Determination of viscosity by viscometer

A viscometer measures the dynamic viscosity $\mu$ of a fluid from a measured relation between shear stress and velocity gradient, using Newton's law $\tau=\mu\,\dfrac{du}{dy}$.

Rotating-cylinder (Couette) viscometer

The fluid fills the thin gap $t$ between a fixed outer cylinder and an inner cylinder (radius $R$, immersed height $h$) turned at angular speed $\omega$. The torque $T$ on the inner cylinder is measured.

   fixed outer cylinder
   |   |  fluid gap t |   |
   |   |<->|  inner   |   |
   |   |   | cylinder |   |
   |   |   | (omega)  |   |

For a thin gap the velocity profile is linear, so

$$\tau=\mu\frac{\omega R}{t},\qquad T=\tau\,(2\pi R h)\,R$$ $$\mu=\frac{T\,t}{2\pi\omega R^{3}h}$$

Capillary-tube (Ostwald) viscometer

The time $t_1$ for a fixed volume of the test liquid to flow through a capillary under its own head is compared with the time $t_2$ for a reference liquid (water) of known viscosity:

$$\frac{\mu_1}{\mu_2}=\frac{\rho_1 t_1}{\rho_2 t_2}$$

For laminar flow (Hagen-Poiseuille), $\mu=\dfrac{\pi\,\Delta p\,r^{4}}{8QL}$.

Falling-sphere viscometer

A small sphere of diameter $d$ and density $\rho_s$ falls at terminal velocity $v$ in the liquid (density $\rho$). Balancing weight, buoyancy and Stokes drag $3\pi\mu d v$:

$$\mu=\frac{d^{2}(\rho_s-\rho)\,g}{18\,v}$$

Other types

The Saybolt (efflux) viscometer gives kinematic viscosity from the time taken for 60 mL of oil to flow out through a standard orifice.