Stokes Law
Drag force on a sphere in viscous flow: F_D = 6πμrv. Terminal velocity v_t = (2r²g(ρ_p−ρ_f))/(9μ). Valid when Reynolds number Re < 1 (creeping flow).
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Re < 1 required for Stokes flow validity. Terminal velocity ∝ r²; larger particles settle faster. Centrifugation: replace g with ω²r. Used in viscometry to measure viscosity.
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Why: Stokes law governs particle settling, sedimentation, centrifugation, and viscometry. Re < 1 ensures laminar creeping flow.
How: Balance drag (6πμrv) with buoyancy-corrected weight. Terminal velocity when forces balance. Settling time = height/velocity.
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Falling Sphere Viscometry
Steel ball (r=2mm) falling through glycerin to measure viscosity. Classic Stokes law application.
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Sedimentation Tank
Clay particles (r=0.01mm) settling in water treatment tank. Calculate settling time for 2m depth.
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Blood Cell Settling
Red blood cells (r=4μm) settling in plasma. Medical application for sedimentation rate.
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Paint Settling
Paint pigment particles (r=5μm) settling in paint base. Important for paint stability.
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Fog Droplet Fall
Water droplets (r=10μm) falling through air. Atmospheric physics application.
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Centrifugation
Plastic beads (r=1mm) in centrifuge at 1000g. Laboratory separation application.
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For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
F_D = 6πμrv for spherical particles
— Stokes (1851)
v_t = 2r²g(ρ_p−ρ_f)/(9μ)
— Terminal velocity
Re < 1: creeping flow; Stokes valid
— Reynolds number
Settling time t = h/v_t
— Sedimentation
What is Stokes Law?
Stokes law describes the drag force experienced by a small spherical particle moving slowly through a viscous fluid. Named after George Gabriel Stokes, this law is fundamental to understanding particle settling, sedimentation, centrifugation, and viscometry. It applies when the Reynolds number is less than 1 (creeping flow regime).
Drag Force
Calculate viscous drag force on small particles: F_D = 6πμrv
Terminal Velocity
Calculate maximum settling speed: v_t = (2r²g(ρ_p - ρ_f))/(9μ)
Settling Time
Calculate time for particles to settle in sedimentation tanks
How Does Stokes Law Work?
Stokes law applies to small particles moving slowly through viscous fluids. At terminal velocity, the drag force equals the net gravitational force (weight minus buoyancy). The law assumes spherical particles, laminar flow (Re < 1), and no wall effects.
🔬 Key Assumptions
Reynolds Number < 1
Stokes law is valid only for creeping flow where inertial forces are negligible compared to viscous forces. Re = (2ρvr)/μ < 1.
Spherical Particles
The law assumes perfectly spherical particles. Non-spherical particles require shape correction factors.
No Wall Effects
Particle must be far from container walls. Wall effects become significant when particle radius approaches container dimensions.
When to Use Stokes Law Calculator
This calculator is essential for engineers, scientists, and researchers working with particle-fluid systems. It's particularly valuable for sedimentation analysis, viscometry, centrifugation, and particle separation processes.
Viscometry
Measure fluid viscosity using falling sphere viscometry technique
Sedimentation
Design sedimentation tanks for water treatment and particle separation
Centrifugation
Calculate separation efficiency in centrifuges and separators
Stokes Law Formulas
Stokes law provides fundamental equations for analyzing particle motion in viscous fluids. Understanding these formulas helps optimize separation processes and predict particle behavior.
📊 Core Stokes Law Equations
Drag Force
Where: μ = dynamic viscosity, r = particle radius, v = velocity
Terminal Velocity
Where: r = radius, g = gravity, ρ_p = particle density, ρ_f = fluid density, μ = viscosity
Settling Time
Where: h = settling height, v_t = terminal velocity
Reynolds Number
Stokes law valid when Re < 1 (creeping flow regime)
Centrifugal Velocity
Where: a_c = centrifugal acceleration. Used in centrifugation processes.
Key Takeaways
- ✓F_D = 6πμrv: Stokes drag force is proportional to viscosity, radius, and velocity - linear relationship valid only for Re < 1.
- ✓Terminal Velocity: When drag equals net gravitational force (weight minus buoyancy), particle reaches constant terminal velocity v_t = (2r²g(ρ_p - ρ_f))/(9μ).
- ✓Reynolds Number < 1: Stokes law applies only to creeping flow where Re = (2ρvr)/μ < 1. For Re ≥ 1, use drag equation instead.
- ✓Settling Time: Time for particle to settle height h is t = h/v_t. Critical for sedimentation tank design and particle separation processes.
- ✓Centrifugation: Replace gravity with centrifugal acceleration - v_c = (2r²a_c(ρ_p - ρ_f))/(9μ). Enables faster separation than gravity settling.
Did You Know?
Falling Sphere Viscometry: Stokes law enables viscosity measurement by timing a sphere's fall through fluid. This technique is used to measure viscosities from 0.001 Pa·s (water) to 1000 Pa·s (pitch).
Blood Sedimentation: ESR (Erythrocyte Sedimentation Rate) test uses Stokes law - red blood cells settle at terminal velocity. Normal rate is 0-20 mm/hour, elevated rates indicate inflammation.
Water Treatment: Sedimentation tanks use Stokes law to design particle removal systems. Clay particles (1 μm) take hours to settle, requiring large tanks or flocculation to increase particle size.
