Stokes Law Interactive Calculator

Stokes Law describes the drag force experienced by spherical particles moving through viscous fluids at low Reynolds numbers (Re < 1), where inertial forces are negligible compared to viscous forces. This fundamental relationship is essential for predicting settling velocities in sedimentation processes, designing centrifuges for particle separation, analyzing aerosol behavior in atmospheric science, and optimizing filtration systems in chemical engineering. The calculator below solves for terminal velocity, particle diameter, fluid viscosity, drag force, and Reynolds number across multiple operational modes used in laboratory analysis, environmental monitoring, and industrial process design.

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Force Diagram: Particle Settling Under Stokes Regime

Stokes Law Interactive Calculator

Governing Equations

Theory & Practical Applications

Fundamental Physics of Stokes Flow

Stokes Law emerges from the solution to the Navier-Stokes equations under conditions where viscous forces dominate over inertial forces—a regime characterized by Reynolds numbers substantially less than unity. When a spherical particle moves through a viscous fluid at low velocity, the flow field around it becomes symmetric and reversible, with no flow separation or wake formation. The drag force arises entirely from viscous shear stresses distributed over the particle surface, producing the elegant linear relationship F_d = 3πηdv. This proportionality to velocity (rather than velocity squared) distinguishes Stokes drag from the quadratic drag encountered at higher Reynolds numbers where form drag and turbulent wake effects dominate.

The terminal velocity equation reveals that settling rate scales with the square of particle diameter—a relationship with profound practical implications. A particle with 10 μm diameter settles 100 times slower than a 100 μm particle of identical density in the same fluid. This quadratic dependence explains why fine sediments remain suspended in water bodies for extended periods while coarser particles rapidly settle, and why air filtration systems must employ electrostatic or impaction mechanisms rather than gravitational settling for sub-micron aerosol removal. The equation also shows that effective particle density (ρ_p - ρ_f) rather than absolute density governs settling—particles with densities approaching the fluid density settle extremely slowly regardless of their absolute mass.

Reynolds Number Limitations and Transitional Behavior

The strict validity of Stokes Law requires Re < 1, with highest accuracy in the creeping flow regime where Re < 0.1. As Reynolds number approaches and exceeds unity, inertial effects introduce asymmetry in the flow field around the particle. The fore-aft symmetry breaks down, a wake region develops downstream of the particle, and form drag begins contributing to total resistance. Empirical correlations such as the Schiller-Naumann equation modify Stokes drag with a correction factor (1 + 0.15Re^0.687) for 1 < Re < 800, while the standard drag curve (C_d = 24/Re) extends Stokes-like behavior to Re ≈ 0.5 with less than 10% error. Beyond Re = 1000, drag coefficients approach constant values characteristic of fully turbulent flow.

For practical calculations, engineers must verify the calculated Reynolds number post-facto to confirm regime validity. A common workflow involves assuming Stokes Law initially, calculating terminal velocity, computing Reynolds number from that velocity, and iterating with corrected drag relationships if Re exceeds 0.5. In water (η = 0.001 Pa·s, ρ = 1000 kg/m³), silica particles (ρ_p = 2650 kg/m³) remain in the Stokes regime up to approximately 100 μm diameter. Larger particles or higher density differences require transitional or Newton's Law drag correlations.

Industrial Sedimentation and Clarification Processes

Stokes Law forms the theoretical foundation for designing sedimentation tanks, clarifiers, and settling basins across water treatment, mineral processing, and chemical manufacturing industries. The critical design parameter is overflow rate (surface loading rate), defined as volumetric flow rate divided by settling basin surface area. For complete removal of particles larger than a target diameter, the overflow rate must not exceed the terminal velocity of that diameter particle. A wastewater clarifier designed to remove 50 μm particles must maintain surface loading rates below the Stokes settling velocity for that size class—typically 0.2-0.4 m/hr in water. Undersized clarifiers with excessive overflow rates allow fine particles to exit with the overflow stream despite adequate residence time.

