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
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
Velocity Profile of the Fluid between
the Container and Cylinder
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 = ωRi at r = Ri and
u = 0 at r = Ro.
Equating the above two equations and integrating from
Ri to Ro gives,
For ω = 55 rev/min, T = 0.9 N-m, L = 0.3 m, Ri =
0.12m and Ro = 0.13 m, the viscosity of the liquid is calculated to be