Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 39C / b / iii / Solution

The polar shear stress in a Newtonian fluid isOn $r=a$, no slip holds for every $\theta$, so $u_{\theta }=0$ and ${\partial }_{\theta }{u}_r=0$. ThereforeDifferentiating the solution from part (ii) givesand hence

The moment arm and line element are both $a$, so the torque exerted by the fluid on the disk per unit axial length isFor $\gamma >0$ the negative sign denotes the clockwise sense of the ambient shear rotation. This agrees with the hydrodynamic torque on a fixed disk in planar shear.