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Flux Integral of a Fluid Rotating about an Axis

  1. May 8, 2016 #1
    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. jcsd
  3. May 9, 2016 #2

    andrewkirk

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    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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