Measuring Viscosity: Stokes Method and Poiseuille Flow

Viscosity is a liquid's internal friction. It measures how strongly the layers of a liquid resist sliding past one another, which is why honey pours slowly while water pours quickly. The SI unit of dynamic viscosity is the pascal-second (Pa·s), and the older unit, the poise, equals 0.1 Pa·s.

Measuring viscosity matters for engine oils, blood flow, and syrups, paints and fuels moving through pipes. Two classic lab methods measure it differently: the Stokes method watches a small sphere sink through the liquid, while Poiseuille's method measures how fast the liquid flows through a narrow tube.

Key Concepts

• Viscosity (η): the ratio of tangential stress to velocity gradient between liquid layers, measured in Pa·s.
• Stokes law: drag on a slow-moving sphere is F = 6πηrv.
• Terminal velocity: the steady speed where drag plus upthrust balance the weight, so acceleration is zero.
• Upthrust: the buoyant force σVg opposing gravity, σ being the liquid's density.
• Poiseuille flow: steady laminar flow through a long narrow tube, with volume rate Q in m³/s.
• Laminar flow: smooth layered flow with no turbulence; both methods need it.

Formulas

Stokes method: η = 2 r² g (ρ − σ) / (9 v). Here r is in metres, g = 9.8 m/s², ρ and σ are the sphere and liquid densities in kg/m³, and v is the terminal velocity in m/s. Assumptions: small smooth sphere, tube diameter at least ten times the sphere's, slow laminar motion, and speed measured after acceleration has ceased.

Poiseuille method: Q = π P r⁴ / (8 η l), so η = π P r⁴ / (8 Q l). Here Q is the volume flowing per second (m³/s), P is the pressure difference driving the flow (Pa), r is the tube's inner radius and l its length (m). Assumptions: steady laminar flow, long tube, liquid sticking to the wall, and constant pressure difference.

Worked Example

A steel ball (r = 1.0 mm = 0.001 m, ρ = 8000 kg/m³) falls through oil (σ = 1200 kg/m³) and reaches a terminal velocity of 5.0 cm/s (0.050 m/s).

1. Write the Stokes formula: η = 2 r² g (ρ − σ) / (9 v).
2. Compute the numerator: 2 × (0.001)² × 9.8 × (8000 − 1200) = 2 × 10⁻⁶ × 9.8 × 6800 = 0.13328.
3. Compute the denominator: 9 × 0.050 = 0.45.
4. Divide: η = 0.13328 / 0.45 = 0.296 Pa·s, roughly 0.30 Pa·s.

Common Misconceptions

• Thick does not mean dense: density is mass per volume, viscosity is resistance to flow. Mercury is dense yet flows almost like water.
• Terminal velocity does not mean the forces vanished: weight still acts, but drag plus upthrust balance it, so net force and acceleration are zero.
• Heating a liquid lowers its viscosity: warm oil flows faster, so the lab temperature must be recorded with the result.

Real-World Applications

• Engine-oil grading (SAE numbers) is based on measured viscosity at standard temperatures.
• Pipeline and pump design for crude oil, syrups, paints and molten polymers depends on viscosity data.
• Doctors and physiologists use viscosity in modelling blood flow through narrow vessels.
• Food and paint factories run viscosity checks for quality control of every batch.

Viscosity is a liquid's internal friction, measured by the Stokes method from a sphere's terminal speed or by the Poiseuille method from flow through a capillary. With laminar flow and careful arithmetic, either route gives a value engineers can use.