Calculate the terminal velocity (v) of a small spherical particle falling through a viscous fluid.
Stokes' law describes the drag force exerted on a small spherical object moving through a viscous fluid. When an object falls under gravity, it eventually reaches a terminal velocity where the drag force and buoyancy exactly balance the gravitational force.
This law is fundamental in fluid mechanics and is used to calculate the settling rate of particles in liquids (sedimentation), the movement of droplets in clouds, and the viscosity of fluids using a falling ball viscometer.
Where:
• v is the terminal velocity (m/s)
• r is the radius of the sphere (m)
• g is the acceleration due to gravity (≈ 9.81 m/s²)
• ρ_p is the density of the particle (kg/m³)
• ρ_f is the density of the fluid (kg/m³)
• η (eta) is the dynamic viscosity of the fluid (Pa·s)
Stokes' law assumes laminar flow (low Reynolds number), a rigid spherical particle, a smooth surface, and an infinite fluid medium (no wall effects).
Viscosity is a measure of a fluid's resistance to flow. Thick fluids like honey have high viscosity, while thin fluids like water have low viscosity.
The net force driving the particle down is its weight minus the buoyancy force. If the particle is less dense than the fluid (ρ_p < ρ_f), the terminal velocity will be negative, meaning the particle will rise (like an air bubble in water).
Stokes' law is generally accurate for Reynolds numbers (Re) less than 0.1. Above this, inertial forces become significant and the drag relationship becomes more complex.
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