6.4 Stokes drag

When a small body moves slowly through a viscous fluid, the resistance it feels is set entirely by viscosity, and takes a simple closed form. This lesson derives the drag on a sphere at low Reynolds number — Stokes drag — and the friction coefficient that the Einstein relation of the next lesson depends on.

Drag in the inertia-free limit

A sphere of radius $a$ moving at velocity $\mathbf{U}$ through a fluid of viscosity $\mu$, slowly enough that the Reynolds number $\mathrm{Re} = \rho U a/\mu \ll 1$, feels a drag force

This is the central numerical result of low-Reynolds-number hydrodynamics. It follows from solving the linear Stokes equation around a no-slip sphere; the factor $6\pi$ is exact for a rigid spherical surface. Three features deserve emphasis. The drag is linear in velocity — double the speed, double the force — unlike the high-speed drag that grows as $U^2$. It is linear in size $a$, not in cross-sectional area $a^2$, because at low Reynolds number the disturbance the sphere creates reaches out many radii into the fluid. And it depends on $\mu$ but not on the fluid density, since inertia plays no role.

The form of the result can be read off from dimensions alone: the only quantities available are $\mu$, $a$, and $U$, and the single combination with units of force is $\mu a U$. Solving the flow merely fixes the dimensionless prefactor at $6\pi$.

The streamlines around the sphere are perfectly fore-aft symmetric — a hallmark of Stokes flow, where the time-reversible equation cannot tell upstream from downstream. There is no wake, no separation, and no pressure recovery asymmetry; the entire force comes from viscous shear and the gentle pressure field draped over the sphere.

The friction coefficient and terminal velocity

Writing the drag as $\mathbf{F}_\text{drag} = -\gamma\,\mathbf{U}$ defines the friction coefficient

A particle pulled by a steady external force $F$ — gravity, say — accelerates only until drag balances the force, then coasts at the terminal velocity $U_\text{term} = F/\gamma$. For a sphere settling under gravity in a fluid, balancing buoyancy-corrected weight against Stokes drag gives a settling speed proportional to $a^2$: larger particles fall far faster, which is why fine silt stays suspended for days while sand drops in seconds.

This same friction coefficient was the lever in Millikan’s oil-drop experiment, where the terminal velocity of a falling droplet fixed its radius and hence — through the balancing electric field — the elementary charge. And it is exactly the $\gamma$ that ties drag to diffusion in the next lesson: the same molecular collisions that resist a particle’s steady motion also drive its random jitter, and the Einstein relation makes that connection quantitative.