2. The rotating-cylinder viscometer shown in the figure below shears the fluid in a narrow clear-

ance ∆r. The torque (M) required to rotate the cylinder at an angular velocity (Ω) is measured

and then used to compute the dynamic viscosity (μ) of the fluid.

(a) Assuming a linear velocity distribution in the gaps, and showing all steps, find an ex-

pression for in terms of the moment (M), radius (R), length of cylinder (L), angular

velocity (Ω), and the gap (∆r) for the following two cases:

i. neglecting bottom friction.

ii. including the bottom friction.

Hint: First write an expression for the shear stress on the inner cylinder, then the force

exerted, and finally the moment caused by that force on a small elemental area. Integrate

to find the net moment.

(b) Now find the numerical value for viscosity (by including the bottom friction) if the

cylinder rotates at 50 RPM, and R = 5 cm, L = 9.85 cm, ∆r = 1.5 mm, and the

measured torque is 6 × 10^-4 N.m.