Biharmonic stream function for a fixed disk in planar shear

For a stationary disk of radius $a$ in the far-field shear $u_{\infty }=\gamma y{e}_x$, using $u_r={\psi }_{\theta }/r$ and $u_{\theta }=-{\psi }_r$, the exterior solution isIt satisfies no slip at $r=a$ and approaches the imposed shear in velocity.

The surface shear stress isThe torque exerted by the fluid on the disk per unit axial length is therefore