A cylinder of radius $a$ falls at speed $U$ without rotating through viscous fluid adjacent to a vertical plane wall, with its axis horizontal and parallel to the wall. The distance between the cylinder and the wall is $h_{0} \ll a$. Use lubrication theory in a frame of reference moving with the cylinder to determine that the two-dimensional volume flux between the cylinder and the wall is

$$q=\frac{2 h_{0} U}{3}$$

upwards, relative to the cylinder.

Determine an expression for the viscous shear stress on the cylinder. Use this to calculate the viscous force and hence the torque on the cylinder. If the cylinder is free to rotate, what does your result say about the sense of rotation of the cylinder?

[Hint: You may quote the following integrals:

$$\left.\int_{-\infty}^{\infty} \frac{d t}{1+t^{2}}=\pi, \quad \int_{-\infty}^{\infty} \frac{d t}{\left(1+t^{2}\right)^{2}}=\frac{\pi}{2}, \quad \int_{-\infty}^{\infty} \frac{d t}{\left(1+t^{2}\right)^{3}}=\frac{3 \pi}{8}\right]$$