Velocity of flow in cylindrical coordinates

[glebovg]

An infinitely long cylindrical bucket with radius $a$ is full of water and rotates with constant angular velocity $\Omega$ about its horizontal axis. The gravity is in the vertical direction. The velocity of the flow in cylindrical coordinates (whose $z$ axis is the horizontal axis of the bucket) is given by $\vec u = \left\langle {{u_r}(r,t),{u_\theta }(r,t),0} \right\rangle$.

Show that ${{u_r}(r,t)} = 0$.

I actually have no idea how to start. Any help will be appreciated.

 

[lippyka]

Since the bucket is rotating with a constant angular velocity \Omega, the Coriolis force is zero. Therefore, the only forces acting on the flow are gravity and the centrifugal force. The centrifugal force is perpendicular to the radial direction, and so it cannot affect the radial component of the velocity. The gravity also cannot affect the radial component as it is in the vertical direction. Therefore, the only force acting on the radial component of the velocity is zero, and hence the radial component must be zero.