Stokes' Law Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding
Last updated: Jan 19, 2023

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 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 has at the end of the tube, you can calculate the viscosity of the fluid. This article will explain how to use the terminal velocity equation to determine the viscosity, as well as elaborate a bit on the viscosity definition.

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 units of viscosity are pascal times second (Pa·s).

If you want to visualize how viscosity affects a liquid, consider a stream of water and honey flowing down the slope. Water has low viscosity, so it will move faster. The honey, on the other hand, will flow very slowly precisely because of its viscosity.

🔎 Learn more about the viscosity of water in our water viscosity calculator.

Terminal velocity equation

Flow lines past a falling particle at terminal velocity
Flow lines past a falling particle at terminal velocity, when the drag force Fd is equal to the force of gravity Fg. (Credit: via Wikimedia Commons/CC BY-SA)

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×(ρpρm)18×μ,v = g \times d^2 \times \frac{(\rho_\text{p} - \rho_\text{m})}{18\times\mu},

Where:

  • vv – Terminal velocity of a spherical particle;
  • gg – Gravitational acceleration - for Earth, equal to 9.80665 m/s²;
  • dd – Particle diameter;
  • ρp\rho_\text{p} – Density of the particle;
  • ρm\rho_\text{m} – Density of the fluid; and
  • μ\mu – 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 input all other values in their respective boxes, and you will receive the answer! If you need help determining the density of your material and fluid, you may find our density calculator handy.

🙋 Expand your knowledge about viscosity by learning how to convert dynamic viscosity (in stokes) to kinematic viscosity (in poise) and vice versa with our poise-stokes converter. You can also read through our kinematic viscosity of air calculator if that topic interests you.

FAQ

What is Stoke's law?

Stoke's law is a dynamics formula that describes the change in motion of objects creeping through a fluid. In its original formulation, Stoke's law calculates the total force acting on a falling sphere. Still, it is commonly used as a way to calculate the viscosity of a fluid by measuring the terminal speed in its fall.

How do I calculate Stoke's law?

To calculate Stoke's law for the terminal velocity of a falling sphere, use the following formula:
v = g × d² × (ρp - ρm)/(18 × μ)
where:

  • v — The terminal velocity;
  • g — The acceleration due to gravity;
  • d — The diameter of the sphere;
  • μ — The dynamic viscosity of the fluid; and
  • ρp and ρm — Respectively the particle and the medium density.

What is the terminal velocity of a 1 cm aluminum sphere in oil?

The terminal velocity of a sphere with a diameter 1 cm in oil, calculated with Stoke's law, is 0.34 m/s.

To calculate this result, follow these steps:

  1. Calculate the **difference between the densities of aluminum and oil:
    ρp - ρm = 2710 - 850 kg/m³ =1860 kg/m³
  2. Calculate the product g × d², where g = 9.81 m/s²:
    g × d² = 9.81 m/s² × 0.01² m² = 0.000981 m³/s²
  3. Divide the product of the results of the point 1. and 2. by 18 × μ, with μ = 0.38 Pa⋅s:
    v = 0.000981 × 1860/(17 × 0.38) = 0.34 m/s

What is the dynamic viscosity of a fluid?

The dynamic viscosity of a fluid is a quantity that measures the intrinsic resistance of a fluid to movement. The higher the dynamic viscosity, the harder it will be for the fluid to move or for a body to move in the fluid. Also known as absolute viscosity, this quantity tells us the force needed to move a fluid at a certain rate. If we are interested in the reaction of a fluid to a force, the kinematic viscosity (the ratio of dynamic viscosity and fluid's density) is more appropriate.

Bogna Szyk
v = gd²(ρp - ρm)/(18μ)
Acceleration of gravity (g)
m/s²
Medium viscosity (μ)
Pa·s
Medium density (ρ_m)
lb/cu ft
Particle density (ρ_p)
lb/cu ft
Particle diameter (d)
in
Terminal velocity (v)
ft/s
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