Centrifugation: Laboratory centrifuges achieve 10,000-100,000g, accelerating sedimentation by factors of 10,000-100,000 compared to gravity. Essential for separating biological particles.
Atmospheric Physics: Fog droplets (10 μm) fall at ~1 cm/s through air. Raindrops (1 mm) fall at ~5 m/s - too fast for Stokes law, requiring drag equation due to Re > 1.
Paint Stability: Paint pigment particles must remain suspended. Stokes law predicts settling rates - particles < 1 μm settle slowly enough for stable paint, larger particles require stabilizers.
Expert Tips
Always Check Reynolds Number: Verify Re < 1 before using Stokes law. For Re ≥ 1, drag force becomes nonlinear - use F_D = 0.5 × C_D × ρ × A × v² instead.
Account for Wall Effects: When particle radius approaches container dimensions, wall effects reduce terminal velocity. Use Faxén correction: v_corrected = v_stokes × (1 - 2.1 × (r/R)) for cylindrical containers.
Non-Spherical Particles: Use shape correction factors - prolate spheroids have higher drag, oblate spheroids have lower drag. For rough estimates, use equivalent spherical radius.
Optimize Centrifugation: Increase RCF (relative centrifugal force) to reduce separation time. Doubling RCF halves settling time. Balance speed with sample stability and equipment limits.
Particle Settling Comparison
| Particle Type | Radius | Terminal Velocity (m/s) | Settling Time (1m) | Application |
|---|---|---|---|---|
| Fog Droplet | 10 μm | 1.2×10⁻⁴ | 2.3 hours | Atmospheric physics |
| Red Blood Cell | 4 μm | 1.8×10⁻⁶ | 6.4 days | Medical diagnostics |
| Clay Particle | 1 μm | 1.1×10⁻⁷ | 106 days | Water treatment |
| Paint Pigment | 0.5 μm | 2.8×10⁻⁸ | 413 days | Paint stability |
| Steel Ball (1mm) | 1 mm | 0.22 | 4.5 s | Viscometry |
| Sand Grain | 0.1 mm | 2.2×10⁻³ | 7.6 min | Sedimentation |
Frequently Asked Questions
What is the Reynolds number limit for Stokes law?
Stokes law is valid for Re < 1 (creeping flow). For 1 ≤ Re < 10, Stokes law has moderate error. For Re ≥ 10, use the drag equation F_D = 0.5 × C_D × ρ × A × v², where C_D depends on Re.
Why does terminal velocity depend on density difference?
Terminal velocity occurs when drag force equals net gravitational force. Net force = weight - buoyancy = (ρ_p - ρ_f) × V × g. If ρ_p < ρ_f, particle floats (no terminal velocity). Larger density difference means faster settling.
How does particle size affect settling time?
Terminal velocity is proportional to r² (v_t ∝ r²), so doubling radius increases velocity by 4× and reduces settling time by 4×. Small particles (μm scale) settle very slowly - this is why water treatment uses flocculation to increase particle size.
What is the difference between sedimentation and centrifugation?
Sedimentation uses gravity (g = 9.8 m/s²), while centrifugation uses centrifugal acceleration (a_c = ω²r, often 1000-100,000g). Centrifugation accelerates separation by factors of 1000-100,000, enabling separation of particles that would take days or years under gravity.
Can Stokes law be used for non-spherical particles?
Stokes law assumes spherical particles. For non-spherical particles, use shape correction factors or equivalent spherical radius. Prolate spheroids (elongated) have higher drag, oblate spheroids (flattened) have lower drag than spheres of same volume.
How do wall effects affect terminal velocity?
When particle radius approaches container dimensions, walls reduce terminal velocity. For cylindrical containers, use Faxén correction: v_corrected = v_stokes × (1 - 2.1 × (r/R)). Wall effects become significant when r/R > 0.1.
What is falling sphere viscometry?
Falling sphere viscometry measures fluid viscosity by timing a sphere's fall through fluid. Rearranging Stokes law: μ = (2r²g(ρ_p - ρ_f))/(9v_t). This technique works for viscosities from 0.001 Pa·s (water) to 1000 Pa·s (pitch), provided Re < 1.
How does temperature affect Stokes law calculations?
Temperature affects viscosity significantly - viscosity typically decreases with temperature (μ ∝ e^(-E/RT)). Higher temperature means lower viscosity, higher terminal velocity, and faster settling. Always use viscosity values at operating temperature.
Stokes Law by the Numbers
Official Sources
National Institute of Standards and Technology - Fluid mechanics reference data and viscosity standards
Comprehensive physics reference on fluid dynamics, drag forces, and Stokes law
MIT OpenCourseWare fluid dynamics course materials covering creeping flow and particle motion
Engineering reference data for Stokes law, drag forces, and terminal velocity calculations
American Society of Mechanical Engineers fluid mechanics reference standards
⚠️ Disclaimer
This calculator is for educational and engineering purposes. Stokes law assumes spherical particles, creeping flow (Re < 1), and no wall effects. Actual particle behavior may differ due to shape, concentration effects, Brownian motion (for very small particles), and container boundaries. For critical applications in viscometry, sedimentation, or centrifugation, verify results with experimental data and consult professional engineers. Always check Reynolds number validity before applying Stokes law.
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