A cylindrical vessel of diameter of height 0.5 m and diameter 0.6 m is filled to a level of 0.4 m with glycerol (density 1260 kg m-3 and dynamic viscosity 1.4 Pa.s). A lid with a central hole of 1 mm in diameter is fitted to the top of the container and then the system is rotated around the central cylindrical axis. Assume that all of the fluid rotates as a forced vortex and that the fluid can be considered incompressible. Although you should clearly show your working, you do not need to derive all equations from first principles.

(a) Sketch the surface profile of the vortex as a function of the distance from the axis (radial position), with relevant labels.

b) Determine the angular speed of rotation when the base of the vortex is exactly halfway up the container, and the lid is in partial contact with the fluid.

c) Calculate the force exerted by the fluid on the lid.

d) The lid with the hole is removed and the container is completely filled with glycerol. The vessel is then sealed with a new lid with no holes in it. The cylinder is then rotated at 100 rpm. Calculate the force on the lid due to the vortex.