The problem is about the famous Newton’s rotating bucket experiment.

(a)

Consider a fluid mass rotating at constant angular speed about the central vertical axis of a cylindrical container. Liquid surface takes the shape as shown in the diagram. What has been shown in the diagram is a vertical cross-section of the rotating fluid. The surface of the fluid is curved. In fact, it is paraboloidal.

We fix the co-ordinate system by measuring vertical distance from the lowest point in the surface of the rotating fluid; it is by symmetry on the central axis.

Fluid below the level is rotating uniformly with angular speed .

We will analyse the dynamics of rotation of the fluid. At a distance r from the central axis we consider a fluid element of cross-section and width . Let be the density of the fluid. The mass of this fluid element will be . As this fluid element is rotating with angular speed at a distance r from the rotation axis, it must obtain a centripetal force of magnitude . As the only source of force in the horizontal direction can be fluid pressure, it implies that in a rotating fluid pressure will vary with depth h and radial distance r. That is the pressure is a function . We will set up the equation of motion of the fluid element and from it obtain a differential equation for. We fill first obtain the variation of p as a function of r. As we are considering the rotation of a fluid element at depth h, in the following we will suppress the dependence of p on h.

Centripetal force is provided by the pressure difference at the two opposite faces of the fluid element.

As by retaining terms of order in , we get

(b)

It is a linear first order differential equation. Its general solution is

Calling ,

We get,

On the central axis fluid is at rest as it does not rotate, variation of pressure on the axis with depth is the hydrostatic relation

where is the atmospheric pressure.

(c)

We now will find out how the radial pressure arises.

Radial pressure is due to the curved shape of the liquid surface. Let the height of the liquid surface at radial distance as measured from the level be y. Then

It is the equation of a parabola. This curve on rotation about the central axis will trace a paraboloid.