 Solution

A Couette rheometer measures the torque required to rotate one coaxial cylinder relative to another. With inner radius $R_i$, outer radius $R_o$, wetted length $L$, inner angular velocity $\Omega$ and stationary outer wall, laminar Newtonian flow hasThe shear stress at the inner wall is ${\tau }_i=\mathcal{T}/(2\pi L{R}_i^2)$. In a narrow gap $d=R_o-R_i\ll R_i$, $\dot{\gamma }\simeq \Omega R_i/d$, so $\eta \simeq \mathcal{T}d/(2\pi L\Omega R_i^3)$.

Measuring torque over a range of speeds gives a flow curve: constant viscosity identifies a Newtonian regime; decreasing or increasing apparent viscosity indicates shear thinning or thickening. A yield-stress fit needs a genuine low-rate regime and must not be confused with bearing drag or wall slip. Small oscillatory strain also probes viscoelasticity, with $\tau ={\gamma }_0[G^{\prime }\cos \omega t-G^{\prime \prime }\sin \omega t]$ for imposed $\gamma ={\gamma }_0\cos \omega t$.

Control temperature, calibrate instrument torque and end effects, and check gap dependence or roughened surfaces for slip. Keep inertia small enough to avoid Taylor vortices. The exact Newtonian profile is a calibration model; a non-Newtonian material needs its own stress–rate relation.