Stokes' Law and Sedimentation

When a small sphere moves slowly through a viscous fluid ($Re \ll 1$), the drag is dominated by viscous forces rather than inertia. Stokes' law gives an exact analytical result:

$F_D = 6\pi\mu r v$

where:

• $\mu$ — dynamic viscosity of the fluid (Pa·s)
• $r$ — radius of the sphere (m)
• $v$ — velocity of the sphere relative to the fluid (m/s)

This is linear in velocity — very different from the quadratic drag seen at high Reynolds numbers.

Terminal Velocity (Sedimentation)

A particle settling through a fluid under gravity reaches a constant terminal velocity when the net downward force (weight minus buoyancy) balances the Stokes drag:

$6\pi\mu r v_t = \frac{4}{3}\pi r^3(\rho_p - \rho_f)g$

Solving for $v_t$:

$v_t = \frac{2r^2(\rho_p - \rho_f)g}{9\mu}$

where $\rho_p$ is the particle density and $\rho_f$ is the fluid density.

Applications

• Viscometry: measuring $\mu$ by timing a sphere's fall
• Sedimentation: designing centrifuges and settling tanks
• Aerosol science: predicting how dust or droplets settle in air

Your Task

Implement:

• stokes_drag(mu, r, v) — Stokes drag force $F_D$ (N)
• stokes_terminal_velocity(r, rho_particle, rho_fluid, mu) — terminal velocity $v_t$ (m/s). Use $g = 9.81$ m/s² inside the function.