Delay of propagation of the pressure in a fluid

[Gh778]

Homework Statement

Does the delay of propagation of pressure gives a torque during a short time ?

A disk full of a liquid turns at a constant angular velocity w around the red axis. At a time, a valve move out, this will increase quickly the pressure inside the liquid.

1/ Draw the lines of equal pressure from the valve
2/ Draw the forces from this additionnal pressure
3/ Is there an additionnal torque ?

Homework Equations

Speed of sound in the fluid

The Attempt at a Solution

1/ & 2/

3/ Yes, the torque from F1 is greater than F2

Is it correct ?

 

[haruspex]

It might help to think of a different model. Instead of a fluid and valve, suppose there is a strut angled across inside the disc, connecting the points on the periphery where you show F1 and F2. Under some influence, heat maybe, the strut attempts to expand, exerting pressure on the disc edges. Would there be a torque on the disc?

 

[Gh778]

I'm not sure, but I can explain the torque like this :

Increase the pressure inside the disk is like launch small balls in all directions. The energy from the spring (or heating) goes to the kinetic energy when balls change their velocity and there is heating too because there are collisions on the wall. The wall of the disk will receive collisions on the curved line $(a,w)$ after the line $(b,x)$, after $(c,y)$ and after $(d,z)$. If I look at the line $(d,z)$, the wall of the disk receive a force that I noted $F$, the torque is $+Fd_1-Fd_2$, it's not 0. and it's the same for the others lines. If I make the comparaison with a translation, "balls" that collide the wall at points $d$ and $z$ transform their energy in heating, and forces canceled themselves. Here, the forces give 2 different moments.

 

[haruspex]

I'm not sure, but I can explain the torque like this :

Increase the pressure inside the disk is like launch small balls in all directions. The energy from the spring (or heating) goes to the kinetic energy when balls change their velocity and there is heating too because there are collisions on the wall. The wall of the disk will receive collisions on the curved line $(a,w)$ after the line $(b,x)$, after $(c,y)$ and after $(d,z)$. If I look at the line $(d,z)$, the wall of the disk receive a force that I noted $F$, the torque is $+Fd_1-Fd_2$, it's not 0. and it's the same for the others lines. If I make the comparaison with a translation, "balls" that collide the wall at points $d$ and $z$ transform their energy in heating, and forces canceled themselves. Here, the forces give 2 different moments.

OK, but consider what is firing these balls. There will be a reaction on that. What will happen to it?

 

[Gh778]

The "spring" for example or the valve is firing the balls.

Reaction from the device that launch the balls ? In the rotation, the valve receives the force $F_v$, this force increases the torque too (counterclockwise rotation):

 

[Gh778]

If someone could say if there is or not a torque and explain ?

 

[Gh778]

Maybe I don't explain my thoughts enough. When the valve is moving quickly, the sum of force is always at 0, so the sum of torque must be at 0. But the density is changing with the speed of sound and the density is not the same at start because there is centrifugal forces in the liquid, the wave of pressure don't change the force but change the density, does this density changes the force on one part more than another part ? The force of pressure depend of the force and the "surface" and with a lower density there is less surface so less pressure. With a gas, it's possible to imagine more molecule at outer radius and less molecules at inner radius. When the valve pushes molecules, there are more molecules push at outer than at inner. So why the pressure could be the same and so the force and the torque ? I don't find a link where the density is link to the force.

 

[haruspex]

This is a most unusual question. Where does it come from?

 

[Gh778]

I saw this exercice in a book, why ?

 

[haruspex]

I saw this exercice in a book, why ?

I feel one needs to know more about the subject matter of the book.
Usually liquids are considered incompressible, but in that theoretical view there can be no propagation delay. The problem title says fluid, but the text says liquid - which is it? Gases are fluids too.