Stokes' law

Remarks

The proof writes the flow explicitly in the frame of the sphere. It is the sum of a uniform stream, a Stokeslet and a potential dipole at the centre, with strengths chosen so that the velocity vanishes on the sphere. Integrating the stress over the sphere gives the drag: one third, $2 \pi \mu a U$, comes from the pressure and two thirds, $4 \pi \mu a U$, from the viscous shear stress. Example: a sphere of density $\rho_s$ settling under gravity $g$ in a fluid of density $\rho_f < \rho_s$ reaches the speed where drag balances weight minus buoyancy, $6 \pi \mu a V = \tfrac{4}{3} \pi a^3 (\rho_s - \rho_f) g$, so $V = \frac{2 (\rho_s - \rho_f) g a^2}{9 \mu}$. In terms of the drag coefficient based on the cross-section $\pi a^2$, the law reads $C_D = 24 / \mathrm{Re}$ with $\mathrm{Re} = 2 a \abs{U} \rho_f / \mu$. The law describes real flows only when the Reynolds number is small; at larger values inertia adds to the drag, and Oseen's correction gives the next term.

Going deeper

For a student

Stokes' law is the classic exact result of Stokes flow, and it calibrates many measurements: falling-ball viscometers and Millikan's oil-drop experiment both rely on it. It is the leading term of the drag for small Reynolds numbers; the next correction comes from the Oseen equations. The analogous problem in two dimensions, a cylinder, has no solution at all, which is Stokes' paradox. Its linearity is behind the Lorentz reciprocal theorem and the scallop theorem.