(Solved): 2. The rotating-cylinder viscometer shown in the figure below shears the fluid in a narrow clearan ...

2. The rotating-cylinder viscometer shown in the figure below shears the fluid in a narrow clearance \( \Delta r \). The torque \( (M) \) required to rotate the cylinder at an angular velocity \( (\Omega) \) is measured and then used to compute the dynamic viscosity \( (\mu) \) of the fluid. (a) Assuming a linear velocity distribution in the gaps, and showing all steps, find an expression for \( \mu \) in terms of the moment \( (M) \), radius \( (R) \), length of cylinder \( (L) \), angular velocity \( (\Omega) \), and the gap \( (\Delta r) \) 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 \( 50 \mathrm{RPM} \), and \( R=5 \mathrm{~cm}, L=9.85 \mathrm{~cm}, \Delta r=1.5 \mathrm{~mm} \).