= End-face torque in narrow-gap cylindrical viscometry
{title2=$\tau=\pi\eta\omega(2R^3H+R^4)/h$}
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> $2\pi\eta\omega R^3H/h$. Each end contributes $\int_0^R(\eta\omega s/h)(2\pi s\,ds)s=\pi\eta\omega R^4/(2h)$. Summing both end faces proves the displayed expression. Shaft area, corner regions, gas slip and viscous heating are neglected.
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