as well (since it starts out with X amount of energy in the battery).

But the one on the right also ends up with an increase in gravitational potential energy (Y amount) while the one on the left does not.

If the answer to why there is X - Y thermal energy in the end for the system on the right is that there is a smaller section of "pipe" and so less fluid moving around the circle, then (if I'm thinking logically correctly) that same dynamic would apply to the system on the left and it would end up with X - Y thermal energy (which cannot be the case).

[ It seems like the answer for why there is X - Y thermal energy in the end for the system on the right must come (somehow/someway) from the moving of the wall and the rising of the fluid.  (But, as I said, that makes no sense to me.) ]

Or, perhaps, I have just misunderstood the idea and the dynamics at work in what you described. : )

Thank you for replying. : ) !!!

 

 

Edited June 9, 2023 by zetetic56

[Genady]

The gravitational energy of the higher level of fluid on the right comes from the battery. The rest of the battery energy goes into kinetic energy of the moving fluid and eventually becomes the thermal energy. While without the changing of the level, all the battery energy goes into the kinetic energy and then becomes the thermal energy.

Edited June 9, 2023 by Genady

[zetetic56]

20 hours ago, Genady said:

The gravitational energy of the higher level of fluid on the right comes from the battery. The rest of the battery energy goes into kinetic energy of the moving fluid and eventually becomes the thermal energy. While without the changing of the level, all the battery energy goes into the kinetic energy and then becomes the thermal energy.

Ah ... I misunderstood your reasoning as to why there would be less fluid moving around the circle in the moveable wall case than in the non-moveable wall case.  I'm sorry about that.

If in the moveable wall case, the energy in the battery both circulates the fluid and raises the fluid on the one side of the wall, then I can easily see how this then leads to less thermal energy in the end.

Thank you. : ) 

---

If you will indulge me a bit further, I'd like to ask about this same question again in the submerged object rising or falling set up.

Say there is a submerged less dense body at the bottom of a more dense column of fluid.  And say there is a barrier wall that the body can rise to and then come to a stop.  And say that the difference in gravitational potential energy between the submerged body at the bottom of the column of fluid and the submerged body at the higher barrier wall is X amount of energy.

 

Here the middle wall cannot move.

The body starts out at the bottom of the column on the right.  It then is allowed to rise.  As it rises the surrounding fluid is displaced downwards.  And when this happens the fluid moves relative to the middle wall.  There is a decrease in pressure on the right side of the middle wall.  The unmovable wall does not move.  And then the body collides with the barrier wall and comes to a stop.

X amount of gravitational potential energy is gone.  (The negative energy becomes less negative.)  And due to the friction between the body and the fluid as it rises and due to the collision between the body and the barrier wall, thermal energy is generated.  And the amount of thermal energy in the end is X amount of energy.

Now the same things is done again.  However this time the middle wall can move from the left to right (but not back again from right to left).

The submerged body is allowed to rise.  There is a decrease in pressure on the right side of the middle wall as the fluid is displaced downwards by the rising body.  There is then a rightward force on the middle wall due to the difference in pressures on the two sides of the wall.  The moveable wall moves to the right.  The fluid on the left side of the wall goes down.  The fluid on the right side of the wall goes up.  (Perhaps by a very small amount, but by some amount.)  There is an increase in gravitational potential energy as the left column of fluid goes down and the right column of fluid goes up.  Say this increase in gravitational potential energy is Y amount.  And the submerged body collides with the barrier wall and comes to a stop.  There is an increase in thermal energy from the friction and from the collision.

For energy to be conserved, given that there is an increase in Y amount of potential energy with the increase in height of the column of fluid on the right, this means the increase in thermal energy in the end must be X - Y amount of thermal energy (as in the other set up with the battery and circulating fluid).

There is no battery here and so the increase in Y amount of gravitational potential energy must come from some other greater decrease in energy in the moving wall case that does not occur here in the non-moving wall case.

Does the rising body rise more slowly when the wall moves (as opposed to when the wall does not move)?  If that were the case, then the collision between the rising body and the barrier wall would produce less thermal energy.  For this to happen the upward buoyant force on the rising body would have to decrease when the wall moves to the right (as opposed to when the wall does not move).  I don't think that would happen.  But maybe it does and I just don't understand the Physics here.  (?)  Or something else?

Thank you for continuing to consider my question.  : )

 

 

[Genady]

1 hour ago, zetetic56 said:

Ah ... I misunderstood your reasoning as to why there would be less fluid moving around the circle in the moveable wall case than in the non-moveable wall case.  I'm sorry about that.

