Cone-and-plate rheometer ($\dot{\gamma }=\Omega /\alpha ,T=2\pi a^3{\sigma }_{\varphi z}/3$)

A small-angle cone rotating above a stationary plate imposes approximately uniform shear rate, since local gap and wall speed both scale with radius. Integrating the uniform shear stress with moment arm r over the plate gives the displayed torque. The leading approximation neglects edge effects and inertia.