# Flux Integral of a Fluid Rotating about an Axis

• vs74043
In summary, the problem involves finding the location of a unit square in the yz-plane such that there is zero net amount of fluid flowing through it. The solution involves positioning the square in a specific location in order to balance the fluid flow in the positive and negative x directions. The angular velocity and position vectors are used to determine this location. However, it is not possible to position the square in a way that completely eliminates fluid flow through it.
vs74043

## Homework Statement

We have a fluid with density ρ which is rotating about the z-axis with angular velocity ω. Where should a unit square, call it S, be placed in the yz-plane such that there is zero net amount of fluid flowing through it?

## Homework Equations

$$\mathbf{v}=\mathbf{\omega}\times\mathbf{r}$$
$$\int\mathbf{F}\cdot d\mathbf{S}$$

## The Attempt at a Solution

My attempt has been to try and show that the velocity vector (cross product of angular velocity and position) is independent of z. This would mean that I can position the square such that one side is on the y-axis at distance δ from the origin. Then just find the value of δ.

I'm not sure how to set-up my angular velocity vector and my position vector. Also, is my method idea correct?

vs74043 said:
that there is zero net amount of fluid flowing through it
There is nowhere you can put it that there is no fluid flowing through it. So focus on that word I have bolded. If the square is in the yz plane, the centre of the square can only be in a very restricted subset of locations for there to be as much fluid flowing through in the positive x direction as there is in the negative x direction.

## 1. What is a flux integral of a fluid rotating about an axis?

A flux integral of a fluid rotating about an axis is a mathematical concept used to calculate the amount of fluid flowing through a surface as it rotates around an axis. It takes into account the velocity and density of the fluid as well as the area of the surface.

## 2. How is the flux integral of a fluid rotating about an axis different from a regular flux integral?

The main difference between the two is that the flux integral of a fluid rotating about an axis takes into account the rotation of the fluid, while a regular flux integral does not. This means that the velocity of the fluid is not constant and must be taken into consideration when calculating the flux integral.

## 3. What is the significance of calculating the flux integral of a fluid rotating about an axis?

Calculating the flux integral of a fluid rotating about an axis is important in understanding the flow of fluids in a rotating system. It can help determine the amount of fluid that is being transported through a surface, which is useful in various engineering and scientific applications such as in fluid dynamics and aerodynamics.

## 4. Can the axis of rotation affect the flux integral of a fluid?

Yes, the axis of rotation can greatly affect the flux integral of a fluid. Depending on the orientation and position of the axis, the velocity and flow of the fluid can vary, resulting in different values for the flux integral. This is why it is important to specify the axis of rotation when calculating the flux integral of a fluid.

## 5. How is the flux integral of a fluid rotating about an axis calculated?

The flux integral of a fluid rotating about an axis is calculated using a mathematical formula that takes into account the velocity and density of the fluid, as well as the area of the surface and the angle of rotation. This formula is derived from the principles of fluid dynamics and can be solved using various numerical and computational methods.

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