# Proving Vorticity of Flow in Rotating Cylinder

• Raees
In summary: If it is a cylinder, then the perpendicular vector to the curved surface is (rcosθ,rsinθ,0) and to the top/bottom surfaces is (0,0,±1). Thus, we both these vectors we get 0.If it is a cylinder, then the perpendicular vector to the curved surface is (rcosθ,rsinθ,0) and to the top/bottom surfaces is (0,0,±1). Thus, we both these vectors we get 0.
Raees
Homework Statement
If liquid contained within a finite closed circular cylinder rotates about the axis k of the cylinder prove that the equation of continuity and boundary conditions are satisfied by u = ΩxR where Ω = Ωk is the constant angular velocity of the cylinder. What is the vorticity of the flow? Here R=xi+yj+zk.
Relevant Equations

Question:
If liquid contained within a finite closed circular cylinder rotates about the axis k of the cylinder prove that the equation of continuity and boundary conditions are satisfied by u = ΩxR where Ω = Ωk is the constant angular velocity of the cylinder. What is the vorticity of the flow? Here R=xi+yj+zk.

u = (-Ωy) i -(-Ωx) j
Therefore: ·u = 0

vorticity: ω = x u = (0) i + (0) j + (-Ω + Ω) k = 0

Raees said:

u = (-Ωy) i -(-Ωx) j
Therefore: ·u = 0
OK. Often, the continuity equation is taken to be ·u) + ∂ρ/∂t = 0. Are you required to show that ·u = 0 implies that ∇·u) + ∂ρ/∂t = 0 for this problem?

vorticity: ω = x u = (0) i + (0) j + (-Ω + Ω) k = 0
Check this. I don't think the vorticity is zero.

What about the boundary conditions at the wall of the container?

TSny said:
OK. Often, the continuity equation is taken to be ·u) + ∂ρ/∂t = 0. Are you required to show that ·u = 0 implies that ∇·u) + ∂ρ/∂t = 0 for this problem?

Check this. I don't think the vorticity is zero.

What about the boundary conditions at the wall of the container?

Are you required to show that ·u = 0 implies that ∇·u) + ∂ρ/∂t = 0 for this problem?
No we are not, I can't find that equation in the notes nor has it been taught to us.

Check this. I don't think the vorticity is zero.
I recalculated the vorticity to be 2Ω.

What about the boundary conditions at the wall of the container?

Has the topic of boundary conditions come up in your course?

Last edited:
TSny said:
Has the topic of boundary conditions come up in your course?

Not yet

Your result for the vorticity looks good.

It's kind of strange that you are asked about the boundary conditions if you haven't covered the topic yet. But you can check your textbook or search the net. For example here.

TSny said:
Your result for the vorticity looks good.

It's kind of strange that you are asked about the boundary conditions if you haven't covered the topic yet. But you can check your textbook or search the net. For example here.

Thanks, would the boundary condition be u·n = 0, where n is the vector perpendicular to the cylinder? Therefore, n = (1,1,0)?

Raees said:
Thanks, would the boundary condition be u·n = 0, where n is the vector perpendicular to the cylinder? Therefore, n = (1,1,0)?
Yes, that is one of the boundary conditions. Another condition is that, at the wall, the component of u that is parallel to the wall should equal the velocity of the wall. In other words, at any point of the wall, the fluid and the wall should have the same velocity so that the relative velocity is zero.

TSny said:
Yes, that is one of the boundary conditions. Another condition is that, at the wall, the component of u that is parallel to the wall should equal the velocity of the wall. In other words, at any point of the wall, the fluid and the wall should have the same velocity so that the relative velocity is zero.

Thanks you!

So, for u·n = 0, we have:

u = (-Ωy) i -(-Ωx) j and n = (1,1,0)

which gives: u·n = Ω(x-y)

This is not equal to zero. Did I do something wrong here?

Raees said:
So, for u·n = 0, we have:

u = (-Ωy) i -(-Ωx) j and n = (1,1,0)
The expression for n is not correct. Sorry I didn't catch that in your post #7. If (x, y, z) are the coordinates of a point on the wall, then what is the expression for n at that point?

TSny said:
The expression for n is not correct. Sorry I didn't catch that in your post #7. If (x, y, z) are the coordinates of a point on the wall, then what is the expression for n at that point?

If it is a cylinder, then the perpendicular vector to the curved surface is (rcosθ,rsinθ,0) and to the top/bottom surfaces is (0,0,±1). Thus, we both these vectors we get 0.

Raees said:
If it is a cylinder, then the perpendicular vector to the curved surface is (rcosθ,rsinθ,0) and to the top/bottom surfaces is (0,0,±1). Thus, we both these vectors we get 0.
OK. If you want n to represent a unit vector, then (rcosθ,rsinθ,0) needs to be normalized so that it has a magnitude of 1. But it won't change the result.

What about the other boundary condition which states that the velocity of the fluid at the wall should equal the velocity of the wall?

## 1. What is vorticity of flow in a rotating cylinder?

Vorticity of flow in a rotating cylinder refers to the circular motion of fluid particles within a cylindrical container that is rotating around its axis. This motion creates a swirling effect and can be measured by the amount of angular rotation per unit time.

## 2. How is vorticity of flow measured in a rotating cylinder?

Vorticity of flow in a rotating cylinder can be measured using a variety of techniques such as flow visualization, velocity measurements, and numerical simulations. These methods allow for the visualization and quantification of the swirling motion of fluid particles.

## 3. What factors affect the vorticity of flow in a rotating cylinder?

The vorticity of flow in a rotating cylinder is affected by several factors including the speed of rotation, the shape and size of the cylinder, and the viscosity of the fluid. These factors can influence the strength and direction of the swirling motion.

## 4. Why is it important to prove vorticity of flow in a rotating cylinder?

Proving vorticity of flow in a rotating cylinder is important because it allows for a better understanding of the behavior of fluids in rotational systems. This knowledge can be applied to various engineering and scientific fields, such as fluid dynamics, meteorology, and geophysics.

## 5. What are some real-world applications of vorticity of flow in a rotating cylinder?

Vorticity of flow in a rotating cylinder has many practical applications, including the study of hurricanes and tornadoes in meteorology, the design of turbine blades in engineering, and the analysis of ocean currents in oceanography. It is also used in various industrial processes, such as mixing and stirring fluids in chemical engineering.

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