A cylinder of radius $a$ rotates about its axis with angular velocity $\Omega$ while its axis is fixed parallel to and at a distance $a+h_{0}$ from a rigid plane, where $h_{0} \ll a$. Fluid of kinematic viscosity $\nu$ fills the space between the cylinder and the plane. Determine the gap width $h$ between the cylinder and the plane as a function of a coordinate $x$ parallel to the surface of the wall and orthogonal to the axis of the cylinder. What is the characteristic length scale, in the $x$ direction, for changes in the gap width? Taking an appropriate approximation for $h(x)$, valid in the region where the gap width $h$ is small, use lubrication theory to determine that the volume flux between the wall and the cylinder (per unit length along the axis) has magnitude $\frac{2}{3} a \Omega h_{0}$, and state its direction.

Evaluate the tangential shear stress $\tau$ on the surface of the cylinder. Approximating the torque on the cylinder (per unit length along the axis) in the form of an integral $T=a \int_{-\infty}^{\infty} \tau d x$, find the torque $T$ to leading order in $h_{0} / a \ll 1$.

Explain the restriction $a^{1 / 2} \Omega h_{0}^{3 / 2} / \nu \ll 1$ for the theory to be valid.

[You may use the facts that $\int_{-\infty}^{\infty} \frac{d x}{\left(1+x^{2}\right)^{2}}=\frac{\pi}{2}$ and $\left.\int_{-\infty}^{\infty} \frac{d x}{\left(1+x^{2}\right)^{3}}=\frac{3 \pi}{8} .\right]$