Stokes' Law Calculator

Fundamental Formulation and Derivation

Stokes' law gives the drag force on a sphere moving through a viscous fluid:

F_d = 6π\eta r v

Where: - F_d = drag force (N) - \eta = dynamic viscosity of the fluid (Pa·s) - r = radius of the sphere (m) - v = velocity of the sphere relative to the fluid (m/s)

The factor 6π arises from the mathematical solution of the Navier-Stokes equations for creeping flow around a sphere.

**Derivation Overview** The derivation involves solving the Navier-Stokes equations under the assumption of: 1. **Creeping flow**: Reynolds number Re = (2\rho vr)/(\eta) \ll 1 2. **Steady flow**: No time dependence 3. **Incompressible fluid**: Constant density 4. **No-slip boundary condition**: Fluid velocity equals sphere velocity at surface

Solving the simplified Navier-Stokes equation:

\nabla P = \eta\nabla^2\mathbfv

with appropriate boundary conditions yields the drag force expression.

**Key Assumptions** - Sphere is rigid and smooth - Flow is laminar (low Reynolds number) - Fluid is Newtonian (linear stress-strain relationship) - Sphere is far from boundaries (infinite fluid) - No rotation of the sphere

**Physical Interpretation** The drag force is: - **Linearly proportional** to velocity (unlike quadratic drag at high speeds) - **Proportional** to viscosity (more viscous fluids create more drag) - **Proportional** to radius (larger spheres experience more drag) - **Independent** of fluid density (unlike inertial drag)

F_d = 6π\eta r v

Validity Conditions and Reynolds Number

Stokes' law is valid only under specific conditions:

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

Re = (\rho v d)/(\eta) = (2\rho v r)/(\eta) \ll 1

where d = 2r is the sphere diameter.

**Practical Validity Range** - **Strict validity**: Re < 0.1 - **Approximate validity**: Re < 1 - **Beyond validity**: For Re > 1, corrections are needed - **High Reynolds number**: Quadratic drag (F_d \propto v^2) dominates

**When Stokes Law Fails** 1. **High velocity**: Inertial effects become significant 2. **Large objects**: Boundary layer separation occurs 3. **Low viscosity**: Flow becomes turbulent 4. **Near boundaries**: Wall effects modify the flow 5. **Non-spherical shapes**: Different drag coefficients 6. **Rotating spheres**: Additional lift forces

**Corrections for Higher Reynolds Numbers** For Re up to ~1000, Oseen's correction applies:

F_d = 6π\eta r v (1 + (3)/(16)Re)

For higher Re, empirical drag coefficients are used:

F_d = (1)/(2)C_D\rhoπ r^2 v^2

where C_D is the drag coefficient, which depends on Re.

Applications in Science and Engineering

Stokes' law finds extensive applications:

**1. Particle Sedimentation** For a sphere settling under gravity, terminal velocity is found by equating drag force to weight minus buoyancy:

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

Solving for terminal velocity:

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

This is used in: - **Sedimentation analysis**: Determining particle size distributions - **Centrifugation**: Separating particles by density - **Water treatment**: Removing suspended particles - **Mining**: Separating minerals

**2. Aerosol Physics** - **Particle deposition**: How particles settle in air - **Filtration efficiency**: Designing air filters - **Respiratory deposition**: Understanding how particles deposit in lungs - **Atmospheric science**: Modeling particle transport

**3. Microfluidics** - **Particle manipulation**: Using drag forces to control particles - **Lab-on-a-chip devices**: Separating and analyzing particles - **Droplet microfluidics**: Understanding droplet motion - **Cell sorting**: Separating cells by size

**4. Rheology and Viscosity Measurement** - **Falling ball viscometers**: Measuring fluid viscosity - **Rheological characterization**: Understanding fluid properties

**5. Biological Systems** - **Cell motility**: Understanding how cells move through fluids - **Blood flow**: Analyzing red blood cell behavior - **Swimming microorganisms**: Modeling bacterial motion

**6. Industrial Processes** - **Spray drying**: Understanding droplet behavior - **Powder processing**: Particle handling and transport - **Paint and coating**: Ensuring proper particle suspension

Terminal Velocity and Settling

When a sphere falls through a fluid under gravity, it accelerates until drag force balances the net gravitational force:

**Force Balance** At terminal velocity:

F_d = F_g - F_b

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

**Terminal Velocity Formula** Equating forces:

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

Solving for terminal velocity:

