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While reading a fluids book for my course and doing some problems I came across this question that the book didn't provide an answer for and was wondering if one of you kind folks could help me out because i haven't the foggiest.

Question:

If a 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 cross product $ u= \omega X R $ where $ \omega=\omega_k$ is the constant angular velocity of the cylinder. What is the vorticity of the flow? Here $ R=x_i+y_j+z_k.$

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  • $\begingroup$ I think I went to the same university as you, your fluids professor is a guy named Johnson, right? And your complex analysis lecturer is a greek guy? On point: what is the boundary of the cylinder? Remember there are two flat tops because it is finite. Once you know the boundary, you can find a normal vector. $\endgroup$
    – Almentoe
    Oct 21, 2015 at 13:55
  • $\begingroup$ Yeah that's right. I'm just completely confused by fluids, it's like a foreign language to me. Even if i could find a normal (doubt that i can) i wouldnt kniw what to with it. We've not discussed problems involving angular velocities in lectures. Could you maybe give me some more hints, i dont want the answer, i just want to understand what's happening $\endgroup$
    – WillBee
    Oct 22, 2015 at 8:58

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To prove that continuity is satisfied you simply sub the proposed form of the solution into the continuity equation (I presume you are dealing with an incompressible fluid since nearly all fluid mechanics is incompressible at undergrad level). So show that the continuity equation $\nabla \cdot \mathbf{u}=0$ is true for this $\mathbf{u}$.

You haven't told us what the boundary conditions are, but you just evaluate $\mathbf{u}$ at the boundary of the cylinder and show that it matches the required conditions (likely to be no slip and impermeability - so the fluid velocity at the boundary must match the velocity of the cylinder's wall / ends.

The vorticity of a flow is defined to be $\nabla \times \mathbf{u}$ so you calculate this and voila! - you have the vorticity of the flow.

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