A closed cylindrical tube, which contains some water but is not...

When the closed cylindrical tube is rotated about its perpendicular bisector, the water inside experiences a centrifugal force directed radially outwards from the axis of rotation.

Due to this centrifugal force, the water moves towards the ends of the tube, away from the axis of rotation.

The moment of inertia of a mass distribution is given by $I = \int r^2 dm$, where $r$ is the perpendicular distance of the mass element $dm$ from the axis of rotation.

Since the water is displaced towards the ends, the effective distance $r$ of the mass elements from the axis increases.

Therefore, the moment of inertia of the water about this axis increases.