Torque

Problem Diagram

The liquid viscosity between the cylinder and container will resist motion. A high viscous liquid will require a high torque to keep it rotating. Thus, as expected, the viscosity of the liquid will be directly related to the torque needed to keep the cylinder rotating.

First, define the toque in terms of the resisting force in the liquid. The torque is given by multiplying the force with its moment arm,

     T = Fr

The force can be expressed in terms of the shear stress and area as follows:

     F = τ (2πrL) = T/r

Rearrange terms to yield

     τ = T / 2πr²L

According to Newton's law of viscosity for Newtonian fluids,

     τ = -μ du/dr

The negative sign is inserted to indicate that u decreases as r increases. Note that the velocity profile is assumed to be linear, as shown in the figure. The end conditions are known and are given as u = ωR_i at r = R_i and u = 0 at r = R_o.

Equating the above two equations and integrating from
r = R_i to R_o gives,

For ω = 55 rev/min, T = 0.9 N-m, L = 0.3 m, R_i = 0.12m and R_o = 0.13 m, the viscosity of the liquid is calculated to be