Stokes' law calculator

This capacity applies that formula, rejects a non-positive viscosity with a clear invalid_input error, and returns structured JSON with the terminal velocity, the echoed inputs, the formula, and the method.

What Stokes' law computes and which quantities you need to send

Delivering radius, particle_density, fluid_density, viscosity, gravity, density_difference, terminal_velocity, settling_velocity, formula, and method as JSON makes the result easy to paste into notes, diff in automated tests, or reuse across languages without reimplementing the two-ninths arithmetic.

How this calculator validates input and applies v = (2/9) r² (ρ_p − ρ_f) g / μ

Numeric strings that parse to finite numbers are accepted so forms and CSV-style pipelines can pass values without an extra cast. Gravity is optional and defaults to 9.81 so ordinary SI work yields metres per second without an extra field. Radius must also be strictly positive so the sphere is a real geometric body. Densities must be non-negative; a negative density is not physical for this model. There is no unit-conversion service, no iterative drag-coefficient table, and no CFD mesh; this is ideal Stokes settling from four known quantities plus gravity. Send a JSON body with required fields radius, particle_density, fluid_density, and viscosity. Every field must be finite: missing fields, empty strings, non-numeric text, NaN, and Infinity all fail immediately with invalid_input before any result is returned, and failed validation is not charged. Aliases are supported: r and sphere_radius for radius; rho_p, density_particle, and particle_rho for particle density; rho_f, density_fluid, and fluid_rho for fluid density; mu, dynamic_viscosity, and eta for viscosity; g and gravitational_acceleration for gravity. Viscosity must be strictly positive: a zero or negative viscosity is always rejected as invalid_input because Stokes' law divides by μ and the product contract names that failure explicitly. After validation, the engine forms the density difference, multiplies two-ninths by r squared times that difference times g, divides by viscosity, and rounds to a stable decimal precision so the same input always yields the same JSON across JavaScript engines. The same pure function powers the free in-browser tool and the API path, which is why a homework check and a production pipeline never disagree on a creeping-flow sphere.

Practical use, limits, pricing, and delivery channels

Use the browser tool when you want an immediate Stokes settling-velocity check while grading homework, verifying a sedimentation lab, sketching whether a silt particle will fall in still water, or sanity-checking the creeping-flow inputs of an aerosol notebook. Choose units consistently — if radius is in metres, densities in kilograms per cubic metre, viscosity in pascal-seconds, and gravity in metres per second squared, the velocity is in metres per second and ready for textbook comparison. There is no language model, no external fluid-property service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Remember that Stokes' law is accurate only while the particle Reynolds number stays much less than one; large, fast spheres need a quadratic-drag treatment instead of this linear viscous formula. Each successful request is one billable item at $0.002; failed validation does not consume balance. Results are fully deterministic: the same four inputs always yield the same terminal_velocity, the same settling_velocity alias, and the same formula and method labels. Wire the endpoint into sedimentation drills, contest answer checkers, droplet-fall sketches, or preprocessing steps where you already prefer explicit domain errors — such as rejecting a non-positive viscosity — over opaque black-box widgets that cannot explain why a bad input was refused.