Stokes’ Law

When a sphere moves through a viscous fluid — think of a raindrop falling through air, or a steel ball sinking through honey — the fluid resists the motion with a drag force. Stokes' Law gives us an exact formula for this drag force, provided the flow around the sphere remains smooth and laminar (not turbulent). Derived by the Irish mathematician George Gabriel Stokes in 1851, the law is one of the most elegant and practically useful results in fluid mechanics. The key insight is that the drag force depends on three things: how sticky (viscous) the fluid is, how large the sphere is, and how fast it moves. Stokes showed that these three factors combine in the simplest possible way — all linearly — to give the drag force. Doubling the radius, doubling the speed, or doubling the viscosity each separately doubles the drag. This linear relationship is what makes laminar flow around a sphere analytically tractable, unlike the chaotic turbulent case. Stokes' Law underpins many real-world applications: determining the viscosity of oils in industry, understanding how red blood cells settle in a test tube (erythrocyte sedimentation rate in medicine), calculating terminal velocity of particles in the atmosphere, and even the historical Millikan oil-drop experiment that measured the charge of a single electron. Whenever small, roughly spherical objects move slowly through a viscous medium, Stokes' Law is the go-to tool.

Key takeaways
  • What is Viscosity?
  • Laminar Flow and the Reynolds Number
  • The Three Contributions to Drag
  • Terminal Velocity
  • Limitations of Stokes' Law

What is Viscosity?

Viscosity (η) is the measure of a fluid's internal resistance to flow — its 'thickness'. Water has low viscosity; honey has high viscosity. Formally, viscosity relates the shear stress between fluid layers to the velocity gradient between them: τ = η (dv/dy). The SI unit is the Pascal-second (Pa·s), also called poiseuille. For example, water at 20°C has η ≈ 0.001 Pa·s, while honey is around 2–10 Pa·s.

Laminar Flow and the Reynolds Number

Stokes' Law only applies when flow is laminar — fluid layers glide smoothly past one another with no turbulent eddies. The Reynolds number Re = ρvd/η (where ρ is fluid density, v is speed, d is sphere diameter) quantifies the competition between inertial and viscous forces. Stokes' Law is valid when Re ≪ 1, meaning viscous forces dominate. For a 1 mm sphere moving at 1 mm/s in water, Re ≈ 1 (borderline), but for a pollen grain or an oil droplet under similar conditions, Re ≪ 1 and Stokes' Law is excellent.

The Three Contributions to Drag

When a sphere moves through a viscous fluid, two distinct physical effects create drag: (1) Pressure drag — the fluid must be pushed aside at the front and fills in at the rear, creating a pressure imbalance. In the Stokes regime this contributes (2/3) of the total drag. (2) Viscous (skin friction) drag — the no-slip condition means fluid touching the sphere moves with it, and viscous stresses in the surrounding fluid resist this. This contributes the remaining (1/3). Stokes solved the full Navier-Stokes equations under low-Reynolds-number assumptions and found both contributions sum to the simple formula F = 6πηrv.

Terminal Velocity

A sphere falling under gravity eventually reaches terminal velocity when the net upward force (drag + buoyancy) exactly balances gravity. Setting forces equal: (4/3)πr³ρ_sphere·g = (4/3)πr³ρ_fluid·g + 6πηrv_t. Solving for terminal velocity: v_t = (2r²(ρ_sphere − ρ_fluid)g) / (9η). This formula is extremely useful — for instance, measuring v_t for a sphere of known radius and density in an unknown fluid lets you calculate the fluid's viscosity directly. This is the principle behind the falling-ball viscometer.

Limitations of Stokes' Law

Stokes' Law assumes: (a) the sphere is rigid and perfectly smooth, (b) the fluid is incompressible and Newtonian (constant viscosity), (c) the flow is steady and laminar (Re ≪ 1), and (d) the sphere is far from any walls. At higher speeds, inertial effects cause the drag to grow faster than linearly with velocity, and empirical corrections (like the drag coefficient C_D) must be used. For bubbles and droplets, the internal circulation of fluid inside the object modifies the drag coefficient from 6π to 4π (Hadamard-Rybczynski correction).

The Key Equation

Fd=6πηrvF_d = 6\pi\eta r v
  • F_d — Drag force (N (Newton)): The resistive force exerted by the fluid on the moving sphere, directed opposite to the velocity.
  • \eta — Dynamic viscosity (Pa·s): A measure of the fluid's resistance to flow. Higher η means thicker, stickier fluid. Sometimes written as μ.
  • r — Radius of sphere (m (metre)): The radius of the spherical object moving through the fluid. Drag is directly proportional to r, not r².
  • v — Speed of sphere (m/s): The speed of the sphere relative to the far-away undisturbed fluid. The direction of drag is always opposite to this velocity.
  • 6\pi — Geometric constant (dimensionless): This numerical factor (≈ 18.85) emerges from solving the Navier-Stokes equations around a sphere; 2/3 comes from pressure drag and 1/3 from viscous skin friction.

Worked Example

Problem. A steel ball bearing of radius r = 2.0 mm and density ρ_s = 7800 kg/m³ is dropped into a container of glycerine (density ρ_f = 1260 kg/m³, viscosity η = 1.49 Pa·s). Find the terminal velocity of the ball bearing. Take g = 9.81 m/s².

vt=2r2(ρs−ρf)g9η≈0.038 m/sv_t = \frac{2r^2(\rho_s - \rho_f)g}{9\eta} \approx 0.038 \ \text{m/s}

Answer. The terminal velocity is approximately 38 mm/s (0.038 m/s), and the Reynolds number check confirms Stokes' Law applies.

Figure
Diagram
Stokes' Law: Forces on a Falling SphereViscous Fluid (η)sphereρ, rF_bF_gF_d=6πηrvv →vBuoyancy (F_b)Gravity (F_g)Stokes Drag (F_d)
Origins · 1 minGeorge Gabriel Stokes and how this was discovered
GG
George Gabriel Stokes
1819–1903 · Irish-British

Stokes derived his famous drag law in 1851 as part of a broader investigation into viscous fluid motion. He also made foundational contributions to the Navier-Stokes equations (the fundamental equations of fluid dynamics), the theory of diffraction, fluorescence (Stokes shift), and asymptotic series. He served as Lucasian Professor of Mathematics at Cambridge — the same chair once held by Newton and later by Stephen Hawking — for over 50 years. Stokes was so meticulous and self-critical that he often delayed publishing his work for years, and sometimes not at all, for fear it was not complete enough. Some of his most important results were only discovered in his private notebooks after his death.

  • Dynamic Viscosity — Viscosity (η) is the central material property in Stokes' Law — understanding Newton's law of viscosity is essential before applying Stokes' result.
  • Terminal Velocity — Stokes' Law is the primary tool for calculating terminal velocity of small spherical particles falling through viscous fluids.
  • Archimedes' Principle and Buoyancy — Buoyancy acts alongside Stokes drag when determining equilibrium forces on a sphere settling through a fluid.
  • Reynolds Number and Flow Regimes — The Reynolds number determines when Stokes' Law is valid (Re ≪ 1) versus when turbulent corrections are needed.