14.7 Exercise 8

A spherical particle falling at a terminal speed in a liquid must have the gravitational force balanced by the drag force and the buoyant force. The buoyant force is equal to the weight of the displaced fluid, while the drag force is assumed to be given by Stokes Law, $F_{\text{s}} = 6 \pi r \eta v .$ Show that the terminal speed is given by $v = \frac{2 R^{2} g}{9 \eta} \left(\rho_{\text{s}} - \rho_{1}\right)$, where R is the radius of the sphere, $\rho_{\text{s}}$ is its density, and $\rho_{1}$ is the density of the fluid, and $\eta$ the coefficient of viscosity.