Fluid mechanics (shape of free surface)

[Mandeep Deka]

Homework Statement

A tube of radius 'R' and height 'H' (placed vertically), is filled with a fluid till a height 'h' (<H). Now it is rotated with an angular velocity 'w' about the central axis. Prove that the shape of the free surface is parabolic.

Homework Equations

The Attempt at a Solution

The proof of this equation is simple, and i have derived it. Now along with this question a statement was mentioned, which i aint able to prove.

Is is said that the initial height of the fluid is the arithmetic mean of the maximum height (i.e the height of the fluid at the end of the tube) and the minimum height (i.e the height of the fluid at the center). I have tried, but failed to prove it. Is the proposition valid?
If yes, I would be grateful if someone could give me some hint as to how i prove it!

 

[fzero]

You will probably want to draw the radial cross-section with the parabola and the original water level crossing it. The volume of water above the original water level at the edges should equal the volume "missing" below it in the center.

 

[Mandeep Deka]

exactly, but how do i do it?
That's what i am asking, How will i calculate the volume of water above and below the original level?

 

[sgd37]

well you integrate your equation for the shape of the free surface from +R/2 to -R/2 and it should give 0 if you have placed the origin of the graph at (R/2,h) in the cylinder frame

 

[paranormal]

I can confirm that the proposition is valid. The reason for this is due to the principle of conservation of energy in fluid mechanics. When the tube is rotated, the fluid experiences a centrifugal force which is balanced by the pressure gradient force. This results in a parabolic shape of the free surface, with the maximum height at the end of the tube and the minimum height at the center.

To prove the statement, we can use the equation for the height of the fluid at any point along the tube:

h = H + R(1-cosθ)

Where h is the height of the fluid at a point, H is the initial height of the fluid, R is the radius of the tube, and θ is the angle of rotation.

At the end of the tube (θ = π/2), the height of the fluid is h_max = H + R. At the center (θ = 0), the height of the fluid is h_min = H. Thus, the average height is:

h_avg = (h_max + h_min)/2 = (H + R + H)/2 = (2H + R)/2 = H + R/2

This is equal to the initial height of the fluid, H. Therefore, the proposition is valid.