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

**Key Observations** - Terminal velocity is **proportional to r^2**: Larger particles fall faster - Depends on **density difference**: Heavier particles fall faster - **Inversely proportional** to viscosity: More viscous fluids slow settling - **Independent** of initial velocity

**Settling Time** Time to settle a distance h:

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

**Practical Applications** - **Water treatment**: Designing settling tanks - **Mining**: Separating particles by size - **Pharmaceuticals**: Ensuring proper suspension - **Environmental**: Modeling pollutant transport

Wall Effects and Corrections

When a sphere moves near a boundary, Stokes' law must be corrected:

**Wall Correction Factor** For a sphere moving parallel to a wall at distance h:

F_d = 6π\eta r v \lambda_w

where \lambda_w is the wall correction factor:

\lambda_w ≈ 1 + (9)/(16)(r)/(h)

for h \gg r.

**Container Effects** In a cylindrical container of radius R:

\lambda_c ≈ 1 + 2.1(r)/(R)

**Multiple Particles** When many particles are present, interactions modify drag: - **Dilute suspensions**: Minimal interaction - **Concentrated suspensions**: Significant interactions - **Hindered settling**: Particles settle slower due to interactions

**Corrections for Non-Spherical Particles** For ellipsoids and other shapes:

F_d = 6π\eta r_eq v K

where r_eq is equivalent radius and K is shape factor: - Sphere: K = 1 - Prolate ellipsoid: K > 1 - Oblate ellipsoid: K < 1

Experimental Verification and Measurement

Stokes' law can be verified experimentally:

**Falling Ball Viscometer** Measures viscosity by timing a ball's fall:

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

**Drag Force Measurement** Direct measurement using: - **Force transducers**: Measure drag force directly - **Balance methods**: Weigh the drag force - **Optical methods**: Track particle motion

**Particle Size Analysis** Using Stokes' law to determine particle size:

r = √(\frac9\eta v_t)2(\rho_s - \rho_f)g

**Precision Considerations** - Ensure low Reynolds number - Account for wall effects - Correct for particle shape - Consider temperature effects on viscosity - Account for particle rotation

**Standard Methods** - **ASTM D445**: Standard test methods - **ISO 3104**: Viscosity measurement - **NIST standards**: Reference measurements

Limitations and Extensions

While Stokes' law is powerful, it has limitations:

**Limitations** 1. **Low Reynolds number only**: Breaks down at higher speeds 2. **Spherical particles**: Non-spherical shapes need corrections 3. **Infinite fluid**: Wall effects modify results 4. **Steady flow**: Unsteady effects not included 5. **Newtonian fluids**: Non-Newtonian fluids behave differently 6. **No rotation**: Rotating spheres experience additional forces

**Extensions**

**1. Oseen Correction** For slightly higher Reynolds numbers:

F_d = 6π\eta r v(1 + (3)/(16)Re)

**2. Non-Spherical Particles** Drag coefficient depends on shape:

F_d = C_D · 6π\eta r_eq v

**3. Unsteady Motion** For accelerating spheres, added mass and history effects:

F_d = 6π\eta r v + (2)/(3)π r^3\rho_f(dv)/(dt) + ...

**4. Non-Newtonian Fluids** For power-law fluids:

F_d = 6π\eta_eff r v

where \eta_eff depends on shear rate.

**5. Electrophoretic Motion** For charged particles in electric fields, additional forces apply.

Worked Example

Inputs

• viscosity: 0.001
• radius: 0.001
• velocity: 0.1

Result: Drag Force: 1.885 × 10⁻⁶ N

Explanation

For a sphere with radius r = 0.001 m moving at velocity v = 0.1 m/s through a fluid with viscosity \eta = 0.001 Pa·s:

Calculate drag force using Stokes' law: F_d = 6π\eta r v F_d = 6π × 0.001 × 0.001 × 0.1 F_d = 6π × 10^-7 F_d ≈ 1.885 × 10^-6 N

This small force is typical for small particles in low-viscosity fluids at low speeds.

Second Scenario

Inputs

• viscosity: 0.001
• radius: 0.001
• velocity: 0.075

Result: Drag Force: 1.885 × 10⁻⁶ N

Explanation

This scenario uses different inputs (viscosity = 0.001, radius = 0.001, velocity = 0.075) to show how changing one variable affects the stokes' law result. Run the calculator above with these values to get the exact updated output with step-by-step work.