4. The rotating-cylinder viscometer shown in the figure below shears the fluid in a narrow clearance Ar. The torque (M) required to rotate the cylinder at an angular velocity (1) is measured and then used to compute the dynamic viscosity (µ) of the fluid. Assuming a linear velocity distribution in the gaps, and showing all steps, find an expression for µ in terms of the moment (M), radius (R), length of cylinder (L), angular velocity (2), and the gap (Ar) for the following two cases: (a) neglecting bottom friction. (b) including the bottom friction.

Fig: 1

Fig: 2

Fig: 3

Fig: 4