Stokes' Law and Terminal Velocity Explained Simply

Drop a steel ball into a jar of honey and it does not keep speeding up like a falling stone. After a moment it settles into a slow, steady descent. The honey pushes back on the ball with a drag force that grows as the ball speeds up, until the drag exactly balances the ball's weight. That final constant speed is called terminal velocity.

The drag force on a slow-moving sphere was worked out by Sir George Stokes in 1851 and is called Stokes' law. It predicts how fast dust settles, how raindrops fall, and how engineers measure the viscosity of thick oils by timing a falling ball.

Key Concepts

• Stokes' law: the viscous drag on a small sphere moving slowly through a fluid is F = 6 x pi x eta x r x v, where eta is the coefficient of viscosity (Pa.s), r is the sphere's radius (m), and v is its speed (m/s).
• Terminal velocity: the constant speed reached when the net force on the falling body becomes zero, so weight = buoyant upthrust + viscous drag.
• Effective weight in the fluid: (4/3) x pi x r^3 x (rho - sigma) x g, where rho is the density of the sphere, sigma is the density of the fluid, and g = 9.8 m/s^2.
• Conditions for Stokes' law: the sphere must be small and smooth, the speed low so the flow stays laminar, the fluid deep and wide compared to the sphere, and there must be no slipping between fluid and sphere.

Formulas

• Stokes' law: F = 6 pi eta r v. Drag is proportional to radius and to speed, not to speed squared.
• Terminal velocity: v_t = [2 r^2 g (rho - sigma)] / (9 eta). All SI units: r in metres, densities in kg/m^3, eta in Pa.s, giving v_t in m/s.
• Assumptions and limits: valid only for low Reynolds numbers (roughly Re below 1), meaning small, slow spheres. For fast or large bodies the drag grows with v^2 and a different law applies.

Worked Example

A steel ball of radius 2 mm falls through glycerine. Density of steel rho = 7800 kg/m^3, density of glycerine sigma = 1260 kg/m^3, and glycerine's viscosity eta = 1.49 Pa.s. Find the terminal velocity.

1. r = 2 mm = 0.002 m, so r^2 = 4 x 10^-6 m^2.
2. Density difference: rho - sigma = 7800 - 1260 = 6540 kg/m^3.
3. Numerator: 2 x 4 x 10^-6 x 9.8 x 6540 = 0.5127 (SI units).
4. Denominator: 9 x 1.49 = 13.41.
5. v_t = 0.5127 / 13.41 = 0.0382 m/s, about 3.8 cm/s.

Common Misconceptions

• Terminal velocity is not the speed of free fall. In a vacuum a body keeps accelerating; terminal velocity only exists because the fluid's drag grows with speed until it cancels the weight.
• Stokes' law does not apply to skydivers or cars. It is valid only for small, slow spheres in laminar flow. Fast motion creates turbulence and the drag then follows a v^2 law instead.

Real-World Applications

• The falling-sphere viscometer measures the viscosity of oils and syrups by timing a steel ball's terminal velocity.
• Centrifuges separate blood cells and cream from milk by spinning them so that an artificial gravity raises their terminal velocity.

Stokes' law gives the drag on a small slow sphere as 6 pi eta r v. When drag plus buoyancy balances weight, the sphere falls at terminal velocity v_t = 2 r^2 g (rho - sigma) / (9 eta), one of the most useful results in fluid mechanics.