Flux Integral of a Fluid Rotating about an Axis

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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?
 

Answers and Replies

  • #2
andrewkirk
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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.
 

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