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Stokes' Law Calculator

Find the drag on a sphere in a viscous fluid.

Result

9.4248e-7 N

How it is calculated

The drag on a small sphere moving slowly through a fluid is F = 6π × η × r × v. Here η is dynamic viscosity in Pa·s, r is the sphere radius in metres and v is its speed relative to the fluid in m/s. The force comes out in newtons, printed in scientific notation with five significant digits because the values are often tiny. The radius must be above zero; viscosity and velocity can be zero but not negative. The law is valid only for slow, smooth (laminar) flow, roughly Reynolds number below 1. Fast or large objects need a different drag model.

Worked example

A 1 mm radius (0.001 m) sphere moves at 0.05 m/s through water with viscosity 0.001 Pa·s. Multiply: 6π ≈ 18.85; 18.85 × 0.001 × 0.001 × 0.05 ≈ 9.4248 × 10⁻⁷. The drag force is 9.4248e-7 N, about a micronewton. Doubling the radius or the speed doubles the force, because it is linear in both.

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Common mistakes

  • Entering the diameter instead of the radius, which makes the force twice as large.
  • Typing the radius in millimetres or centimetres rather than metres; 1 mm must be 0.001.
  • Entering viscosity in centipoise or mPa·s without converting; water is about 0.001 Pa·s, not 1.
  • Using the formula for fast or large spheres where flow is turbulent; the law then underestimates drag.
  • Reading the result as total force on a falling sphere; weight and buoyancy are separate.

About this calculator

This calculator applies Stokes' law, F = 6πηrv, to find the viscous drag force on a small sphere moving slowly through a fluid, where η is the fluid's dynamic viscosity, r is the sphere's radius, and v is its velocity. The law holds in the laminar (low Reynolds number, Re < 1) regime, where drag is dominated by viscous forces rather than fluid inertia — it's also the basis for terminal velocity calculations of a falling sphere, where drag balances gravity minus buoyancy.

Physics and engineering students use it in fluid mechanics coursework, while chemists and lab scientists apply the same principle to sedimentation and centrifugation analysis, particle-size determination (falling-ball viscometers), and estimating settling velocities of particles in suspensions. Enter the fluid viscosity, sphere radius and velocity to get the drag force.

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