If in the moveable wall case, the energy in the battery both circulates the fluid and raises the fluid on the one side of the wall, then I can easily see how this then leads to less thermal energy in the end.

Thank you. : ) 

---

If you will indulge me a bit further, I'd like to ask about this same question again in the submerged object rising or falling set up.

Say there is a submerged less dense body at the bottom of a more dense column of fluid.  And say there is a barrier wall that the body can rise to and then come to a stop.  And say that the difference in gravitational potential energy between the submerged body at the bottom of the column of fluid and the submerged body at the higher barrier wall is X amount of energy.

 

Here the middle wall cannot move.

The body starts out at the bottom of the column on the right.  It then is allowed to rise.  As it rises the surrounding fluid is displaced downwards.  And when this happens the fluid moves relative to the middle wall.  There is a decrease in pressure on the right side of the middle wall.  The unmovable wall does not move.  And then the body collides with the barrier wall and comes to a stop.

X amount of gravitational potential energy is gone.  (The negative energy becomes less negative.)  And due to the friction between the body and the fluid as it rises and due to the collision between the body and the barrier wall, thermal energy is generated.  And the amount of thermal energy in the end is X amount of energy.

Now the same things is done again.  However this time the middle wall can move from the left to right (but not back again from right to left).

The submerged body is allowed to rise.  There is a decrease in pressure on the right side of the middle wall as the fluid is displaced downwards by the rising body.  There is then a rightward force on the middle wall due to the difference in pressures on the two sides of the wall.  The moveable wall moves to the right.  The fluid on the left side of the wall goes down.  The fluid on the right side of the wall goes up.  (Perhaps by a very small amount, but by some amount.)  There is an increase in gravitational potential energy as the left column of fluid goes down and the right column of fluid goes up.  Say this increase in gravitational potential energy is Y amount.  And the submerged body collides with the barrier wall and comes to a stop.  There is an increase in thermal energy from the friction and from the collision.

For energy to be conserved, given that there is an increase in Y amount of potential energy with the increase in height of the column of fluid on the right, this means the increase in thermal energy in the end must be X - Y amount of thermal energy (as in the other set up with the battery and circulating fluid).

There is no battery here and so the increase in Y amount of gravitational potential energy must come from some other greater decrease in energy in the moving wall case that does not occur here in the non-moving wall case.

Does the rising body rise more slowly when the wall moves (as opposed to when the wall does not move)?  If that were the case, then the collision between the rising body and the barrier wall would produce less thermal energy.  For this to happen the upward buoyant force on the rising body would have to decrease when the wall moves to the right (as opposed to when the wall does not move).  I don't think that would happen.  But maybe it does and I just don't understand the Physics here.  (?)  Or something else?

Thank you for continuing to consider my question.  : )

 

 

I also don't think that the body rises more slowly when the wall moves. But I think that there is another term in the equation that needs to be taken in account, namely, the kinetic energy of the fluid replacing the volume occupied by the body.

When the wall does not move, this fluid has to come from above the body down to under the body. But when the wall moves, part of the replacing fluid comes from the side, horizontally, because the volume there decreases. Thus, less replacing fluid comes from above, which, I think, reduces the kinetic energy of the moving fluid.

IOW, when the wall moves, the energy of the impact of the body and the barrier is the same, but the thermal energy produced by the moving fluid, is smaller.

I am guessing here. Does it make sense?

[zetetic56]

22 hours ago, Genady said:

Does it make sense?

Yes it does.

And it cuts the right way.

However, its only part of something larger that I overlooked.

And the larger overall aspect cuts the wrong way.

In the the moveable wall case there is an increase in Y amount of gravitational potential energy that does not occur in the non-moveable wall case.  And this is part of how I originally framed the question.

 

But after reading your last post I realized there was something else I did not include.

When the wall moves, the body of fluid on the left shifts down and to the right and the body of fluid on the right shifts up and to the right and the wall itself moves from left to right.

There is an increase in kinetic energy in the moveable wall case that does not occur in the non-moveable wall case (W).

And this additional kinetic energy in the moveable wall case will end up as as additional thermal energy in the end.

My guess is that your guess is right.

To fill in the gap below the rising body there will probably be more movement of the fluid in the non-moveable wall case than in the moveable wall case since some of the gap below the rising body can be filled in by the fluid displaced by the nearby moving wall.  And more movement means more kinetic energy.

We could call this additional amount of kinetic energy in the non-movable wall case Z amount of kinetic energy.

 

And so the question is: Is Z greater than W or is W greater than Z?