Lamella settlers and tube settlers exploit Stokes Law to achieve high clarification rates in compact footprints. By installing inclined plates or tubes at angles between 45-60° from horizontal, these devices create multiple shallow settling zones. Particles need only settle the perpendicular distance between plates (typically 25-50 mm) rather than the full basin depth, reducing required settling time by factors of 10-20. The theoretical settling area becomes the projected horizontal area of all inclined surfaces combined, far exceeding the plan view footprint. This geometry allows wastewater treatment plants to retrofit existing tanks for higher capacity without physical expansion.

Centrifugal Separation and Enhanced Gravity

Centrifuges apply artificial gravitational fields to accelerate settling rates for particles that would settle impractically slowly under normal gravity. The terminal velocity equation remains valid when g is replaced by centrifugal acceleration ω²r, where ω is angular velocity (rad/s) and r is radial position. Industrial centrifuges commonly generate 3000-15000 g, reducing settling times by three to four orders of magnitude. A protein molecule with Stokes diameter 8 nm and density 1350 kg/m³ sedimenting in water experiences terminal velocity 6.4 × 10⁻¹⁰ m/s under earth gravity—requiring 73 days to settle 4 cm. At 50,000 g in an ultracentrifuge, the same particle settles the same distance in 2.1 hours, enabling practical biochemical separations.

The sigma factor (Σ) provides a standardized metric for comparing centrifuge performance independent of geometry. Defined as the equivalent settling area that would achieve the same separation under normal gravity, sigma factors typically range from 50 m² for small laboratory centrifuges to 15,000 m² for large industrial disc-stack separators. This allows process engineers to scale separations from laboratory tests: if a pilot centrifuge with Σ₁ = 100 m² achieves target separation at 500 L/hr, a production unit with Σ₂ = 5000 m² can process 25,000 L/hr while maintaining identical removal efficiency.

Aerosol Physics and Atmospheric Applications

Stokes Law governs the behavior of atmospheric aerosols, respirable dust, and pharmaceutical aerosols in the 0.5-100 μm diameter range where gravitational settling competes with turbulent dispersion and Brownian motion. For air at 20°C (η = 1.81 × 10⁻⁵ Pa·s, ρ = 1.2 kg/m³), aerosol particles below 1 μm diameter exhibit terminal velocities under 0.1 mm/s and remain airborne for hours to days even in still air. This size range encompasses many industrial dusts, combustion aerosols, and bioaerosols that pose inhalation hazards. HVAC filtration design must account for the reality that gravitational settling contributes negligibly to removal of sub-micron particles—such systems rely on impaction, interception, and diffusion mechanisms instead.

The Cunningham slip correction factor becomes essential for particles below 1 μm where molecular mean free path (68 nm in air at STP) becomes comparable to particle radius. Gas molecules no longer behave as a continuum around such particles; instead, they "slip" at the particle surface, reducing drag below the Stokes prediction. The correction factor C_c = 1 + (2λ/d)[1.257 + 0.400 exp(-1.10d/2λ)] increases drag reduction as particle size decreases, reaching C_c ≈ 2 for 100 nm particles and C_c ≈ 15 for 10 nm particles. Neglecting this correction produces significant errors in calculating terminal velocities and aerodynamic diameters for sub-micron aerosols.

Worked Example: Wastewater Clarifier Design

Problem: A municipal wastewater treatment plant requires a secondary clarifier to remove suspended solids following biological treatment. The design flow rate is 8500 m³/day, and regulations require removal of all particles with diameter ≥ 65 μm. The suspended solids have density ρ_p = 1240 kg/m³, wastewater density is ρ_f = 998 kg/m³, and dynamic viscosity is η = 1.15 × 10⁻³ Pa·s at the operating temperature of 18°C. Determine: (a) minimum required clarifier surface area, (b) diameter for a circular clarifier, (c) particle size that will be 50% removed, and (d) Reynolds number to verify Stokes regime validity.

