Viscosity and Stokes' law

Laminar and turbulent flow

• Laminar (streamline) flow: the fluid moves in smooth layers that do not cross or mix; the velocity at any point stays constant.
• Turbulent flow: the fluid swirls in eddies and mixes, and the velocity at a point keeps changing. Turbulence happens at higher speeds and greatly increases drag.

Viscosity

The viscosity η (eta) of a fluid describes its resistance to flow. Its unit is Pa s (equivalent to N s m^-2). Honey and glycerol have high viscosity; water has low viscosity. Viscosity is temperature dependent: liquids become much less viscous when warmed.

Stokes' law

The viscous drag on a sphere of radius r moving at speed v through a fluid of viscosity η is F = 6πηrv. It only applies to small spheres moving at low speeds with laminar flow.

Falling through a fluid

A sphere falling through a liquid has three forces on it: weight (down), upthrust (up) and viscous drag (up). At terminal velocity: weight = upthrust + viscous drag.

Worked example

A steel ball of radius 1.0 mm falls at 0.050 m/s through glycerol (η = 1.5 Pa s). Find the viscous drag.

1. r = 1.0 x 10^-3 m
2. F = 6π x 1.5 x 1.0 x 10^-3 x 0.050

Answer: F = 1.4 x 10^-3 N

Check yourself

1. A sphere of radius 2.0 mm moves at 0.10 m/s through oil of viscosity 0.80 Pa s. Find the viscous drag in mN, to 3 s.f.
   Show answer

   3.02 mN. F = 6π x 0.80 x 2.0 x 10^-3 x 0.10 = 3.02 x 10^-3 N = 3.02 mN.

2. A viscous drag of 2.0 x 10^-3 N acts on a sphere of radius 1.5 mm moving at 0.040 m/s. Find the viscosity in Pa s, to 3 s.f.
   Show answer

   1.77 Pa s. η = F / (6πrv) = 2.0 x 10^-3 / (6π x 1.5 x 10^-3 x 0.040) = 1.77 Pa s.

3. Stokes' law applies only to
   • any object moving through a liquid
   • large spheres in turbulent flow
   • small spheres moving at low speeds with laminar flow
   • spheres moving at terminal velocity in air only
   Show answer

   small spheres moving at low speeds with laminar flow. These are the conditions stated in the specification.

4. Which describes laminar flow?
   • The fluid forms eddies and swirls
   • Layers of fluid move smoothly without crossing or mixing
   • The velocity at a point changes constantly
   • The fluid does not move at all
   Show answer

   Layers of fluid move smoothly without crossing or mixing. Laminar (streamline) flow is smooth, layered flow.

5. Oil in a tube is warmed. What happens to the terminal velocity of a ball falling through it?
   • It decreases, because the viscosity increases
   • It stays the same
   • The ball stops falling
   • It increases, because the viscosity decreases
   Show answer

   It increases, because the viscosity decreases. Warm liquids are less viscous, so the drag at any speed is smaller and the ball falls faster.

6. A ball falls at terminal velocity through a liquid. Which equation is correct?
   • Weight = viscous drag
   • Weight = upthrust + viscous drag
   • Upthrust = weight + viscous drag
   • Viscous drag = 0
   Show answer

   Weight = upthrust + viscous drag. Upthrust and drag both act upwards and together balance the weight.

7. What is the unit of viscosity?
   • N s
   • Pa/s
   • N/m
   • Pa s
   Show answer

   Pa s. η = F / (6πrv) has units N / (m x m/s) = N s m^-2 = Pa s.

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)

Last updated 7 November 2026. Spotted a mistake? Tell us.