= 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 has
$$
v_\theta(r)=\frac{\Omega R_i^2}{R_o^2-R_i^2}\left(\frac{R_o^2}{r}-r\right),\qquad
\boxed{\mathcal T=\frac{4\pi\eta L\Omega R_i^2R_o^2}{R_o^2-R_i^2}.}
$$
The <shear stress> at the inner wall is $\tau_i=\mathcal T/(2\pi LR_i^2)$. 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 $\tau=\gamma_0[G'\cos\omega t-G''\sin\omega t]$ 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.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2014/paper-1-couette.png]
{title=Original Couette geometry and dimensionless Newtonian velocity and shear-rate profiles}
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