A rigid sphere of radius $a$ falls under gravity through an incompressible fluid of density $\rho$ and viscosity $\mu$ towards a rigid horizontal plane. The minimum gap $h_{0}(t)$ between the sphere and the plane satisfies $h_{0} \ll a$. Find an approximation for the gap thickness $h(r, t)$ between the sphere and the plane in the region $r \ll a$, where $r$ is the distance from the axis of symmetry.

For a prescribed value of $\dot{h}_{0}=d h_{0} / d t$, use lubrication theory to find the radial velocity and the fluid pressure in the region $r \ll a$. Explain why the approximations of lubrication theory require $h_{0} \ll a$ and $\rho h_{0} \dot{h}_{0} \ll \mu$.

Calculate the total vertical force due to the motion that is exerted by the fluid on the sphere. Deduce that if the sphere is settling under its own weight (corrected for buoyancy) then $h_{0}(t)$ decreases exponentially. What is the exponential decay rate for a solid sphere of density $\rho_{s}$ in a fluid of density $\rho_{f}$ ?