# Flux Integral of a Fluid Rotating about an Axis

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1. May 8, 2016

### vs74043

1. The problem statement, all variables and given/known data
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?

2. Relevant equations
$$\mathbf{v}=\mathbf{\omega}\times\mathbf{r}$$
$$\int\mathbf{F}\cdot d\mathbf{S}$$

3. 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?

2. May 9, 2016

### andrewkirk

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.