Stokes’ viscosity lab report

1. Title and aim

Viscosity by Stokes’ falling-sphere method. Determine the dynamic viscosity of a liquid from the terminal speeds of small spheres and the gradient of a speed–radius-squared graph.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of dynamic viscosity of 1.1999 Pa s, differing by 0.004% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Liquid column with two timing marks: Provide a 0.100 m timing distance below the initial settling region.
• Spherical beads: Provide smooth spheres of known density and different measured radii.
• Micrometer and ruler: Measure sphere diameter and vertical separation of the timing marks.
• Timer and thermometer: Record transit time and monitor liquid temperature, which affects viscosity.
• Sphere density ρₛ = 2500 kg m⁻³; liquid density ρₗ = 1260 kg m⁻³.
• Gate separation s = 0.100 m; gravity g = 9.80665 m s⁻².
• Timing starts after terminal speed is reached and is rounded to 0.01 s.

4. Theory and graph choice

A small sphere sinking through a thick liquid reaches a steady speed when gravity is balanced by buoyancy and viscous drag. Time several sphere sizes across the same marked distance, convert times to speeds, and investigate the predicted square dependence on radius. The virtual fluid has density 1260 kg/m³ and model viscosity 1.200 Pa·s; sphere density is 2500 kg/m³. Radii range from 0.300 to 0.900 mm.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units m⁻¹ s⁻¹.

5. Method

1. Record sphere density 2500 kg/m³, liquid density 1260 kg/m³ and g = 9.80665 m/s². In a physical lab also record temperature and confirm the liquid is suitable for a falling-sphere method.
2. Measure each sphere diameter in several orientations, halve it to obtain radius, and convert to metres before calculating r². Select virtual radii within 0.300–0.900 mm.
3. Set the vertical mark separation to 0.100 m below the release region. Release the sphere gently away from walls and verify steady motion before the upper mark.
4. Time the sphere centre between the marks and record radius, r², distance and physical elapsed time. Repeat for several radii; the animation accelerates viewing while retaining modeled physical time.
5. For each row calculate v = H/t. Fit v against r² and inspect the points and intercept. Calculate a trial viscosity and Reynolds number to check the creeping-flow condition.
6. Use the gradient in η = 2(ρs − ρl)g/(9s). State viscosity in Pa·s, report the low Reynolds numbers and discuss temperature, walls, radius and timing as uncertainty sources.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0045%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 1.3765e-14; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.11772. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 5.6487e-08.

Propagating only the relevant fit coefficient gives a fit-only standard error of 6.2726e-05 Pa s. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Viscosity is proportional to density difference and inversely proportional to the v-versus-r² gradient. Radius errors enter r², so horizontal-axis uncertainty matters. A naive equal-weight vertical least-squares fit does not account for it. Gate spacing, timing, liquid temperature and wall/end effects also contribute in a real apparatus.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Use a clean, smooth sphere, avoid trapped bubbles, and release gently near the column centre.
• Begin timing below the initial settling region and use the same sphere-centre crossing rule at both marks.
• Maintain and record liquid temperature; allow disturbances from a previous fall to settle before repeating.
• In a real setup assess wall and end effects, check sphere size and density, and verify low Reynolds number before using the unbounded Stokes formula.

The calculation assumes an isolated rigid sphere in an unbounded, incompressible Newtonian fluid with constant density and viscosity and creeping flow. The illustrated column is a schematic timing gate; its walls exert no modeled correction. Initial acceleration, wall and end effects, bubbles, nonspherical beads and temperature drift are omitted. Timing starts after terminal motion is established. Virtual animation time is accelerated; reported elapsed time is physical model time. A fit error or perfect simulated line is not a complete experimental uncertainty estimate. The Newtonian liquid has a model viscosity of 1.20 Pa s at a fixed temperature. An effectively unbounded liquid is assumed. The simulated Reynolds number stays below 0.004 across the allowed radii. There are no wall/end corrections or temperature changes. Pre-terminal motion is not timed; the displayed fall is accelerated and not to scale.

12. Conclusion

The worked example illustrates how to obtain dynamic viscosity from measurements, graph variables and a fitted coefficient. The estimate is 1.1999 Pa s. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.