Last updated: 2026-07-01

Stokes' Law

Stokes' law describes the drag force that a viscous fluid exerts on a small sphere moving slowly through it. Unlike the drag on a fast-moving car or aircraft, this force grows in direct proportion to speed, because it is dominated by the fluid's viscosity rather than its inertia. The law underpins sedimentation analysis, aerosol science, the falling-ball method for measuring viscosity, and the physics of raindrops and settling particles.

The equation

For a small, rigid sphere moving slowly through an unbounded Newtonian fluid, the viscous drag force is:

$F = 6\pi\, \mu\, r\, v$

where $F$ is the drag force, $\mu$ the dynamic viscosity of the fluid, $r$ the sphere's radius, and $v$ its velocity relative to the fluid. The force acts opposite to the direction of motion. Because it is linear in $v$, doubling the speed doubles the drag — a hallmark of the low-speed, viscosity-dominated regime.

Symbol table

Worked example

Why the drag is linear in speed

At everyday scales, drag grows with the square of speed because the moving object must push fluid out of the way, and that inertial cost rises steeply. For a tiny, slow sphere in a viscous fluid the picture is different: inertia is negligible and the drag comes almost entirely from viscous shearing of the fluid layers around the sphere. The result is a force that rises only linearly with speed. The dividing line between these regimes is set by the Reynolds number, which compares inertial to viscous forces. Stokes' law holds when that number is well below 1.

Settling and terminal velocity

The most common use of Stokes' law is to find how fast a particle settles under gravity in a fluid. As the particle falls, drag increases with speed until it balances the net downward force — weight minus buoyancy. At that point the particle moves at a constant terminal (settling) velocity:

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

where $\rho_p$ and $\rho_f$ are the densities of the particle and fluid and $g$ is gravitational acceleration. This settling-velocity relation is used to size particles in sedimentation tanks, to interpret centrifuge data, and to estimate how long dust or droplets remain suspended in air.

Where it applies, and where it does not

Stokes' law assumes creeping (Stokes) flow, meaning a Reynolds number well below 1. The sphere must be small, rigid, and smooth; the fluid Newtonian and incompressible; and the flow steady and laminar. It also assumes the sphere is far from any walls and from other particles, so the surrounding fluid is effectively unbounded.

When these conditions break down — larger particles, higher speeds, or crowded suspensions — the law loses accuracy. As the Reynolds number approaches 1, inertial effects make the true drag larger than Stokes' law predicts, and corrections such as the Oseen approximation or an empirical drag curve are needed. At high Reynolds number the flow is inertia-dominated and the quadratic drag law, $F = \tfrac{1}{2}\rho v^2 C_d A$, takes over entirely.

Related tools

To check whether the creeping-flow assumption holds, compute the Reynolds Number Calculator for the sphere and confirm it is below 1. To find the constant settling speed of a particle balancing drag against gravity, use the Terminal Velocity Calculator. And for inertia-dominated flow at higher speeds, the general Drag Force Calculator is the companion relation.