Hydrodynamic torque on a fixed disk in planar shear
= Hydrodynamic torque on a fixed disk in planar shear
The surface shear stress is
$$
\sigma_{r\theta}(a,\theta)
=-\mu\gamma+2\mu\gamma\cos2\theta.
$$
The torque exerted by the fluid on the disk per unit axial length is therefore
$$
\mathcal T_z
=a^2\int_0^{2\pi}\sigma_{r\theta}(a,\theta)\,d\theta
=-2\pi\mu\gamma a^2.
$$