Solution Part (a) - Terminal Velocity and Minimum Surface Area:

First calculate the terminal settling velocity for the 65 μm design particle using Stokes Law:

v_t = [d²g(ρ_p - ρ_f)] / (18η)

Convert diameter to meters: d = 65 μm = 65 × 10⁻⁶ m

v_t = [(65 × 10⁻⁶)² × 9.81 × (1240 - 998)] / (18 × 1.15 × 10⁻³)

v_t = [4.225 × 10⁻⁹ × 9.81 × 242] / (0.0207)

v_t = 1.003 × 10⁻⁵ / 0.0207 = 4.847 × 10⁻⁴ m/s

Converting to more practical units: v_t = 4.847 × 10⁻⁴ m/s × 3600 s/hr = 1.745 m/hr

The overflow rate (surface loading rate) must not exceed this terminal velocity. Minimum surface area is:

A_min = Q / v_t = (8500 m³/day) / (1.745 m/hr × 24 hr/day)

A_min = 8500 / 41.88 = 203.0 m²

Solution Part (b) - Circular Clarifier Diameter:

For a circular clarifier with area A = πD²/4:

D = √(4A/π) = √(4 × 203.0 / π) = √(258.3) = 16.07 m

Standard engineering practice adds 25-40% safety margin for non-ideal flow patterns and temperature variations. Specifying D = 18 m provides A = 254.5 m² (25% margin).

Solution Part (c) - 50% Removal Particle Size:

In an ideal clarifier with plug flow, 50% removal occurs for particles with terminal velocity equal to half the overflow rate. However, actual clarifiers exhibit short-circuiting and dead zones. Using the conservative assumption that 50% removal requires terminal velocity equal to the overflow rate:

v₅₀ = Q / A_actual = 8500 / (24 × 254.5) = 1.390 m/hr = 3.861 × 10⁻⁴ m/s

Solving for diameter from v_t = d²g(ρ_p - ρ_f) / 18η:

d₅₀² = (18ηv₅₀) / [g(ρ_p - ρ_f)]

d₅₀² = (18 × 1.15 × 10⁻³ × 3.861 × 10⁻⁴) / (9.81 × 242)

d₅₀² = 7.986 × 10⁻⁶ / 2374.0 = 3.364 × 10⁻⁹ m²

d₅₀ = 5.800 × 10⁻⁵ m = 58.0 μm

Solution Part (d) - Reynolds Number Verification:

Calculate Reynolds number for the 65 μm design particle at terminal velocity:

Re = (ρ_fv_td) / η

Re = (998 × 4.847 × 10⁻⁴ × 65 × 10⁻⁶) / (1.15 × 10⁻³)

Re = 3.146 × 10⁻⁵ / 1.15 × 10⁻³ = 0.0274

Since Re = 0.0274 ≪ 0.1, the flow is deep in the Stokes regime and the calculation is highly accurate. The design clarifier with 18 m diameter and 254.5 m² surface area will reliably remove all particles ≥ 65 μm at the design flow rate of 8500 m³/day.

Non-Spherical Particles and Shape Factors

Real particles rarely exhibit perfect sphericity, requiring corrections to Stokes Law for elongated, flaky, or irregular geometries. The shape factor ψ (psi) modifies drag as F_d = 3πηd_eqvψ, where d_eq is the diameter of a sphere with equivalent volume. Elongated particles like fibers orient with their long axis perpendicular to flow direction, maximizing drag; thin flakes settle with broad faces perpendicular to gravity. Typical shape factors range from ψ = 0.85 for rounded sand grains to ψ = 2.5 for mica flakes and ψ = 5-10 for asbestos fibers. Particle characterization instruments report equivalent spherical diameter based on settling velocity, optical projected area, or sieve opening—these metrics produce different values for non-spherical particles, requiring careful specification of the measurement principle employed.

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