Stokes Law
Calculate terminal velocity of a sphere in a viscous fluid using Stokes law v = g × d² × (ρp − ρm) / (18 × μ). Free physics calculator with material presets, drag force, and Reynolds number.
About This Calculator
The Stokes Law Calculator computes the terminal velocity of a spherical particle falling through a viscous fluid under gravity. Based on George Gabriel Stokes' 1851 solution to the Navier-Stokes equations for creeping flow, this calculator is an essential tool for fluid dynamics, sedimentology, chemical engineering, and materials science.
The core formula is v = g × d² × (ρp − ρm) / (18 × μ), where v is terminal velocity, g is gravitational acceleration (default 9.80665 m/s²), d is the sphere diameter, ρp and ρm are the particle and fluid densities respectively, and μ is the dynamic viscosity of the fluid. The calculator automatically verifies whether the flow falls within the Stokes regime by computing the Reynolds number Re = ρmvd/μ. When Re < 0.1, Stokes law is accurate to within 1%. For transitional flow (0.1 < Re < 10), a corrected velocity is provided using the Schiller-Naumann drag correlation Cd = (24/Re)(1 + 0.15Re0.687).
Material presets include common engineering materials (aluminum, steel, copper, glass, etc.) from the physics constants database with their standard densities. Fluid presets cover air, water, oil, glycerin, ethanol, and mercury with their respective densities and dynamic viscosities. Custom values can be entered for any parameter when the presets do not match the user's specific application.
How to Use This Calculator
Select a particle material and fluid medium from the dropdown presets, or enter custom density and viscosity values manually. Specify the particle diameter in millimeters and adjust gravitational acceleration if needed. Click Calculate to obtain the terminal velocity, drag force, and Reynolds number with a regime classification. The breakdown table shows all input values alongside computed quantities for verification.
Frequently Asked Questions
What is Stokes law?
Stokes law describes the drag force on a spherical particle moving through a viscous fluid at low Reynolds numbers (Re < 0.1). It relates the terminal velocity of the sphere to its diameter, densities of the particle and fluid, fluid viscosity, and gravitational acceleration. The formula is v = g × d² × (ρp − ρm) / (18 × μ).
How do I calculate terminal velocity using Stokes law?
To calculate terminal velocity using Stokes law, use the formula v = g × d² × (ρp − ρm) / (18 × μ). Enter the particle density, fluid density, particle diameter in millimeters, dynamic viscosity of the fluid in Pa·s, and gravitational acceleration. The calculator also computes the drag force and Reynolds number to verify the Stokes flow regime applies.
What is the terminal velocity of a 1 cm aluminum sphere in oil?
The terminal velocity of a 1 cm diameter aluminum sphere (density 2710 kg/m³) falling through vegetable oil (density 850 kg/m³, viscosity 0.38 Pa·s) under Earth gravity is approximately 0.27 m/s. This is calculated as v = 9.81 × 0.01² × (2710 − 850) / (18 × 0.38) = 0.27 m/s.
When is Stokes law valid?
Stokes law is valid when the Reynolds number (Re) is less than 0.1, meaning the flow around the sphere is laminar and viscous forces dominate. For Re between 0.1 and 10, a correction factor using the drag coefficient Cd = (24/Re)(1 + 0.15Re^0.687) gives more accurate results. For Re above 10, the full drag equation should be used instead.
What units does this Stokes law calculator use?
This calculator uses SI units: density in kg/m³, diameter in millimeters (converted internally to meters), dynamic viscosity in pascal-seconds (Pa·s), gravitational acceleration in m/s², and results in m/s for terminal velocity and newtons (N) for drag force. Particle and fluid presets provide real-world values for common materials and fluids.
What is the difference between Stokes law and the drag equation?
Stokes law (F = 6πμrv) applies only for creeping flow at very low Reynolds numbers (Re < 0.1), where viscous forces dominate and inertial forces are negligible. The general drag equation (F = 0.5 × ρ × v² × A × Cd) applies across all flow regimes but requires knowing the drag coefficient Cd, which varies with shape and Reynolds number. Stokes law is an exact solution derived from the Navier-Stokes equations for creeping flow past a sphere.
Can Stokes law be used for particles falling in air?
Yes, Stokes law applies for very small particles falling in air, such as dust particles, pollen, or water droplets with diameters under about 0.1 mm. For example, a 0.05 mm water droplet in air has Re ≈ 0.03, well within the Stokes regime. Larger objects like raindrops or skydivers have Re values in the thousands and require the general drag equation instead.
What is the drag force calculated by Stokes law?
The drag force according to Stokes law is F = 3πμdv, where μ is dynamic viscosity, d is sphere diameter, and v is velocity. This is the resistance force the fluid exerts on the sphere. At terminal velocity, this drag force equals the net gravitational force (buoyancy corrected weight), so F = (π/6) × d³ × (ρp − ρm) × g. The calculator verifies both approaches match.