The Stokes' law calculator is a tool for analyzing a spherical particle's motion in a falling ball viscometer. This device is just a vertical tube filled with a viscous liquid. When dropped into the tube, a small particle is subject to a drag force resulting from the fluid's resistance. If you measure the velocity it reaches at the end of the tube, you can calculate the viscosity of the fluid.
This article explains how to use the terminal velocity equation to determine viscosity, as well as elaborating a bit on the viscosity definition. The calculator supports both the metric (SI) and the American (Imperial) system of units.
Viscosity definition
The viscosity of a fluid (gas or liquid) describes its resistance to shearing stresses. For example, honey, which is "thicker" than water, has a much higher viscosity and so is more resistant to shear stresses. The SI unit of (dynamic) viscosity is the pascal-second (Pa·s).
If you want to visualize how viscosity affects a liquid, consider a stream of water and a stream of honey flowing down a slope. Water has a low viscosity, so it moves faster. Honey, on the other hand, flows very slowly precisely because of its viscosity.
Terminal velocity equation
When a particle falls through a viscous fluid it accelerates until the drag force Fd equals the net of gravity and buoyancy. At that point its velocity becomes constant — this is the terminal velocity. Our Stokes' law calculator finds the terminal velocity of a particle in a viscometer filled with a viscous fluid according to the following formula:
v = g · d² · (ρp − ρm) / (18 · μ)
where:
- v — Terminal velocity of a spherical particle;
- g — Gravitational acceleration — for Earth, equal to 9.80665 m/s²;
- d — Particle diameter;
- ρp — Density of the particle;
- ρm — Density of the fluid (medium); and
- μ — Dynamic viscosity of the fluid.
You can use this terminal velocity calculator to find any of these values. If you need to determine the viscosity, simply select it in the "Solve for" box, input all other values in their respective boxes, and you will receive the answer instantly. The calculator rearranges Stokes' law automatically:
- Viscosity: μ = g · d² · (ρp − ρm) / (18 · v)
- Particle diameter: d = √[ 18 · μ · v / (g · (ρp − ρm)) ]
- Particle density: ρp = 18 · μ · v / (g · d²) + ρm
- Fluid density: ρm = ρp − 18 · μ · v / (g · d²)
When is Stokes' law valid?
Stokes' law assumes creeping (laminar) flow around the sphere, which holds only for very small Reynolds numbers. The Reynolds number for a settling particle is:
Re = ρm · v · d / μ
As a rule of thumb, Stokes' law is accurate when Re < 0.1. Between 0.1 and 1 it remains a reasonable approximation, but above Re ≈ 1 inertial effects become significant and the predicted velocity is increasingly inaccurate. The calculator reports the Reynolds number with every result so you can check the validity of the assumption.
Common viscosities and densities
| Fluid (at ~20 °C) | Dynamic viscosity μ (Pa·s) | Density ρ (kg/m³) |
|---|---|---|
| Air | 0.0000181 | 1.2 |
| Water | 0.001 | 998 |
| Olive oil | 0.085 | 900 |
| Glycerin | 1.412 | 1260 |
| Honey | 10 | 1420 |
FAQs
What is Stokes' law?
Stokes' law describes the drag force experienced by a small spherical object moving slowly through a viscous fluid: Fd = 3πμdv. When the drag force balances gravity minus buoyancy, the particle reaches a constant terminal velocity v = g·d²·(ρp − ρm) / (18·μ).
How do I calculate viscosity with a falling ball viscometer?
Drop a sphere of known diameter and density into a tube of the fluid, measure its terminal velocity, and select "Dynamic viscosity" in the "Solve for" box. The calculator returns μ = g·d²·(ρp − ρm) / (18·v).
What units does the calculator use?
You can work in the metric (SI) system — m/s, mm, kg/m³, Pa·s — or the American (Imperial) system — ft/s, in, lb/ft³, lb/(ft·s). Every result is shown in both systems at once, and the viscosity is additionally given in mPa·s (cP), poise, and lbf·s/ft².
Why must the particle be denser than the fluid?
If the particle density ρp is greater than the fluid density ρm, the particle sinks and the terminal velocity is positive (downward). If it is less dense, the particle rises (the velocity becomes negative). The classic falling ball viscometer relies on a particle that is denser than the fluid.
Does Stokes' law work for large objects or fast flows?
No. Stokes' law is only valid for low Reynolds numbers (creeping flow), typically Re < 0.1. For larger particles or higher velocities, drag is dominated by inertial effects and you should use the general drag equation instead.