If both rising bodies encounter the same amount of friction as they rise and if both bodies have the same collisions with the barrier wall, then there is same amount of thermal energy (X) in the end in both cases from this.

And then, in the end, in the non-moveable wall case there is more thermal energy from the displaced fluid filling in the gap below the rising body (Z) while, in the end, in the moveable wall case there is more thermal energy from the shifting columns of fluid and the movement of the wall (W).

And, again, there is also an increase in gravitational potential energy in the moveable wall case (Y).

And so, the question is: Is Z greater than Y plus W?

---

My guess is that Z would not be greater than the increase in gravitational potential energy (Y) plus the additional kinetic energy (then turned into thermal energy) from the shifting columns of fluid and moving wall (W).

But, of course and obviously, that is just a guess.

Thank you helping me see an aspect of this question I overlooked before.

But, if my guess is right (and it, of course, may not be) then this cuts the wrong way and points to a greater energy imbalance between the two cases.

Thank you. : )

[ Edit: And I may have missed your point again in your last post.  You talked about the fluid moving down with the force of gravity in the non-moveable wall case versus the fluid moving horizontally in the moveable wall case, when it comes to replacing the fluid in the gap below the rising body.  And so yes this makes sense to me there would be more kinetic energy, the fluid would be moving faster as it moves with gravity, in the non-moveable wall case.  I was thinking in terms of the fluid being more "nearby" the gap below the rising body.  It ends up with the same thing, that there is an amount of energy (Z) in the non-moveable wall case that is not present in the moveable wall case.  But, I just want to apologize again for my misunderstanding of your replies.  Thank you for continuing to consider my question : ) ]

 

Edited June 11, 2023 by zetetic56

[Genady]

You can safely ignore W. It can be, in this ideal experiment, equal zero, because the wall mass doesn't play any role and can be equal 0.

OTOH, the mass of the fluid cannot be zero, because in that case the body would not rise.

[zetetic56]

27 minutes ago, Genady said:

You can safely ignore W. It can be, in this ideal experiment, equal zero, because the wall mass doesn't play any role and can be equal 0.

OTOH, the mass of the fluid cannot be zero, because in that case the body would not rise.

Okay.  So a massless wall.  And so if a massless wall moves then there is no kinetic energy.  That makes the thought experiment more simple.

So what I wrote above would need to be changed to " W is the kinetic energy in the fluid moving down and to the right in the left column and up and to the right in the right column " and my analysis still stays the same (?)  or no (?)

Thank you. : )

[ Edit: Also, even if the mass of the fluid could be hypothetically 0 in a thought experiment with a rising body (which you said it cannot be), it could not be 0 in this thought experiment because the Z kinetic energy would need a fluid with a mass to try to offset the Y increase in gravitational potential energy.  And so, to then be consistent, W would need to include a fluid with a mass. ]

Edited June 11, 2023 by zetetic56

[Genady]

3 hours ago, zetetic56 said:

Okay.  So a massless wall.  And so if a massless wall moves then there is no kinetic energy.  That makes the thought experiment more simple.

So what I wrote above would need to be changed to " W is the kinetic energy in the fluid moving down and to the right in the left column and up and to the right in the right column " and my analysis still stays the same (?)  or no (?)

Thank you. : )

[ Edit: Also, even if the mass of the fluid could be hypothetically 0 in a thought experiment with a rising body (which you said it cannot be), it could not be 0 in this thought experiment because the Z kinetic energy would need a fluid with a mass to try to offset the Y increase in gravitational potential energy.  And so, to then be consistent, W would need to include a fluid with a mass. ]

z should be equal to y+w rather than greater than.

I don't have any idea how to calculate these dynamics, but thinking about an infinitesimal step in the process:

A body rising by some small distance allows a small amount of fluid to move down, which releases some energy. It also creates a pressure difference on the wall causing it to move a bit, but this motion is not free as it works to move the water. The energy is somehow split between a small rise in the fluid column on the right and the kinetic energy of the moving fluid, but the wall can move only so that the sum of the latter two is equal to the former. As this equality holds at each infinitesimal step, it will hold at the end after all the steps are summed up.

[zetetic56]

18 hours ago, Genady said:

z should be equal to y+w rather than greater than.

Yep, I got that part wrong.

18 hours ago, Genady said:

each infinitesimal step

I'm guessing you are right and the "infinitesimal steps" of the different movements and shifts would need to be analyzed to resolve the overall process.

Obviously, beyond anything I can do.

Thank you for considering my question and thank you for all of your insights and thank you (especially) for your patience with me. : )

Take care and 'till another time. : ) !!!