 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 stressAn annulus has area $2\pi rdr$, and its tangential force acts at lever arm $r$. Integrating its torque givesThe 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.