Terminal Velocity (Stokes) Calculator

Derivation from Force Balance

Terminal velocity occurs when forces balance:

**Force Balance at Terminal Velocity** At terminal velocity, acceleration is zero:

F_net = F_g - F_b - F_d = 0

where: - F_g = (4)/(3)π r^3\rho g = gravitational force - F_b = (4)/(3)π r^3\rho_f g = buoyant force - F_d = 6π\eta r v_t = drag force (Stokes' law)

**Net Gravitational Force** The effective weight (gravity minus buoyancy):

F_net = (4)/(3)π r^3(\rho - \rho_f)g

**Equating Forces** At terminal velocity:

(4)/(3)π r^3(\rho - \rho_f)g = 6π\eta r v_t

Solving for terminal velocity:

v_t = (2r^2(\rho - \rho_f)g)/(9\eta)

**Key Dependencies** - **Proportional to r^2**: Larger particles fall much faster - **Proportional to density difference**: Heavier particles fall faster - **Inversely proportional to viscosity**: More viscous fluids slow settling - **Independent of initial velocity**: Terminal velocity is determined by properties, not initial conditions

v_t = (2r^2(\rho - \rho_f)g)/(9\eta)

Validity Conditions and Reynolds Number

Stokes' law terminal velocity is valid only under specific conditions:

**Reynolds Number Criterion** The Reynolds number must be much less than 1:

Re = (2\rho_f v_t r)/(\eta) \ll 1

**Practical Validity Range** - **Strict validity**: Re < 0.1 - **Approximate validity**: Re < 1 - **Beyond validity**: For Re > 1, corrections needed

**Checking Validity** Substituting terminal velocity into Reynolds number:

Re = (2\rho_f)/(\eta) · (2r^2(\rho - \rho_f)g)/(9\eta) · r = (4\rho_f r^3(\rho - \rho_f)g)/(9\eta^2)

For validity, this must be \ll 1.

**When Stokes Law Fails** 1. **Large particles**: Boundary layer separation 2. **High velocities**: Inertial effects dominate 3. **Low viscosity**: Flow becomes turbulent 4. **Non-spherical particles**: Different drag coefficients 5. **Near boundaries**: Wall effects modify flow

**Corrections for Higher Reynolds Numbers** For Re up to ~1000, empirical drag coefficients:

F_d = (1)/(2)C_D\rho_fπ r^2 v^2

where C_D depends on Reynolds number.

Applications in Science and Industry

Terminal velocity calculations are crucial in many fields:

**1. Sedimentation Analysis** - **Particle size distribution**: Measure settling times - **Centrifugation**: Separate particles by size/density - **Water treatment**: Remove suspended particles - **Mining**: Separate minerals - **Pharmaceuticals**: Ensure proper suspension

**2. Aerosol Physics** - **Particle deposition**: How particles settle in air - **Air quality**: Modeling pollutant transport - **Respiratory deposition**: Understanding lung particle behavior - **Filtration**: Designing air filters

**3. Microfluidics** - **Particle manipulation**: Control particle motion - **Lab-on-a-chip**: Separating and analyzing particles - **Cell sorting**: Separating cells by size - **Droplet microfluidics**: Understanding droplet motion

**4. Industrial Processes** - **Spray drying**: Droplet behavior - **Powder processing**: Particle handling - **Paint and coating**: Ensuring suspension - **Food processing**: Separating components

**5. Environmental Science** - **Pollutant transport**: Modeling particle movement - **Oceanography**: Sediment settling - **Atmospheric science**: Aerosol behavior - **Soil science**: Particle sedimentation

Settling Time and Distance

Understanding settling behavior:

**Time to Settle Distance h**

t = (h)/(v_t) = (9\eta h)/(2r^2(\rho - \rho_f)g)

**Distance Settled in Time t**

h = v_t t = (2r^2(\rho - \rho_f)gt)/(9\eta)

**Key Observations** - Settling time is **inversely proportional to r^2** - Larger particles settle much faster - More viscous fluids slow settling - Greater density difference accelerates settling

**Practical Examples** - **Water treatment**: Design settling tanks - **Centrifugation**: Determine spin time - **Particle analysis**: Measure size from settling time - **Environmental**: Model pollutant deposition

Effect of Fluid Properties

Fluid properties significantly affect terminal velocity:

**Viscosity Effects** - **Higher viscosity**: Lower terminal velocity - **Temperature dependence**: Viscosity decreases with temperature - **Non-Newtonian fluids**: Complex behavior

**Density Effects** - **Buoyancy**: Reduces effective weight - **Dense fluids**: Lower terminal velocity - **Negative density difference**: Particles float instead of sink

**Practical Considerations** - **Water vs air**: Much different terminal velocities - **Oil vs water**: Viscosity differences - **Temperature effects**: Viscosity changes - **Pressure effects**: Density changes

Particle Shape and Size Effects

Stokes' law assumes spherical particles. Real particles:

**Non-Spherical Particles** - Different drag coefficients - Orientation-dependent drag - Shape factors needed - More complex calculations

**Size Distribution** - Particles have size distributions - Different terminal velocities - Separation possible - Size analysis from settling

**Agglomeration** - Particles may clump together - Effective size increases - Terminal velocity increases - Affects separation efficiency

Worked Example

Inputs

• radius: 0.001
• density: 2000
• fluidDensity: 1000
• viscosity: 0.001

Result: Terminal Velocity: 2.18 m/s

Explanation

For a sphere with radius r = 0.001 m, density \rho = 2000 kg/m³, falling through a fluid with density \rho_f = 1000 kg/m³ and viscosity \eta = 0.001 Pa·s:

Calculate terminal velocity: v_t = (2r^2(\rho - \rho_f)g)/(9\eta) v_t = (2 × (0.001)^2 × (2000 - 1000) × 9.8)/(9 × 0.001) v_t = \frac2 × 10^-6 × 1000 × 9.89 × 10^-3 v_t ≈ 2.18 m/s

This particle will reach a terminal velocity of approximately 2.18 m/s.

Second Scenario

Inputs

• radius: 0.001
• density: 1500
• fluidDensity: 1000
• viscosity: 0.001

Result: Terminal Velocity: 2.18 m/s

Explanation

This scenario uses different inputs (radius = 0.001, density = 1500, fluidDensity = 1000, viscosity = 0.001) to show how changing one variable affects the terminal velocity (stokes) result. Run the calculator above with these values to get the exact updated output with step-by-step work.