Viscosity Lab: Falling Ball, Couette Start-up and Pipe Flow

Measure the same fluid property three ways: how fast a ball sinks, how momentum diffuses between a moving and a still plate, and how much flows down a pipe.

Fluid

Experiment

Live readouts

Chart

Viscosity vs temperature (all fluids)

Independent numerical checks

Controls

Pick a fluid and temperature: viscosity follows an Andrade law and changes by a factor of 10 000 between water and cold honey.
Falling ball: drop a ball and read the viscosity back from the stopwatch between the two marks; the page shows how wall and inertia corrections change the answer.
Plate flow: the top plate is dragged; watch the profile relax from rest. Pipe flow: set the pressure drop and read the flow rate.

The science

Viscosity. Newton's law τ = η du/dy. For liquids η falls steeply with temperature, modelled by Andrade η = A·exp(B/T) (T in kelvin), with A and B fitted to two published points per fluid (water 20 and 60 °C, glycerol 20 and 40 °C, honey 20 and 40 °C, SAE 30 oil 40 and 100 °C, olive oil 20 and 40 °C; values are typical and vary by batch).

Falling ball. (m + ½ρ_fV) dv/dt = (ρ_s − ρ_f) V g − 6π η r v · F_Re/K(λ). Stokes drag is multiplied by the Schiller-Naumann inertia factor F_Re = 1 + 0.15 Re^0.687 (Re = ρ_f v 2r/η) and divided by the Faxén wall factor K(λ) = 1 − 2.10444λ + 2.08877λ³ − 0.94813λ⁵ − 1.372λ⁶ + 3.87λ⁸ − 4.19λ¹⁰ for a ball of diameter ratio λ = d/D in a tube. Stokes terminal speed: v_s = 2 r² (ρ_s − ρ_f) g / (9 η); a viscometer inverts it to η = 2 r² Δρ g / (9 v), which is only exact for small Re and λ.

Plate flow. Between plates a gap h apart, with the top plate at speed U and a pressure gradient G = −dp/dx, ∂u/∂t = ν ∂²u/∂y² + G/ρ with ν = η/ρ. Steady state is u = U y/h + G y (h − y)/(2η). Start-up from rest has the series solution u = U y/h + (G/2η) y(h−y) − Σ b_n sin(nπy/h) e^{−n²π²νt/h²}, so the settling time scales as h²/(π² ν). The page integrates the PDE with an implicit finite-difference scheme and checks it against that series.

Pipe flow. Hagen-Poiseuille: u(r) = ΔP (R² − r²) / (4 η L), Q = π R⁴ ΔP / (8 η L), mean speed Q/πR² = u_max/2, wall shear stress τ_w = ΔP R / (2L). Valid for Re = ρ u_m D / η below about 2000; above that the page flags that the real flow would be turbulent and only shows the Blasius estimate f = 0.316 Re^−1/4 for the pressure drop.

Simplifications: Newtonian fluids, constant density, no temperature gradient or shear heating, ball starts on the axis and stays there, no tube end effects. Fluid properties are rounded textbook values. Pure Canvas 2D; no Blender assets used.