A very long cylinder of radius a translates steadily at speed $V$ in a direction perpendicular to its axis and parallel to a plane boundary. The centre of the cylinder remains a distance $a+b$ above the plane, where $b \ll a$, and the motion takes place through an incompressible fluid of viscosity $\mu$.

Consider the force $F$ per unit length parallel to the plane that must be applied to the cylinder to maintain the motion. Explain why $F$ scales according to $F \propto \mu V(a / b)^{1 / 2}$.

Approximating the lower cylindrical surface by a parabola, or otherwise, determine the velocity and pressure gradient fields in the space between the cylinder and the plane. Hence, by considering the shear stress on the plane, or otherwise, calculate $F$ explicitly.

[You may use

$$\int_{-\infty}^{\infty}\left(1+x^{2}\right)^{-1} d x=\pi, \quad \int_{-\infty}^{\infty}\left(1+x^{2}\right)^{-2} d x=\frac{1}{2} \pi \quad \text { and } \quad \int_{-\infty}^{\infty}\left(1+x^{2}\right)^{-3} d x=\frac{3}{8} \pi$$