Stokes Law Calculator - Terminal Velocity & Fluid Dynamics

What is Stokes Law?

Definition and Formula
Physical Principles
Assumptions and Limitations

Stokes Law describes the terminal velocity of a spherical particle falling through a viscous fluid under the influence of gravity. This fundamental principle in fluid dynamics is essential for understanding particle sedimentation, filtration processes, and various industrial applications.

The Stokes Law Formula

The terminal velocity (v) of a spherical particle is given by: v = (2/9) × (ρp - ρf) × g × r² / μ where ρp is particle density, ρf is fluid density, g is gravitational acceleration, r is particle radius, and μ is fluid viscosity.

Physical Interpretation

Stokes Law balances three forces: gravitational force (downward), buoyant force (upward), and drag force (upward). When these forces are in equilibrium, the particle reaches its terminal velocity and falls at a constant speed.

Practical Examples

A sand particle (ρp = 2650 kg/m³) falling through water (ρf = 1000 kg/m³) with radius 1mm reaches terminal velocity of approximately 0.18 m/s
An oil droplet in air demonstrates much lower terminal velocity due to the large density difference and low air viscosity

Step-by-Step Guide to Using the Stokes Law Calculator

Input Parameters
Calculation Process
Result Interpretation

Using the Stokes Law calculator requires understanding the physical parameters involved in the calculation. Each input parameter affects the terminal velocity in specific ways, making accurate measurements crucial for reliable results.

Required Input Parameters

Particle density and fluid density determine the buoyant force. Particle radius affects both gravitational and drag forces. Fluid viscosity controls the drag force magnitude. Gravitational acceleration varies with location but is typically 9.81 m/s² on Earth.

Calculation Verification

The calculator automatically validates inputs, checks for physical constraints, and provides error messages for invalid parameters. Results include terminal velocity, drag force, and Reynolds number for comprehensive analysis.

Calculation Examples

For a 1mm sand particle in water: terminal velocity ≈ 0.18 m/s, drag force ≈ 0.00034 N, Reynolds number ≈ 180
For a 10μm dust particle in air: terminal velocity ≈ 0.0003 m/s, drag force ≈ 0.0000000001 N, Reynolds number ≈ 0.0004

Real-World Applications of Stokes Law

Industrial Processes
Environmental Science
Medical Applications

Stokes Law has numerous practical applications across various fields, from industrial processes to environmental monitoring and medical research. Understanding particle behavior in fluids is crucial for optimizing processes and predicting outcomes.

Industrial Applications

In wastewater treatment, Stokes Law helps design sedimentation tanks and predict particle removal efficiency. In pharmaceutical manufacturing, it's used for drug particle size analysis and formulation optimization. Mining operations use it for ore separation and tailings management.

Environmental Monitoring

Atmospheric scientists use Stokes Law to model dust particle deposition and air quality. Oceanographers apply it to study sediment transport and marine particle dynamics. Climate researchers use it for aerosol behavior modeling.

Application Examples

Wastewater treatment plants use sedimentation rates calculated from Stokes Law to design efficient settling tanks
Pharmaceutical companies use particle size analysis based on Stokes Law for drug formulation and quality control

Common Misconceptions and Correct Methods

Reynolds Number Limitations
Non-Spherical Particles
Turbulent Flow Effects

While Stokes Law is a powerful tool, it has specific limitations and assumptions that must be understood for accurate application. Misunderstanding these constraints can lead to significant calculation errors and incorrect predictions.

Reynolds Number Constraints

Stokes Law is valid only for Reynolds numbers less than 1, indicating laminar flow around the particle. For higher Reynolds numbers, turbulent effects become significant, and more complex drag coefficient models must be used.

Particle Shape Effects

Stokes Law assumes perfectly spherical particles. Non-spherical particles experience different drag forces, requiring shape factor corrections or alternative calculation methods.

Limitation Examples

A 1cm particle falling through water violates Stokes Law assumptions due to high Reynolds number (>1000)
Irregularly shaped particles like sand grains require shape factor corrections for accurate terminal velocity calculation

Mathematical Derivation and Examples

Force Balance Analysis
Derivation Steps
Numerical Examples

The mathematical derivation of Stokes Law involves analyzing the forces acting on a falling particle and solving for the equilibrium condition. This derivation provides insight into the physical principles underlying the law and its limitations.

Force Balance Derivation

The gravitational force Fg = (4/3)πr³ρpg, buoyant force Fb = (4/3)πr³ρfg, and drag force Fd = 6πμrv. At terminal velocity, Fg = Fb + Fd, leading to the Stokes Law formula.

Numerical Verification

For a 1mm glass bead (ρp = 2500 kg/m³) in water (ρf = 1000 kg/m³, μ = 0.001 Pa·s): v = (2/9) × (2500-1000) × 9.81 × (0.001)² / 0.001 = 0.33 m/s

Derivation Examples

Derivation shows that terminal velocity is proportional to particle radius squared and inversely proportional to fluid viscosity
Numerical calculations demonstrate that small particles (μm scale) have very low terminal velocities in air

Sand Particle in Water

Oil Droplet in Air

Glass Bead in Glycerin

Dust Particle in Air