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.$
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.