End-face torque in narrow-gap cylindrical viscometry ($\tau =\pi \eta \omega (2R^{3}H+R^4)/h$)

A rotating inner cylinder of radius $R$ and height $H$ has a uniform gap $h$ from a fixed housing at its side and both ends. For $h\ll R,H$, no-slip boundary condition and laminar Couette flow give side torque $2\pi \eta \omega R^{3}H/h$. Each end contributes ${\int }_0^R(\eta \omega s/h)(2\pi sds)s=\pi \eta \omega R^4/(2h)$. Summing both end faces proves the displayed expression. Shaft area, corner regions, gas slip and viscous heating are neglected.