= Solution
In the thin-gap <Couette flow> approximation, the tangential <velocity> varies linearly through the gap. At radius $r$, its difference between the two discs is $r\omega$, so a <Newtonian fluid> exerts <shear stress>
$$
\tau_{\theta z}=\eta\frac{r\omega}{d}.
$$
An annulus has area $2\pi r\,dr$, and its tangential <force> acts at lever arm $r$. Integrating its <torque> gives
$$
\tau_{\rm drive}=\int_0^a r\frac{\eta r\omega}{d}\,2\pi r\,dr,\qquad \boxed{\tau_{\rm drive}=\frac{\pi\eta\omega a^4}{2d}.}
$$
The driving <torque> is in the direction of rotation; the liquid's resisting <torque> has the opposite sign. This <parallel-disc viscometer> result assumes laminar shear with no slip, negligible edge effects and negligible secondary circulation. Its fourth-power radius dependence comes from the combined radial dependence of speed, annular area and lever arm.
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